Supergravity

Antoine Van Proeyen

Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,Celestijnenlaan 200D B-3001 Leuven, Belgium.

Lectures in the Amsterdam-Brussels-Paris doctoral school, Paris 2008;Based on notes for a forthcoming book in collaboration with D. Freedmane-mail: antoine.vanproeyen@fys.kuleuven.be

1 Introduction

A few words on supergravity. In the 60’s and early 70’s all particle interactions ex-cluding gravity had found a consistent description in the context of quantum mechanicsand special relativity. This series of successful discoveries was mainly due to the use of‘gauge symmetries’, i.e. the idea that a symmetry may act differently at each spacetimepoint. When it became clear that bosons and fermions could be related by the conceptof supersymmetry, researchers looked for a way to promote also this new symmetry to agauge symmetry. This was achieved in 1976 by Ferrara, Freedman and van Nieuwenhuizen.These newly built theories were shown to automatically incorporate gravity. Hence the name‘supergravity ’.

In the following years, research in supergravity became very intense. Several versions,depending on the amount of supersymmetry (simple N = 1 to a maximum of N = 8), werediscovered. Furthermore, it turned out to be convenient to consider supergravity theoriesin spacetimes of dimension greater than 4. In this direction, it was shown that field theoryimposes a maximum: 11 dimensions. However, maximal theories did not have much successat that time because they could not include the symmetries of the standard model. Phe-nomenologists rather preferred to exploit the many possibilities offered by various matterfields coupled to minimal supergravity in a supersymmetric way.

In 1984, important results in superstring theory were found, leading to the general ideathat a consistent quantum theory of gravity should be a superstring theory. Such a theoryis not a field theory for physical particles, it describes them instead as the vibration modesof a string. However, when one takes the limit that the string is so small that it lookslike a particle, the theory reduces to a supergravity (field) theory. Since the main stringtheory results were not found from such an approximation, though, research in supergravitydiminished.

Superstring theories, first believed to be nearly unique, were shown to also exist in manyvarieties. Ten years later, it was proven that all the known superstring theories can berelated to each other by the concept of duality. This unification made again contact with 11-dimensional supergravity. Later on, other supergravity theories became useful to understanddifferent aspects of the full picture. The whole concept of dualities was first clarified in thecontext ofN = 2 supersymmetry by Seiberg and Witten. This discovery revived the researchactivity in various supergravity theories. A new understanding of the physics of black holes,arising as solutions of supergravity theories, and of their thermodynamical properties furtherstimulated this revival, eventually leading, a few years later, to the understanding of blackhole thermodynamics from the point of view of string microstates.

In 1997, a duality between conformal field theory and anti-de Sitter theories was discov-ered. The latter are solutions of supergravity, which can be treated classically to get infor-mation on the quantum aspects of the dual field theory. The applications of this so-calledAdS/CFT duality and its extensions became in the following years more and more successfuland led recently to e.g. the study of the quark-gluon plasma and of non-perturbative aspectsof QCD using supergravity duals.

Meanwhile, phenomenological models were built using compactifications of the original

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10-dimensional string theories on a 6-dimensional ‘Calabi-Yau’ manifold. This procedureleads to different 4-dimensional theories, characterized by ‘special geometries’. Furthermore,extensions have been built such that all sort of other supergravities also play a role now forconstructing realistic models.

In the last 8 years cosmologists also showed some interest for various stringy models,using mostly the supergravity limit. Many successful models, using data from the cosmicmicrowave background, have now been built starting from supergravity.

In conclusion, various supergravity theories have been and are still being used for manyapplications. This is the case for phenomenological applications as well as for the duality-based treatment of gauge theories. They are often inspired (or derived in some limit) fromsuperstring theory. But there are also supergravity theories that cannot (at least currently)be related to any string theory.

Supergravity in the coming years. In the coming years applications of supergravitymight be expected to get heavily boosted by the upcoming new-generation experiments.At CERN, the newly-built accelerator LHC will open a new energy range and the searchfor supersymmetry is one of the main first goals of its colliding experiments. Moreover, incosmology the precision measurements of the cosmic microwave background will be furtherimproved and gravitational wave experiments will reach a crucial stage. For new modelbuilding, researchers will very likely use supergravity.

Open problems in supergravity. Supergravity is not a unique theory. There are manyversions and a complete catalogue does not exist yet. Moreover, many existing versions havenot been completely constructed and thus cannot be considered full-fledged theories.

And of course, the last year there is even hope again that supergravity could be finite.Who knows ?

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2 Scalar Field Theory and its Symmetries

Assuming that you have studied basic field theory in another course, most of this chaptercan be skipped. However, it is useful to acquaint yourself with the treatment of symmetries.

Consider a set of fields such as the φi(x). A general symmetry of the system is a mappingof the configuration space, φi(x) → φ′i(x), with the property that if the original field con-figuration φi(x) is a solution of the equations of motion, then the transformed configurationφ′i(x) is also a solution. For scalar fields and for most other systems of interest in this book,we can restrict to symmetry transformations that leave the action invariant. Thus we requirethat the mapping has the property

S[φi] = S[φ′i] . (2.1)

We consider both spacetime symmetries, which involve a motion in Minkowski spacetimesuch as the global translation of the exercise, and internal symmetries, which do not. Internalsymmetries are simpler to describe. So we start with them.

2.1 General internal symmetry

It will be useful to establish the notation for the general situation of linearly realized internalsymmetry under an arbitrary connected Lie group G, usually a compact group, of dimensiondimG. We will be interested in an n-dimensional representation of G in which the generatorsof its Lie algebra are a set of n×n matrices (tA)ij, A = 1, 2, . . . , dimG. Their commutationrelations are1

[tA, tB] = fABCtC , (2.2)

and the fABC are structure constants of the Lie algebra. The representative of a general

element of the Lie algebra is a matrix Θ which is a superposition of the generators with realparameters θA, i.e.

Θ = θAtA. (2.3)

An element of the group is represented by the matrix exponential

U(Θ) = e−Θ = e−θAtA . (2.4)

We consider a set of scalar fields φi(x) which transforms in the representation just de-scribed. The fields may be real or complex. If complex, the complex conjugate of everyelement is also included in the set. A group transformation acts by matrix multiplication onthe fields:

φi(x) → φ′i(x) = U(Θ)ijφj(x) . (2.5)

1Although it is common in the physics literature to insert the imaginary i in the commutation rule, we donot do this in order to eliminate i’s in most of the formulas of the book. This means that compact generatorstA in this book are anti-Hermitian matrices.

4

An infinitesimal transformation of the group is defined as the truncation of the exponen-tial power series in (2.4) to first order in Θ. This gives the field variation (matrix and vectorindices suppressed)

δφ = −Θφ , (2.6)

which defines the action of a Lie algebra element on the fields.It is important to define iterated Lie algebra variations carefully. The action of a trans-

formation δ2 with Lie algebra element Θ2 followed by a transformation δ1 with element Θ1

is defined byδ1δ2φ ≡ −Θ2δ1φ = Θ2Θ1φ . (2.7)

The second variation acts only on the dynamical variables of the system, the fields φi, andis not affected by the matrix Θ2 which multiplies φi. In detail,

δ1δ2φ = θA1 θB2 tBtAφ . (2.8)

The commutator of two symmetry variations is then

[δ1, δ2]φ = −[Θ1,Θ2]φ ≡ δ3φ ,

Θ3 = [Θ1,Θ2] = fABCθA1 θ

B2 tC . (2.9)

The commutator of two Lie algebra transformations is again an algebra transformation bythe element Θ3 = [Θ1,Θ2].

It follows that finite group transformations compose as

φθ2−→ φ′ = U(Θ2)φ

θ1−→ φ′′ = U(Θ2)U(Θ1)φ . (2.10)

2.2 Spacetime symmetries – the Lorentz and Poincare groups

The Poincare group acts on coordinates as

xµ = Λµνx′ν + aµ . (2.11)

where aµ parametrize translations and Λµν the Lorentz rotations that should satisfy

ΛµρηµνΛ

νσ = ηρσ . (2.12)

The Lorentz rotations have an infinitesimal action using parameters λµν and generatorsmµν such that

Λµν =

(e

12λρσmρσ

)µν . (2.13)

Here the matrices mρσ are

mρσµν ≡ δµρηνσ − δµσηρν = −mσρµν . (2.14)

Note the distinction between the coordinate plane labels in brackets and the row and columnindices. In this basis, a finite proper Lorentz transformation is specified by a set of 1

2D(D−1)

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real parameters λρσ. With the representation (2.14), the full Lorentz transformation is givenby the series

Λµν = δµν + λµν + 1

2λµρλ

ρν + . . . . (2.15)

The commutators of the generators defined in (2.14) are

[mµν,mρσ] = ηνρmµσ − ηµρmνσ − ηνσmµρ + ηµσmνρ . (2.16)

These equations specify the structure constants of the Lie algebra which may be written as

fµνρσκτ = 8η[ρ[νδ

[κµ]δ

τ ]σ] . (2.17)

Note that antisymmetrization is always done with ‘weight 1’, such that the right hand sidecan be written as 8 terms with coefficients ±1.

Ex. 2.1 Check that (2.16) leads to (2.17). To compare with (2.2), you have to replace eachof the indices A,B,C by antisymmetric combinations, e.g. A→ µν. Moreover, you haveto insert a factor 1

2each time that you sum over such a combined index to avoid double

counting, similar to the factor 12

in (2.13). Therefore we rewrite (2.16) as

[mA,mB] = fABCmC → [mµν,mρσ] = 1

2fµνρσ

κτmκτ . (2.18)

The global transformations act on scalar fields as φ′(x′) = φ(x), i.e.

φi(x)a,Λ−→ φ′i(x) = U(a,Λ)φi(x) ≡ φi(Λx+ a) , (2.19)

where U(a,Λ) is an operator that acts on the fields. It splits as

U(a,Λ) ≡ U(Λ)U(a) , U(a) = eaµ∂µ ,

U(Λ) = e−12λρσLρσ , Lρσ ≡ xρ∂σ − xσ∂ρ . (2.20)

By expanding the exponential, the latter defines the infinitesimal Lorentz transformation ofthe scalars:

δ(λ)φi(x) = φ′i(x)− φi(x) = φi(Λx)− φi(x) = λµνxν∂µφ

i(x) = −12λµνLµνφ

i(x) . (2.21)

This definition does give a homomorphism of the group; namely the product of maps, firstwith Λ2, then with Λ1, produces the compound map associated with the product Λ1Λ2. Thiscan be seen from the sequence of steps

φi(x)Λ2−→ φ′i(x) = U(Λ2)φ

i(x)Λ1−→ φ′′i(x) = U(Λ2)U(Λ1)φ

i(x)

= U(Λ2)φi(Λ1x)

= φi(Λ1Λ2x) . (2.22)

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Ex. 2.2 It is instructive to check (2.22) for Lorentz transformations which are close to theidentity. Specifically, use the definition (2.20) to show that the order λ1 λ2 terms in theproduct U(Λ2)U(Λ1)φ

i of differential operators acting on φi agrees with terms of the sameorder in φi(Λ1Λ2x).

Perform now the calculation in the form of commutators of infinitesimal transforma-tions. The second transformation acts on φi(x), and not on the x-dependent factor in (2.21).The commutator of two transformations is a Lorentz transformation with parameter-matrix[λ1, λ2].

The Noether current of spacetime translations is the energy-momentum tensor, and theone for Lorentz rotations is related. For a real scalar field we have

T µν = ∂µφ ∂νφ+ δµνL , Jµρσ = −xρT µσ + xσTµρ . (2.23)

These define the conserved charges for translations and Lorentz rotations:

Pµ =

∫dD−1xT 0

µ , Jρσ =

∫dD−1x J0

ρσ . (2.24)

Ex. 2.3 Using also the canonical momenta π = ∂0φ, use the Poisson bracket techniquesto show that these lead to the infinitesimal transformations for translations and Lorentztransformations, e.g. for translations

δφ = aµ∂µφ = aµ Pµ, φ . (2.25)

Then check that the commutators of these transformations lead to the Poincare algebra

[Jµν, Jρσ] = ηνρJµσ − ηµρJνσ − ηνσJµρ + ηµσJνρ ,

[Jρσ, Pµ] = Pρησµ − Pσηρµ ,

[Pµ, Pν ] = 0 . (2.26)

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3 The Dirac Equation

3.1 Introduction to spinors and gamma matrices

For students that have an elementary knowledge of the physics of the Dirac equation, thischapter is mostly important for manipulations of gamma matrices and knowledge of theClifford algebra in general dimensions, including the definition of Majorana spinors.

γ-matrices are defined byγµγν + γνγµ = 2ηµν (3.1)

where is the identity matrix in the spinor indices, which we usually do not write. Onecan easily construct such matrices as products of the σ-matrices. Let us first consider theEuclidean case:

γ1 = σ1 ⊗ ⊗ ⊗ . . .

γ2 = σ2 ⊗ ⊗ ⊗ . . .

γ3 = σ3 ⊗ σ1 ⊗ ⊗ . . .

γ4 = σ3 ⊗ σ2 ⊗ ⊗ . . .

γ5 = σ3 ⊗ σ3 ⊗ σ1 ⊗ . . .

. . . = . . . . (3.2)

This is for even dimensions a representation of dimension 2D/2. For odd dimensions, e.g.D = 5, one does not need the last σ1 factor in γ5, and the dimension is still 2(D−1)/2.Therefore we write in general that the dimension is 2[D/2], where [D/2] means the integerpart of D/2.

These matrices are all hermitian. For a timelike direction, we just have to multiplythe corresponding matrix by i. We will always consider Hermitian gamma matrices for thespacelike directions, and anti-Hermitian for the timelike direction, i.e.

(γµ)† = γ0γµγ0 . (3.3)

Further we use their antisymmetric products (always with ‘total weight 1)

γµ1...µn = γ[µ1 . . . γµn] , e.g. γµν = 12γµγν − 1

2γνγµ . (3.4)

All these matrices are traceless, except the limiting case n = 0, which is the unit matrix,and for odd dimensions, the matrix with n = D as we will see below.

The following is a basis for all matrices in spinor space in even dimensions:

ΓA = , γµ, γµ1µ2 , γµ1µ2µ3 , · · · , γµ1,···µD , (3.5)

with the restriction µ1 < µ2 <, . . . < µr at each rank r.

Ex. 3.1 Count the number of elements in this basis and see that this is appropriate togenerate all the matrices in spinor space. Check your counting for odd dimensions, andobtain that we already have enough matrices if we go up to γµ1...µD/2−1

.

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To test your knowledge of Clifford algebras, consider the following exercise:

Ex. 3.2 Define a ‘reverse-order’ basis

ΓA = , γµ, γµ2µ1 , γµ3µ2µ1 , · · · , γµD,···µ1 , (3.6)

again with the restriction µ1 < µ2 <, . . . < µr at each rank r. Derive the trace orthogonalityproperty Tr(ΓAΓB) = d δAB, where d = Tr .

The Lorentz transformations of Dirac fermions are determined from

Ψ′(x) = L(λ)−1Ψ(Λ(λ)x) , L(λ) = e12λµνΣµν , Σµν = 1

2γµν . (3.7)

Ex. 3.3 Prove that this leads to the infinitesimal transformations

δ(λ)Ψ = −1

2λρσ(Σρσ + Lρσ)Ψ , (3.8)

and that these satisfy the Lorentz algebra.

3.1.1 Levi-Civita tensor

We need a small technical intermezzo to introduce the Levi-Civita tensor for students whoare not familiar with this quantity and its manipulations. In 4 dimensions the totally an-tisymmetric Levi-Civita tensor density is εµνρσ. Because an antisymmetric tensor of rank 5necessarily vanishes when D = 4, this satisfies the Schouten identity

0 = 5δµ[νερστλ] ≡ δµ

νερστλ + δµρεστλν + δµ

σετλνρ + δµτελνρσ + δµ

λενρστ , (3.9)

where antisymmetrization as in [µ1 . . . µp] is always done with ‘weight 1’, e.g. δ[ρ1[ν1δρ2]ν2] =

12

(δρ1ν1 δ

ρ2ν2− δρ2ν1 δ

ρ1ν2

). The anticommuting Levi-Civita tensor can be generalized to any dimen-

sion. It is real, and taken to be

ε012(D−1) = 1 , ε012(D−1) = −1 , (3.10)

where the − sign is of course related to the one timelike direction.

Ex. 3.4 Prove the contraction identity for these tensors

εµ1...µnν1...νpεµ1...µnρ1...ρp = − p!n! δ

[ρ1···νp]

[ν1···νp, p = D − n . (3.11)

The anti-symmetric p-index Kronecker δ is in turn defined by

δν1...νqρ1···ρp

≡ δν1[ρ1δν2ρ2 · · · δ

νp

ρp] , (3.12)

which includes a signed sum over p! permutations of the lower indices, but each with acoefficient 1/p!, such that the ‘total weight’ is 1

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3.1.2 Gamma matrix manipulations

There are some easy tricks to multiply gamma matrices and their extensions (3.4). Considerfirst sums over indices. E.g.

γµνγν = (D − 1)γµ . (3.13)

You can remember this rule, but it is easier to remember the logic behind: ν runs over allvalues except µ. Thus there are (D − 1) terms in the sum. Similar is

γµνργρ = (D − 2)γµν , (3.14)

or even more general

γµ1...µrν1...νsγνs...ν1 =(D − r)!

(D − r − s)!γµ1...µr . (3.15)

Indeed, first we can split γνs...ν1 in γνs . . . γν1 as the antisymmetry is guaranteed by the firstfactor. Then the index νs has (D− (r− s− 1)) values, then νs−1 has (D− (r− s− 2) values,and that continues to (D−r) values for the last one. Note that the second γ in the left-handside has its indices in opposite order, such that there are no signs in contracting the indices.Reordering r indices in opposite order gives a sign factor (−)r(r−1)/2.

Even if one does not make sums, such combinatorial tricks can be used. E.g. whencalculating

γµ1µ2γν1...νD, (3.16)

one knows that the values µ1 and µ2 must appear in the set of ν. There are D possibilitiesfor µ2, and then remain D − 1 possibilities for µ1. Hence the result is

γµ1µ2γν1...νD= D(D − 1)δµ2µ1

[ν1ν2γν3...νD] . (3.17)

Note that such generalized δ functions are always with ‘weight one’, i.e.

δµ2µ1ν1ν2

= 12

(δµ2ν1δµ1ν2− δµ1

ν1δµ2ν2

). (3.18)

This makes substitutions easy, e.g. we obtain

γµ1µ2γν1...νDεν1...νD = D(D − 1)εµ2µ1ν3...νDγν3...νD

. (3.19)

We now consider general products of gamma matrices. The very simplest case is

γµγν = γµν + ηµν . (3.20)

This follows directly from the definition, and is thus simple. But it shows the general pattern:first one combines the gamma matrices to the complete antisymmetric gamma matrix, andthen one makes contractions. Consider another example:

γµνργστ = γµνρστ + 6γ[µν[τδ

ρ]σ] + 6γ[µδν [τδ

ρ]σ] . (3.21)

This follows the same pattern. We put the indices στ down to make the indication of theantisymmetry easier. The second term contains one contraction. One could choose three

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indices from the first gamma and two indices from the second gamma, which gives thefactor 6. For the third term one could also have 6 ways to make two contractions. The deltafunctions contract indices that were adjacent, or separated by already contracted indices, sothat there appear no minus signs.

Ex. 3.5 As a similar exercise, derive

γµ1...µ4γν1ν2 = γµ1...µ4ν1ν2 + 8γ[µ1...µ3

[ν2δµ4]

ν1] + 12γ[µ1µ2δµ3[ν2δ

µ4]ν1] . (3.22)

Finally let us consider that there are sums involved in such constructions. Consider theproduct γµ1...µ4ργρν1ν2 . The result should contain similar terms as in (3.22). But each termhas an extra multiplication factor reflecting the number of values that ρ can take in thissum. E.g. if the result contains one contraction between a µ and a ν index, then ρ can runover all D values except the 4 values µ1, . . . µ4, and the remaining ν that is different. Thiscounting thus gives

γµ1...µ4ργρν1ν2 = (D − 6)γµ1...µ4ν1ν2 + 8(D − 5)γ[µ1...µ3

[ν2δµ4]

ν1] + 12(D − 4)γ[µ1µ2δµ3[ν2δ

µ4]ν1] .

(3.23)

3.1.3 Even Spacetime Dimension D = 2m.

In even dimensions D = 2m, we use the generalization of the γ5 matrix as

γ∗ ≡ (−i)m+1γ0γ1 . . . γD−1 , (3.24)

which satisfies γ2∗ = in every even dimension and is Hermitian.

The highest rank subspace of the algebra has dimension 1, and for any order of compo-nents µi, one can write

γµ1µ2···µD= im+1εµ1µ2···µD

γ∗ , (3.25)

where εµ1µ2···µDis the Levi-Civita tensor (with ε01...(D−1) = −ε01...(D−1) = 1).

Ex. 3.6 Show that γ∗ commutes with all even rank elements of the Clifford algebra andanti-commute with all odd rank. Thus, for example

γ∗, γµ = 0 , (3.26)

[γ∗, γµν ] = 0 . (3.27)

The gamma matrices satisfy a duality relation:

γµ1µ2...µrγ∗ = −(−i)m+1 1

(D − r)!εµrµr−1...µ1ν1ν2...νD−rγν1ν2...νD−r

. (3.28)

The γ∗ matrix also allows to make chiral projections

PL =1

2( + γ∗) , PR =

1

2( − γ∗) . (3.29)

Ex. 3.7 Show that these are orthogonal projections, i.e. PLPL = PL and PLPR = 0.

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3.1.4 Odd spacetime dimension D = 2m+ 1.

The basic idea needed is that the Clifford algebra for dimension D = 2m+1 can be obtainedby reorganizing the matrices in the previously discussed Clifford algebra for dimension D =2m. In particular we can define two sets of 2m + 1 generating elements by adjoining thehighest rank γ∗ as follows

γ±µ = (γ0, γ1, . . . , γ(2m−1), γ2m = ±γ∗) . (3.30)

This gives us two sets of matrices, which each satisfy the Clifford algebra for dimensionD = 2m + 1. At the end they lead to the same representations of the Lorentz group, andcan therefore both be chosen. The product of all gamma matrices in odd dimensions is then± . In fact, the rank r and rank D − r sectors are related by the duality relations

γµ1...µr± = ±im+1 1

(D − r)!εµ1...µDγ±µD...µr+1 . (3.31)

Note that the order of the indices in the γ matrix in the right-hand side is inverted. Otherwisethere would be different sign factors.

3.1.5 Majorana spinors

In supersymmetric theories, we want the smallest possible representations of the gamma-matrix algebra, which are often real spinors. ‘Real’ means that there is a relation betweenthe complex conjugate spinor and the spinor itself. We will use the terminology ‘Majoranaspinors’ for spinors that satisfy such a condition. The condition is defined by a matrix thatwe call B:

Ψ∗ = BΨ . (3.32)

This matrix enters in the complex conjugation of gamma matrices, which is also related totransposes of these matrices:

γµ∗ = −t0t1BγµB−1 , (3.33)

γµT = t0t1CγµC−1 . (3.34)

Here t0t1 is a sign ±1. The reason why we write it like this will be clarified later. B and Care unitary (e.g. B† = B−1) and have definite transposition symmetry (we do not give theproof here).

BT = −t1B , CT = −t0C , t0 = ±1, t1 = ±1 . (3.35)

Ex. 3.8 Show from the fact that spacelike gamma-matrices are hermitian, and timelikegamma matrices are anti-hermitian that C = Bγ0 up to multiplication by an arbitrary phasefactor. There are other equivalent expressions.

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We will choose that B and C relate as

B = it0Cγ0 . (3.36)

There is a consistency condition for (3.32). Take the complex conjugate and use the equationagain to obtain Ψ = B∗BΨ = −t1Ψ. This gives a very direct argument that Majoranaspinors can exist only if t1 = −1. We will see that this equation can be satisfied in 4spacetime dimensions, but not for arbitrary dimensions. Using (3.36) we find the equivalentdefinition for Majorana spinors:

Ψ = Ψ†iγ0 = ΨTC . (3.37)

In classical text of quantum field theory, the first expression is a definition, and the sec-ond equality is a property for Majorana spinors, i.e. spinors that satisfy this equation, orequivalently (3.32). However, for purposes of supersymmetry the definition using C is moreconvenient, even for spinors that are not Majorana. Hence, from now on we will use thedefinition

Ψ ≡ ΨTC , (3.38)

Ex. 3.9 Prove that with both definitions a spinor bilinear of the form λχ that is a Lorentzscalar.

3.2 Spinors and spinor bilinears

For applications in supersymmetry (3.38) is more convenient, as it allows us to use symme-tries of bilinears, which are the most relevant ones for proving symmetries of an action.

For any dimension and choice of Clifford matrices, there is a symmetry or antisymmetry2

λγµ1...µnχ = tnχγµ1...µnλ . (3.39)

where tn = ±1 are sign factors. These depend on the spacetime dimension D modulo 8 andon n modulo 4, and are given in table 1. For even dimensions there are two possibilities.One can go from one to the other by replacing the charge conjugation matrix to C ′ = Cγ∗.For applications in supersymmetry we use the ones indicated in boldface.

The same sign factors can be used in a longer chain of gamma matrices:

λΓ(n1)Γ(n2) · · ·Γ(np)χ = tp−10 tn1tn2 · · · tnp χΓ(np) · · ·Γ(n2)Γ(n1)λ , (3.40)

where Γ(n) stands for a combined gamma matrix γµ1...µn . The prefactor tp−10 is thus not

relevant in 4 dimensions, where t0 = 1.

2These relations hold for anticommuting spinors. Thus a minus sign has been taken into account forchanging the order of these spinors. The matrices Cγµ1...µn or γµ1...µnC

−1 have symmetry (−tn).

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Table 1: Symmetries of gamma matrices. The entries contain the numbers n mod 4 forwhich tn = ±1. For even dimensions, in boldface are the choices that are most convenientfor supersymmetry.

D (mod 8) tn = −1 tn = +1

0 0, 3 2, 10, 1 2, 3

1 0, 1 2, 32 0, 1 2, 3

1, 2 0, 33 1, 2 0, 34 1, 2 0, 3

2, 3 0, 15 2, 3 0, 16 2, 3 0, 1

0, 3 1, 27 0, 3 1, 2

The previous relations imply also the following rule. When we have a relation betweenspinors with gamma matrices, there is a corresponding relation for the barred spinors:

χµ1...µn = γµ1...µnλ → χµ1...µn = t0tnλγµ1...µn , (3.41)

and similar for longer chains:

χ = Γ(n1)Γ(n2) · · ·Γ(np)λ → χ = tp0tn1tn2 · · · tnp λΓ(np) · · ·Γ(n2)Γ(n1) . (3.42)

In even dimensions we define left and right spinors using the projection matrices (3.29).The definition (3.38) implies that

χ = PLλ → χ =

λPL for D=0,4,8,. . . ,λPR for D=2,6,10,. . . .

(3.43)

Finally, we often use Fierzing of expressions. A general formula for any matrix in spinorspace is

M =

[D]∑k=0

1k!

Γa1...akTr (Γak...a1M) where

[D] = D for even D ,[D] = (D − 1)/2 for odd D .

(3.44)

Ex. 3.10 Prove the following chiral Fierz identities for D = 4:

PLχλPL = −12PL

(λPLχ

)+ 1

8PLγ

µν(λγµνPLχ

), PLχλPR = −1

2PLγ

µ(λγµPLχ

). (3.45)

You will need (3.28) to combine terms in (3.44).

14

3.3 Reality

In section 3.2 we have not yet defined a complex conjugation. We will now give the relevantformulae for any dimension, (with Minkowski signature).

Taking the complex conjugate of an expression can look complicated having to take allthe equations of this chapter into account, but in practice it is much easier. The easy methodconsists in replacing the complex conjugation by a charge conjugation. A charge conjugationof a scalar is just its complex conjugate. On a spinor bilinear, the rule is

(χMλ)∗ = (χMλ)C = −t0t1χCMCλC . (3.46)

Here M is any matrix in spinor space. The charge conjugation of a spinor, λC is B−1λ∗,hence equal to λ for Majorana spinors (and then also χC = χ). For the complex conjugationof a product of fermions, we have chosen as definition that this operation interchanges theorder of the fermions, but the C-operation takes this already into account, as shown in (3.46).The number (−t0t1) is e.g. +1 in 2,3,4, 10 or 11 dimensions. Hence, e.g. χλ is real in thesecases. Finally, MC = B−1M∗B. In practice, it is determined by the same number:

(γµ)C = (−t0t1)γµ . (3.47)

Charge conjugation does not change the order of matrices: (MN)C = MCNC , thus in 4dimensions all spinor bilinears λγµ1...µnχ are real.

Majorana spinors appear when t0 = 1 and t1 = −1. This can be easily extended to theso-called pseudo-Majorana case, where also real spinors are possible. In that case t0 = −1,and gamma matrices cannot be constructed that are completely real. However, completelyimaginary gamma matrices exist in that case.

When t1 = 1 we cannot define Majorana spinors, but can define ‘symplectic Majoranaspinors’. They are defined using a set of spinors λi, where i runs over an even number ofvalues, and an antisymmetric matrix εij, defining also εijεkj = δik. The symplectic Majoranaspinors satisfy

λi = (λi)C ≡ εijB−1(λj)∗ . (3.48)

We have thus modified for these cases the definition of the charge conjugation. The rule(3.46) remains valid, but matrices Mi

j transform also according to this index under chargeconjugation

(Mij)C = εikB

−1(Mk`)∗Bεj` . (3.49)

Ex. 3.11 Check that in 5 dimensions with symplectic Majorana spinors, λiχi ≡ λiχjεji ispure imaginary while λiγµχi is real.

As a general summary we can thus link the existence of different types of spinors to the signst0 and t1:

t1 = −1 , t0 = 1 : Majoranat0 = −1 : pseudo-Majorana

t1 = 1 symplectic Majorana .(3.50)

15

The difference between pseudo-Majorana and Majorana is not relevant further, and we willthus not any more make a distinction between them.

The other relevant rule is that

(γ∗)C = (−)D/2+1γ∗ . (3.51)

Majorana-Weyl spinors (or symplectic Majorana-Weyl spinors) are possible when this signfactor is 1.

A useful overview is table 2, where according to the number of dimensions it is indicated

Table 2: Irreducible spinors, number of components and symmetry properties.

Dim Spinor min # components antisymmetric2 MW 1 13 M 2 1,24 M 4 1,25 S 8 2,36 SW 8 37 S 16 0,38 M 16 0,19 M 16 0,110 MW 16 111 M 32 1,2

whether Majorana (M), Majorana-Weyl (MW) symplectic (S) or symplectic-Weyl (SW)spinors can be defined, and the corresponding number of components of a minimal spinor isgiven (the table is for Minkowski signature and has a periodicity of 8). In the final columnis indicated which spinor bilinears are antisymmetric, e.g. a 0 indicates that ε2ε1 = −ε1ε2,and a 2 indicates that ε2γµνε1 = −ε1γµνε2. This entry is modulo 4, i.e. a 0 indicates also a 4or 8 if applicable. For the even dimensions, when there are Weyl-like spinors, the symmetrymakes only sense between two spinors of the same chirality, which occurs for spinor bilinearswith an odd number of gamma matrices in these dimensions D = 2 mod 4. In the other evendimensions, D = 4 mod 4, there are always two possibilities for reality conditions and wegive here the one that includes the ‘1’ as we will need this property for the supersymmetryalgebra.

3.4 Spinor indices

For most of the book we do not need spinor indices. The spinors appear in Lorentz covariantexpressions. However, in some cases it is necessary, e.g. to write (anti)commutation relationsof supersymmetry generators. The components of the basic spinor λ are indicated as λα.The components of the barred spinor defined in (3.38) are indicated with upper indices: λα.

16

Sometimes we write λα to stress that it are the components of the barred spinor, but in factthe bar can be omitted. We introduce the raising matrix Cαβ such that

λα = Cαβλβ . (3.52)

Comparing with (3.38) this thus means that Cαβ are the components of the matrix CT .Note that the summation index β in (3.52) appears in a NW-SE line in the equation whenthe quantities are written in the order such that contracted indices appear next to eachother. Therefore, this convention is often indicated as the NW-SE spinor convention. This isrelevant when the raising matrix is antisymmetric (t0 = 1 in the terminology of table 1). Mostapplications in the book are for dimensions where this is the case, e.g. D = 2, 3, 4, 5, 10, 11.

We then also introduce a lowering matrix such that (again contraction NW-SE)

λα = λβCβα . (3.53)

In order for these two equations to be consistent, we should have

CαβCγβ = δγα , CβαCβγ = δα

γ . (3.54)

Hence Cαβ are the components of C−1.If we translate a covariant expression to spinor matrix notation, the gamma matrices

have components (γµ)αβ. E.g. for the simplest case:

χγµλ = χα(γµ)αβλβ , (3.55)

where again all contractions are NW-SE.One can now raise or lower indices consistently. E.g. one can define

(γµ)αβ = (γµ)αγCγβ . (3.56)

These gamma matrices with indices at equal level have a definite symmetry or antisymmetryproperty:

(γµ1...µn)αβ = −tn(γµ1...µn)βα . (3.57)

An interesting property is that

λαχα = −t0λαχα , (3.58)

In 4 dimensions, raising and lowering an index gives thus a minus sign. This can be usedanywhere, thus also when the contracted indices for which we flip the positions sit on gammamatrices.

Ex. 3.12 Using this property and (3.57) prove the relation (3.40). Do not forget the signfor changing the order of two (anticommuting) spinors.

Ex. 3.13 Show that using the index raising and lowering conventions, Cαβ = δβα, and forD = 4 that Cαβ = −δαβ .

17

4 The Maxwell and Yang-Mills Gauge Fields.

In this chapter we will discuss the classical field theories of the abelian and non-abeliangauge vector fields.

4.1 The abelian gauge field Aµ(x)

4.1.1 Abelian gauge transformations and the curvature

We consider first a Dirac spinor field of charge q, for which the gauge transformation is alocal change of the phase of the complex field, is

Ψ(x) → Ψ′(x) ≡ eiqθ(x)Ψ(x). (4.1)

The goal is to formulate field equations that transform covariantly under the gauge trans-formation. This requires the introduction of a new field, the vector gauge field or vectorpotential Aµ(x), which is defined to transform as

Aµ(x) → A′µ(x) ≡ Aµ(x) + ∂µθ(x). (4.2)

One then defines the covariant derivative

DµΨ(x) ≡ (∂µ − iqAµ(x))Ψ(x), (4.3)

which transforms with the same phase factor as Ψ(x), namely DµΨ(x) → eiqθ(x)DµΨ(x).

Ex. 4.1 Derive from (4.3) that

[Dµ, Dν ]Ψ ≡ (DµDν −DνDµ)Ψ = −iqFµνΨ . (4.4)

The action functional for the electromagnetic field interacting with a field of charge q,which we take to be a massive Dirac field, is the sum of two terms, each gauge invariant,

S[Aµ, Ψ,Ψ] =

∫dDx

[−1

4F µνFµν + Ψ(γµDµ −m)Ψ

]. (4.5)

The Euler variation of (4.5) with respect to the gauge potential Aν is

δLδAν

= ∂µFµν − iqΨγνΨ = 0 . (4.6)

Ex. 4.2 Consider the gauge covariant translation, defined by δAµ = aνFνµ and δΨ =aνDνΨ. Show that they differ from a conventional translation by a gauge transformationwith gauge-dependent parameter θ = −aνAν. Gauge covariant translations are a symmetryof the action (4.5).

18

4.1.2 Dual tensors

Using the ε-tensor introduced in Sec. 3.1.1 one can define in 4 dimensions the dual of thefield strength tensor

F µν ≡ −12iεµνρσFρσ . (4.7)

In our conventions the dual tensor is imaginary. The equations above are valid in Minkowskispace but must be modified in curved spacetimes as we will discuss later.

Ex. 4.3 Prove that the dual of the dual is the identity, specifically that

− 12iεµνρσFρσ = F µν . (4.8)

You will need (3.11).

Ex. 4.4 Define the quantity Lµν = γµνPL. Show that this is anti-selfdual. Hint: check firstthat γµνγ∗ = 1

2iεµνρσγ

ρσ.

Ex. 4.5 Show that the Bianchi identity

∂µFνρ + ∂νFρµ + ∂ρFµν = 0 , (4.9)

can be expressed as ∂µFµν = 0.

Ex. 4.6 Show that the quantity FµνFµν is a total derivative, i.e.

FµνFµν = −i∂µ(εµνρσAνFρσ) . (4.10)

Show, using the Lorentz transformation law of a vector, i.e.

Aµ(x) → A′µ(x) ≡ Λ−1µνAν(Λx) , (4.11)

that under a Lorentz transformation(FµνF

µν)

(x) → det Λ−1(FµνF

µν)

(Λx) . (4.12)

Thus FµνFµν transforms as a scalar under proper Lorentz transformations but changes sign

under space or time reflections. Use the Schouten identity (3.9) to prove that

FµρFνρ = 1

4ηµνFρσF

ρσ . (4.13)

Given any antisymmetric tensor such as Fµν , we introduce the selfdual and anti-selfdualtensors

F±µν = 12

(Fµν ± Fµν

), F±µν =

(F∓µν

)∗. (4.14)

Ex. 4.7 Prove the following relations between (anti)-selfdual tensors

G+µνH−µν = 0 , G±ρ(µH±ν)ρ = −14ηµνG±ρσH±ρσ , G+

ρ[µH−ν]ρ = 0 . (4.15)

19

4.1.3 Duality transformations and the symplectic group

The standard kinetic terms of spin 1 fields are proportional to FµνFµν . More general kinetic

Lagrangians frequently appear in supergravity, especially when the system contains a set ofAbelian gauge fields AAµ (x), indexed by A = 1, 2, . . . ,m, together with scalar fields φi. IffAB(φ) = fBA(φ) are arbitrary complex functions of the scalars, we can consider the action

S =

∫d4xL , L = −1

4(Re fAB)FA

µνFµν B + 1

4i(Im fAB)FA

µνFµν B , (4.16)

which is real since, as defined in (4.7), FµνFµν is pure imaginary. Although FµνF

µν is atotal derivative, the second term does contribute to the equations of motion when Im fABis a function of other fields of the system. Our discussion will not depend on the specificproperties of fAB(φ). However, one usually requires that Re fAB is a positive definite matrix,which ensures that gauge field kinetic energies are positive.

Using the self-dual tensors of (4.14), we then rewrite the Lagrangian (4.16) as

L(F+, F−) = −12

Re(fABF

−Aµν F µν−B)

= −14

(fABF

−Aµν F µν−B + f ∗ABF

+Aµν F µν+B

), (4.17)

and define the new tensors

GµνA = εµνρσ

δS

δF ρσ A= −(Im fAB)F µν B − i(Re f)ABF

µν B = Gµν+A +Gµν−

A , (4.18)

Gµν−A = −2i

∂S(F+, F−)

∂F−Aµν

= ifABFµν−B , Gµν+

A = 2i∂S(F+, F−)

∂F+Aµν

= −if ∗ABFµν+B .

Since the field equation for the action containing (4.17) is

0 =δS

δAAν= −2∂µ

δS

δFAµν

, (4.19)

the Bianchi identity (see Ex. 4.5) and the equation of motion can be expressed in the conciseform

∂µ ImFA−µν = 0 : Bianchi identities,

∂µ ImGµν−A = 0 : Equations of motion. (4.20)

(The same equations hold for ImFA+ and ImGA+.)

Duality transformations are linear transformations of the 2m tensors FAµν and GµνA

(accompanied by transformations of the fAB) which mix Bianchi identities and equations ofmotion, but preserve the structure which led to (4.20). Typically electric F0i and magneticFij components of the field strengths mix under the transformations; hence the name electric-magnetic duality. Since the equations (4.20) are real, we can mix them by a real 2m × 2mmatrix. We extend this transformations to the (anti)self-dual tensors, and consider(

F ′−

G′−

)= S

(F−

G−

)=

(A BC D

) (F−

G−

), (4.21)

20

with real m ×m sub-matrices A, B, C, D. Due to the reality of these matrices, the samerelations hold for the selfdual tensors F+ and G+.

We require that the transformed field tensors F ′A, G′A are also related by the definitions(4.18), with appropriately transformed fAB. We work out this requirement in the followingsteps:

G′− = (C + iDf)F− = (C + iDf)(A+ iBf)−1F ′−

→ if ′ = (C + iDf)(A+ iBf)−1 . (4.22)

The last equation gives the symmetry transformation relating f ′AB to fAB. If G′−µν is to be thevariational derivative of a transformed action, as (4.18) requires, then the matrix f ′ must besymmetric. For a generic3 symmetric f , this requires that the matrices A,B,C,D satisfy

ATC − CTA = 0 , BTD −DTB = 0 , ATD − CTB = . (4.23)

These relations among A,B,C,D are the defining conditions of a matrix of the symplecticgroup in dimension 2m so we reach the conclusion that

S =

(A BC D

)∈ Sp(2m,R) . (4.24)

The conditions (4.23) may be summarized as

STΩS = Ω where Ω =

(0− 0

). (4.25)

The matrix Ω is often called the symplectic metric. This is the main result: duality trans-formations in 4 spacetime dimensions are transformations of the group Sp(2m,R) , which isa non-compact group.

Ex. 4.8 The dimension of the group Sp(2m,R) is the number of elements of the matrix S,namely 4m2 minus the number of independent conditions contained in (4.25). Show that thedimension is m(2m+ 1).

The symplectic transformations thus transform solutions of (4.20) into solutions. How-ever, they are not invariances of the action. Indeed, writing

L = −12

Re(fABF

−Aµν F µν−B)

= −12

Im(F−Aµν Gµν−

A

), (4.26)

we obtain

ImF ′−G′− = Im(F−G−

)+ Im

(2F−(CTB)G− + F−(CTA)F− +G−(DTB)G−

). (4.27)

3If the initial fAB is non-generic, then the matrix in the last equation can be replaced by any matrixwhich commutes with fAB . For generic fAB , this must be a constant multiple of the unit matrix. Theconstant, which should be positive to preserve the sign of the kinetic energy of the vectors, can be absorbedby rescaling the matrices A,B,C,D.

21

If C 6= 0, B = 0 the Lagrangian is invariant up to a four-divergence, since ImF−F− =−1

4εµνρσFµνFρσ and the matrices A and C are real constants. For B 6= 0 neither the La-

grangian nor the action is invariant.If sources are added to the equations (4.20), the transformations can be extended to the

dyonic solutions of the field equations by letting the magnetic and electric charges

(qAmqeA

)transform as a symplectic vector too. The Schwinger-Zwanziger quantization condition re-stricts these charges to a lattice. Invariance of this lattice restricts the symplectic transfor-mations to a discrete subgroup Sp(2m,Z).

Finally, the transformations with B 6= 0 will be non–perturbative. This can be seen fromthe fact that they do not leave the purely electric charges invariant, or from the fact that(4.22) shows that these transformations invert f , which plays the role of the gauge couplingconstant.

The important properties of the matrix fAB are that it be symmetric and that Re fABdefine a positive definite quadratic form in order to have positive gauge field energy. Theseproperties are preserved under symplectic transformations defined by (4.22).

Ex. 4.9 The simplest case of electromagnetic duality occurs with one Abelian vector Aµ(x)and f = Z, where Z(x) is a complex scalar field. The action is

L = −14(ReZ)FµνF

µν + 18(ImZ)εµνρσFµνFρσ . (4.28)

Verify that the set of Bianchi identities and field equations for Aµ is invariant under thesymplectic transformation (4.21), if Z transforms as

iZ ′ =C + iDZ

A+ iBZwhere AD −BC = 1 . (4.29)

In supergravity there will an additional kinetic term for Z, which must also be invariantunder this transformation.

The transformations (4.29) define the Sp(2,R) = S`(2,R) symmetry. The S`(2,Z) sub-group, often called the modular group, is generated by(

1 01 1

),

(0 1−1 0

)Z ′ = Z − i , Z ′ =

1

Z. (4.30)

Ex. 4.10 Consider in the previous exercise that Z is not a field, but a real coupling con-stant g−2. Then the first transformation of (4.30) does not preserve this reality restriction.However, the second one does. Prove that this transformation is the interchange of electricand magnetic fields in the simplest electromagnetic theory. It transforms g to its inverse,and thus relates the strong and weak coupling description of the theory.

Finally, let us consider the meaning of the transformation (4.22). In some cases the trans-formations can be induced by transformations of scalar fields E.g. in the example of ex.

22

4.9, it can be implemented by the transformation of Z as given in (4.29). However, in othercases, as the one in ex. 4.10 we have to change constants. These constants that transformare sometimes called ‘spurionic quantities’. The transformations thus relate two differenttheories. Solutions of one theory are mapped in solutions of the other one. This is the basicidea of dualities in M-theory.

4.2 Non-abelian Gauge Symmetry

Yang-Mills theory is based on a non-abelian generalization of the U(1) gauge symmetry. Itis the fundamental idea underlying the standard model of elementary particle interactions.We follow the pattern of starting with the global symmetry and then gauging it. The focusof our discussion is the derivation of the basic formulas of the classical gauge theory.

4.2.1 Global internal symmetry

We consider a symmetry group G. We will mostly restrict our attention to infinitesimaltransformations, related to a real algebra g. It is represented by a basis of matrix generatorstA, A = 1, . . . , dimG, which are anti-Hermitian in the case of a compact gauge group. Thecommutator of the generators is of the form

[tA, tB] = fABCtC . (4.31)

The array of real numbers fABC are structure constants of the algebra. They obey the Jacobi

identityfAD

EfBCD + fBD

EfCAD + fCD

EfABD = 0 . (4.32)

For simple algebras, the indices can be lowered by the Cartan-Killing metric, and then thefABC are totally antisymmetric. There is one important representation, the adjoint represen-tation of dimension, in which the representation matrices are closely related to the structureconstants by (tA)DE = fAE

D. Note that the labels DE denote row and column indices of thematrix tA. The adjoint representation is a real representation; the representation matricesare real and antisymmetric for compact algebras. For complex representations we will usethe notation (tA)αβ. Anti-Hermiticity then requires (tA)α∗β = −(tA)βα. The row and columnindices will often be suppressed when no ambiguity arises.

Ex. 4.11 Use (4.32) to show that the matrices (tA)DE = fAED satisfy (4.31) and therefore

give a representation.

The general element of g is represented by a superposition of generators θAtA where the θA

are real parameters.On the fields the algebra is realized by infinitesimal transformations, e.g.

δΨ = −θAtAΨ ,

δΨ = ΨθAtA ,

δφA = θCfBCAφB . (4.33)

23

The first two relations are related by hermitian conjugation, assuming a compact algebra.The last relation is the infinitesimal transformation of a field in the adjoint representation,taken here as the set of real scalars φA. Of course, scalars could be assigned to any repre-sentation.

Actions, such as the kinetic action for massive fields,

S[Ψ,Ψ] =

∫dDxΨ[γµ∂µ +m]Ψ , (4.34)

are required to be invariant under (4.33).

Ex. 4.12 Show that (4.34) is invariant under the transformation (4.33). Consider the con-served current

JAµ = ΨtAγµΨ , A = 1, . . . , dimG . (4.35)

Show that it transforms as a field in the adjoint representation, i.e.

δJAµ = θCfCABJBµ . (4.36)

4.2.2 Gauging the symmetry

In gauged non-abelian internal symmetry, the group parameter θA(x) is promoted to anarbitrary function in the transformation rules of matter fields. The first step in the systematicformulation of gauge invariant field equations is to introduce the gauge field with infinitesimaltransformation rule

δAAµ (x) = ∂µθA + θC(x)ABµ (x)fBC

A . (4.37)

This gauge field appears in covariant derivatives. For the fields Ψα, Ψα, and φA of (4.33) wewrite

DµΨ = (∂µ + tAAAµ )Ψ ,

DµΨ = ∂µΨ− ΨtAAAµ ,

DµφA = ∂µφ

A + fBCAABµ φ

C . (4.38)

Note that the gauge transformation (4.37) can be written as δAAµ (x) = DµθA using the

covariant derivative for the adjoint representation.

Ex. 4.13 Show that the covariant derivatives of the three fields in (4.38) transform in thesame way as the fields themselves, and with no derivatives of the gauge parameters. Forexample δDµΨ = −θAtADµΨ.

Given this result it is easy to see that any globally symmetric action for scalar and spinormatter fields becomes gauge invariant if one replaces ∂µ → Dµ for all fields.

24

4.2.3 The Yang-Mills action

The next step in the development is to define the quantities necessary for the dynamics of thegauge field itself. The simplest way to proceed is to compute the commutator of two covariantderivatives acting on a field in the representation R. We would get the same information, nomatter which representation, so we will study just the case [Dµ, Dν ]Ψ ≡ (DµDν − DνDµ)Ψ.A careful computation gives

[Dµ, Dν ]Ψ = FAµνtAΨ , (4.39)

whereFAµν = ∂µA

Aν − ∂νA

Aµ + fBC

AABµACν . (4.40)

The properties of the covariant derivative guarantee that the right side of (4.39) transformsas a field in the same representation as Ψ. Thus FA

µν should have simple transformationproperties. Indeed, one can derive

δFAµν = θCFB

µνfBCA . (4.41)

We see that FAµν is an anti-symmetric tensor in spacetime, which transforms as a field in the

adjoint representation of g.

Ex. 4.14 Derive (4.41).

The gauge-invariant action is a natural generalization of (4.5)

S[AAµ , Ψα,Ψα] =

∫dDx

[−1

4FAµνFA

µν + Ψα(γµDµ −m)Ψα]. (4.42)

Implicitly, a metric for the gauge algebra is used here. This is the Cartan-Killing metric,which for simple gauge algebras can be taken to be

Tr(tAtB) = −αδAB . (4.43)

for some positive α.

4.3 Internal Symmetry for Majorana Spinors

Majorana spinors play a central role in supersymmetric field theories. In many applicationsthey transform in a representation of a non-abelian internal symmetry group. For example,the spinor fields of super-Yang-Mills theory are denoted as λA and transform in the adjointrepresentation of the gauge group. Since the structure constants are real, this transforma-tion rule is consistent with the fact that Majorana spinors obey a reality constraint. Onecan consider the more general situation of a set of Majorana spinors Ψα transforming asδΨα = −θA(tA)αβ Ψβ. The transformation must also satisfy the Majorana condition, andthis requires that the matrices tA are those of a really real representation of the group G.(Obviously there is a similar requirement on the symmetry transformation of a set of realscalars).

25

In D = 4 dimensions, the requirement that Majorana spinors transform in a real repre-sentation of the gauge group can be bypassed because internal symmetries can include chiraltransformations, which involve the highest rank element γ∗. This matrix is imaginary underthe C-operation, see (3.51). We use the chiral projectors PL and PR as in (3.29). Supposethat the matrices tA are generators of a complex representation of the Lie algebra. Thenthe complex conjugate matrices t∗A are generators of the conjugate representation. Let χα

denote a set of Majorana spinors to which we assign the group transformation rule

δχα = −θA(tAPL + t∗APR)αβχβ . (4.44)

The matrices tAPL + t∗APR are generators of a representation of an explicitly real represen-tation of the Lie algebra, so the transformed spinors also satisfy the Majorana condition.

By applying the projectors to (4.44), one can see that the chiral and anti-chiral compo-nents of χα transform as

δPLχα = −θA(tA)αβPLχ

β ,

δPRχα = −θA(t∗A)αβPRχ

β . (4.45)

Ex. 4.15 What is the covariant derivative Dµχ? Show that the kinetic Lagrangian densityχγµDµχ is invariant under (4.44) for anti-Hermitian tA, and that the variation of the massterm is

δ(χχ) = −θAχ(tA + tTA)γ∗χ . (4.46)

The mass term is invariant only for the subset of generators that are antisymmetric, andthus real. This condition defines a subalgebra of the original Lie algebra g of the theory,specifically the subalgebra which contains only parity conserving vector-like gauge transfor-mations. For the case g = su(N), the subalgebra is isomorphic to so(N). Non-invariance ofthe Majorana mass term is a special case of the general idea that chiral symmetry requiresmassless fermions.

26

5 The free Rarita-Schwinger field

In this chapter we will begin to assemble the ingredients of supergravity by studying thefree spin 3

2field. Supergravity is the gauge theory of global supersymmetry, which we will

usually abbreviate as SUSY. The key feature is that the symmetry parameter of global SUSYtransformations is a constant spinor εα. In supergravity it becomes a general function inspacetime, εα(x). The associated gauge field is a vector-spinor Ψµα(x). This field and thecorresponding particle have acquired the name ‘gravitino’.

Supergravity theories necessarily contain the gauge multiplet, the set of fields required togauge the symmetry in a consistent interacting theory, and may contain matter multiplets,sets of fields realizing global SUSY. The gauge multiplet contains the gravitational field, oneor more vector-spinors, and sometimes other fields. In this chapter we are concerned with thefree limit, in which the various fields do not interact, and we can consider them separately.In particular we consider Ψµ(x) as a free field, subject to the gauge transformation

δΨµ(x) = ∂µε(x) . (5.1)

Furthermore we will assume that Ψµ and ε are complex spinors with 2[D2

] spinor componentsfor spacetime dimension D. This is fine for the free theory in any dimension D, but inter-acting supergravity theories are more restrictive as to the spinor type permitted in a givenspacetime dimension (and such theories exist only for D ≤ 11). We will need to use therequired Majorana and/or Weyl spinors when we study these theories in later chapters. The

number 2[D2

] that is mentioned here, may thus change according to these conditions.It is consistent with the pattern set in the previous chapter that the gauge field Ψµ(x)

is a field with one more vector index than the gauge parameter ε(x). Furthermore, asin the the case of electromagnetism, the anti-symmetric derivative ∂µΨν − ∂νΨµ is gaugeinvariant. An important difference arises because we now seek a gauge invariant first orderwave equation for the fermion field. It is advantageous to start with the action, which mustbe a) Lorentz invariant, b) first order in spacetime derivatives, c) invariant under the gaugetransformation (5.1) and the simultaneous conjugate transformation of Ψµ, and d) the Euler-Lagrange equation for Ψµ must be the Dirac conjugate of that for Ψµ. It is easy to see thatthe expression

S =

∫dDx Ψµγ

µνρ∂νΨρ , (5.2)

which contains the third rank Clifford algebra element γµνρ, has all these properties. Notethat the action is gauge invariant but the Lagrangian density is not. Instead its variationis the total derivative δL = ∂µ(εγµνρ∂νΨρ). The reason is that the fermionic gauge sym-metry shares some features of a space-time symmetry. It is related to global SUSY, whichtransforms particles of spin s into particles of spin s ± 1

2. A change of spin is a change in

a spacetime property. Historically, Rarita and Schwinger invented a wave equation for amassive spin 3

2particle in 1941. The massless limit of the action is a transformed version of

(5.2), and Rarita and Schwinger simply noted that it possesses a fermionic gauge symmetry.The equation of motion obtained from (5.2) reads

γµνρ∂νΨρ = 0 . (5.3)

27

One can immediately see that it shares some of the properties of the analogous electro-magnetic equation, which reads ∂µFµν = 0. Gauge invariance is manifest, and the left side

vanishes identically when the derivative ∂µ is applied. Thus (5.3) comprises 2[D2

](D − 1)

independent equations, which is enough to determine the 2[D2

]D components of Ψρ up to thefreedom of a gauge transformation.

Ex. 5.1 Show directly that for D = 3, the field equation (5.3) implies that ∂νΨρ−∂ρΨν = 0.This means that the field has no gauge invariant degrees of freedom and thus no propagatingparticle modes. This is the supersymmetric counterpart of the situation in gravity for D = 3,where the field equation Rµν = 0 implies that the full curvature tensor Rµνρσ = 0. Hence nodegrees of freedom.

5.1 Degrees of freedom

We are now going to show that this equation describes the right number of degrees of freedomfor a massless spin 3/2 field. But to clarify what we really count, we will repeat what wemean also with this for the lower spin.

We will again take a general dimensional viewpoint, but let’s begin the discussion in 4dimensions with some remarks about the particle representations of the Poincare group andthe fields usually used to describe elementary particles. A particle is classified by its massm and spin s, and a massive particle of spin s has 2s + 1 helicity states. Massless particlesof spin s = 0 or s = 1/2 have one or two helicity states, respectively, in agreement with thecounting for massive particles. However, massless particles of spin s ≥ 1/2 have 2 helicitystates, for all values of s.

Helicity is defined as the eigenvalue of the component of angular momentum in thedirection of motion. For a massless particle of spin s, the two helicity states have eigenvalues±s. For a massive particle of spin s the helicity eigenvalues, called λ, range in integer stepsfrom λ = s to λ = −s.

Let us compare the count of the helicity states with the number of independent functionsthat must be specified as initial data for the Cauchy initial value problem of the associatedfield. The first number can be considered to be the number of quantum degrees of freedomwhile the second is the number of classical degrees of freedom.

Let’s do the counting for massless particles that are identified with their anti-particles.The associated fields are real for bosons and satisfy the Majorana condition for fermions. Thecounting is similar for complex fields. We assume that the equations of motion are secondorder in time for bosons and first order for fermions. A unique solution of the Cauchyproblem for the scalar φ(x) requires the initial data φ(~x, 0) and φ(~x, 0), the time derivative.For Ψα(x), we must only specify the initial values Ψα(~x, 0) of all 4 components, because theDirac equation gives then Ψα(~x, 0) = −γ0γi∂iΨα(~x, 0). The number of helicity states is 1for φ(x) and 2 for Ψα(x). The number of classical degrees of freedom is twice the number ofhelicity states. We will denote this number of helicity states further as the ‘on-shell numberof degrees of freedom.’

28

We continue this counting, in a naive fashion, for vector Aµ(x), vector-spinor Ψµα(x), andsymmetric tensor hµν(x) fields, the latter describing gravitons in Minkowski space. Followingthe earlier pattern we would expect to need 8, 16, and 20 functions, respectively, as initialdata. These numbers greatly exceed the two helicity states for spin 1, spin 3

2and spin 2

particles. Something new is required to resolve this mismatch.Before continuing with the technical discussion we reiterate that the classical degrees of

freedom we are concerned with are the independent functions required as initial data forthe Cauchy problem of hyperbolic equations. In the analysis we also find that certain fieldcomponents satisfy elliptic equations, such as the basic Laplace equation ∇2φ = 0. Thespecification of a solution requires supplying data on a boundary 2-surface rather than the3-dimensional Cauchy surface. Thus a field component satisfying an elliptic equation doesnot contain any degrees of freedom.4

5.1.1 The counting for Maxwell fields

The free electromagnetic field satisfies

∂µFµν = 0 . (5.4)

Since ∂ν∂µFµν vanishes identically, (5.4) comprises D − 1 independent components in D-dimensional Minkowski spacetime. This is not enough to determine the D components ofAµ, which is not surprising because of the gauge symmetry. One can change Aµ → Aµ +∂µθwithout affecting Fµν . So far, we did not yet use the field equations. Therefore, we will callthis number, i.e. (D− 1) for the gauge vectors, the ‘off-shell number of degrees of freedom’.

One deals with this situation by ‘fixing the gauge’. This means that one imposes onecondition on the D components of Aµ, which eliminates the freedom to change gauge. Dif-ferent gauge conditions illuminate different physical features of the theory. We will lookfirst at the condition ∂iAi(~x, t) = 0, which is called the Coulomb gauge condition. Althoughnon-covariant it is a useful gauge to explore the structure of the initial value problem anddetermine the true degrees of freedom of the system. Note that the time-space split implicitin the initial value problem is also non-covariant.

First let’s show that this condition does eliminate the gauge freedom. We check whetherthere are gauge functions θ(x) with the property that ∂iA

′i = ∂i(Ai+∂iθ) = 0 when ∂iAi = 0.

This requires that ∇2θ = 0. As explained above, this implies that the gauge freedom hasbeen fixed.

Let’s write out the time (µ→ 0) and space (µ→ i) components of the Maxwell equation(5.4). One finds

∇2A0 − ∂0(∂iAi) = 0 ,

2Ai − ∂i∂0A0 − ∂i(∂jAj) = 0 . (5.5)

4It is instructive to note that the Fourier transform of the harmonic condition ∇2φ(x) = 0 is ~k2φ(~k) = 0whose only smooth solution vanishes identically.

29

In the Coulomb gauge, the first equation simplifies to ∇2A0 = 0, and we see that5 A0 doesnot contain any degrees of freedom. The second equation in (5.5) then becomes 2Ai = 0,so the spatial components Ai satisfy the massless scalar wave equation.

We can now count the classical degrees of freedom, which are the initial data Ai(~x, 0),Ai(~x, 0) required for a unique solution of 2Ai = 0. There are, due to the gauge condition, atotal of 2(D−2) independent degrees of freedom, because the initial data must be constrainedto obey the Coulomb gauge condition.

This number thus agrees for D = 4 with the rule that the classical degrees of freedom aretwice the number of on-shell degrees of freedom counted as helicity states. In general, wefind for the gauge vectors (D − 1) off-shell degrees of freedom and (D − 2) on-shell degreesof freedom. It turns out that this is part of a general rule for massless fields of higher spin:the off-shell degrees of freedom are representations of SO(D− 1) and the on-shell degrees offreedom are representations of SO(D − 2). Here we find for both the vector representation.

5.1.2 The counting for the Rarita-Schwinger field

The Rarita Schwinger field has a priori D 2[D2

] components (where the 2[D2

] stands for thenumber of component of a spinor and may thus change according to reality or chiralityconditions). The gauge degree of freedom takes away one spinor, such that there are (D −1)2[D

2] off-shell degrees of freedom. Let’s now study the initial value problem for (5.3) and

thus count the on-shell number of degrees of freedom. We need to fix the gauge, so weimpose the non-covariant condition

γiΨi = 0 , (5.6)

which will play the same role as the Coulomb gauge condition above.The ν → 0 and ν → i components of (5.3) are

γij∂iΨj = 0 ,

−γ0γij (∂0Ψj − ∂jΨ0) + γijk∂jΨk = 0 . (5.7)

Using γij = γiγj − δij and the gauge condition, the first equation implies that

∂iΨi = 0 . (5.8)

Multiplying the second equation from the left with γi and using the gauge condition and thefirst equation, leaves us with

− 2γ0γj∂jΨ0 = 0 → ∇2Ψ0 = 0 → Ψ0 = 0 , (5.9)

where we first applied γiδi, and then applied the arguments according to the discussion inthe introduction to this section. Therefore, the second equation of (5.7) reduces to (usingγijk = γiγjk − δijγk + δikγj and the first two terms can be dropped due to previous results)

γ0∂0Ψi + γj∂jΨi = γµ∂µΨi = 0 . (5.10)

5When a source current is added to the Maxwell equation (5.4), A0 no longer vanishes, but it is alge-braically determined by the source. Thus it is not a degree of freedom of the gauge field system.

30

Hence, the remaining components of Ψi satisfy a Dirac equation, such that the time deriva-tives of the initial conditions are no independent initial data. Hence the number of classicaldegrees of freedom after imposing (5.6) and (5.8) is (D − 3)2[D

2], i.e. 4 in 4 dimensions with

Majorana conditions. Hence, the on-shell degrees of freedom are one half of this, whichcorrespond to the helicity states ±3/2 in 4 dimensions. In general dimension, it should be arepresentation of SO(D− 2) as we mentioned before. Indeed, a vector-spinor representationis an irreducible representation when one subtracts the γ-trace, giving again as number ofcomponents (D − 3)2[D

2].

5.2 Dimensional Reduction on MinkowskiD × S1.

Our aim in this section is quite narrow, but the approach will be broad. The narrow goalis to extend the Rarita-Schwinger equation to describe massive gravitinos, but we wish todo it by introducing the important technique of dimensional reduction, which is also calledKaluza-Klein theory. The main idea is that a fundamental theory, perhaps supergravityor string theory, that is formulated in D′ spacetime dimensions can lead to an observablespacetime of dimension D < D′. In the most common variant of this scenario, there is astable solution of the equations of the fundamental theory that describes a manifold of thestructure MD′ = MD ×Xd with d = D′ −D. The factor MD is the spacetime in which wemight live, thus non-compact with small curvature, while Xd is a tiny compact manifold ofspatial extent L. The compact space Xd can be thought of as hidden dimensions of spacetimethat are not accessible to direct observation because of basic properties of wave physics thatare coded in quantum mechanics as the uncertainty principle. This principle asserts thatit would take wave excitations of energy E ≈ 1/L to explore structures of spatial scaleL. If L is sufficiently small, this energy scale cannot be achieved by available apparatus.Nevertheless, the dimensional reduction might be confirmed since the presence of Xd hasimportant indirect effects on physics in MD.

In this section we use an elementary version of dimensional reduction, which still hasinteresting physics to teach. Instead of obtaining the structure MD×Xd from a fundamentaltheory including gravity, we will simply explore the physics of the various free fields we havestudied, assuming that the (D + 1)-dimensional spacetime is MinkowskiD ⊗ S1. The mainfeature is that Fourier modes of fields on S1 are observed as infinite sets of massive particles.

5.2.1 Dimensional reduction for scalar fields.

Let’s change to a more convenient notation and rename the coordinates of the (D + 1)-dimensional product spacetime x0 = t, x1, . . . , xD−1, y, where y is the coordinate of S1 withrange 0 ≤ y ≤ 2πL. It is more accurate to take y of infinite range, but to assume that pointsrelated by translation by a multiple of 2πL are identified; i.e. y ≡ y + 2πL. We consider amassive complex scalar field φ(xµ, y) that obeys the Klein-Gordon equation

[2D+1 −m2]φ =

[2D +

(∂

∂y

)2

−m2

]φ = 0 . (5.11)

31

Acceptable solutions must be single-valued on S1 and thus have a Fourier series expansion

φ(xµ, y) =∞∑

k=−∞

eikyL φk(x

µ). (5.12)

It is immediate that the spacetime function associated with the kth Fourier mode, namelyφk(x

µ), satisfies [2D −

(k

L

)2

−m2

]φk = 0 . (5.13)

Thus it describes a particle of mass mk2 =

(kL

)2+ m2. So the spectrum of the theory, as

viewed in MinkowskiD, contains an infinite tower of massive scalars!There is an even simpler way to find the mass spectrum. Just substitute the plane wave

eipµxµ eikyL directly in the (D + 1)-dimensional equation (5.11). The D-component energy-

momentum vector must satisfy pµpµ = (k/L)2 +m2. The mass shift due to the Fourier waveon S1 is immediately visible. This observation applies to all other fields discussed below.

5.2.2 Dimensional reduction for spinor fields.

We will consider the dimensional reduction process for a complex spinor Ψ(xµ, y) for evenD = 2m (i.e. such that the spinors in D+ 1 dimensions have the same size). Two new ideasenter the game. The first just involves the Dirac equation in D dimensions. We remark thatif Ψ(x) satisfies

[/∂D −m]Ψ(x) = 0 , (5.14)

then the new field Ψ ≡ e−iγ∗βΨ, obtained by applying a chiral phase factor, satisfies

[/∂D −m(cos 2β + iγ∗ sin 2β)]Ψ = 0 . (5.15)

Physical quantities are unchanged by the field redefinition, so both equations describe par-ticles of mass m. One simple implication is that the sign of m in (5.14) has no physicalsignificance, since it can be changed by field redefinition with β = π/2.

The second new idea is that a fermion field can be either periodic or anti-periodicΨ(xµ, y) = ±Ψ(xµ, y + 2πL)). Anti-periodic behavior is permitted because a fermion fieldis not observable. Rather, bilinear quantities such as the energy density T 00 = Ψγ0∂0Ψare observables and they are periodic even when Ψ is anti-periodic. Thus we consider theFourier series

Ψ(xµ, y) =∑k

eikyL Ψk(x

µ), (5.16)

where the mode number k is integer or half-integer for periodic or anti- periodic fields,respectively. In either case when we substitute (5.16) in the (D + 1)-dimensional Diracequation [/∂D+1 −m]Ψ((xµ, y) = 0, we find that Ψk(x

µ) satisfies6[/∂D −

(m+ iγ∗

k

L

)]Ψk(x

µ) = 0 . (5.17)

6Recall that for odd spacetime dimension D = 2m+ 1, γD = ±γ∗ where γ∗ is the highest rank Cliffordelement in D = 2m dimensions.

32

By applying the appropriate chiral phase, namely tan 2β = k/(mL), we see that Ψk(xµ)

describes particles of mass mk2 = ( k

L)2 + m2. Again we would observe an infinite tower of

massive spinor particles with distinct spectra for the periodic and anti-periodic cases.

5.2.3 Dimensional reduction for the vector gauge field.

We now apply circular dimensional reduction to Maxwell’s equation

∂νFνµ = 2D+1Aµ − ∂µ(∂νAν) = 0 (5.18)

in D + 1 dimensions, and we assume a periodic Fourier series representation

Aµ(x, y) =∑k

eikyL Aµk(x) , AD(x, y) =

∑k

eikyL ADk(x) , (5.19)

with k an integer. The analysis simplifies greatly if we assume the gauge conditions ADk(x) =0 for k 6= 0 and vector component D tangent to S1. It is easy to see that this gauge canbe achieved and uniquely fixes the Fourier modes θk(x), k 6= 0 of the gauge function. Thegauge invariant Fourier mode AD0(x) remains a physical field in the dimensionally reducedtheory. A quick examination of the µ→ D component of (5.18) shows that it reduces to

k = 0 : 2D+1AD0 = 2DAD0 = 0 ,

k 6= 0 : ∂µAµk = 0 , (5.20)

so the mode AD0(x) simply describes a massless scalar in D dimensions. For µ ≤ D− 1, thewave equation (5.18) implies that the vector modes Aµk(x) satisfy[

2D −k2

L2

]Aµk − ∂µ(∂νAνk) = 0 . (5.21)

For mode number k = 0 this is just the Maxwell equation in D dimensions with its gaugesymmetry under Aµ0 → Aµ0 + ∂µθ0 intact, since the Fourier mode θ0(x) remained unfixedin the process above. For mode number k 6= 0, (5.21) is just the standard equation for amassive vector field with mass m2

k = k2/L2, as the last term vanishes due to (5.20).The Lagrangian that describes a massive spin-1 field should give rise to the field equation

(5.21) with this mass. Therefore, a massive spin-1 field is described by the action

S =

∫dDx

[−1

4FµνF

µν − 12m2AµA

µ]. (5.22)

We have the D component field Aµk subject to the single constraint (5.20) and thus givingD − 1 quantum degrees of freedom for each Fourier mode k 6= 0. The D − 1 particle statesfor each fixed energy- momentum pµ transform in the vector representation of SO(D − 1)as appropriate for a massive particle. Note that there are 3 states for D = 4, which agreeswith 2s+ 1 for spin s = 1. The count of states is the same in the massless k = 0 sector also,where we have the gauge vector Aµ0 plus the scalar AD0 with D− 2 and 1 on-shell degree offreedom respectively.

33

5.2.4 Finally Ψµ(x, y).

Let’s apply dimensional reduction to the massless Rarita-Schwinger field in D+1 dimensionswith D = 2m. We will therefore concentrate at the k 6= 0 part of the Fourier series, whichcan be done e.g. by assuming that the field Ψµ(x, y) is anti-periodic in y so that its Fourier

series involves the modes eikyL Ψµk(x) with half-integral k.

We would like to derive the wave equation of such a massive gravitino, and a gauge choicemakes this task much easier. We impose the condition ΨDk(x) = 0 on all Fourier modes.Anti-periodicity eliminates the complication of zero-modes that we faced in Sec. 5.2.3,because these gauge conditions now completely eliminate the field component ΨD(x, y).

Let’s write out the µ → D and µ ≤ D − 1 components of (5.3) with ΨD = 0 as (usingγD = γ∗)

γνρ∂νΨρk = 0 ,[γµνρ∂ν − i

k

Lγ∗γ

µρ

]Ψρk = 0 . (5.23)

Ex. 5.2 Apply ∂µ to the second equation of (5.23) to show that it already includes the first.Apply γµ to obtain γµΨµk = 0, and finally split γµνρ = γµγνρ + . . . (and similar for γµρ) anduse the previous results to derive [

/∂ + ik

Lγ∗

]Ψµk = 0 . (5.24)

As for the Dirac particles, we can get rid of the iγ∗ factor by a chiral redefinition Ψµk(x) ≡e−iγ∗

π4 Ψµ(x). Identifying again m = |k|/L (note that the sign of m or k can be changed by a

similar rotation as it was the case for the Dirac particle) we find that the field equation canbe written as

γµνρ∂νΨρ(x)−mγµνΨν(x) = 0 . (5.25)

We now interpret the m as an arbitrary mass. The last equation is the Euler-Lagrangeequation of the action

S =

∫dDx Ψµ [γµνρ∂ν −mγµρ] Ψρ . (5.26)

It is useful to recapitulate other equations that we obtained during the analysis (using thesame chiral rotation, or one can derive them again directly from (5.25)

γµΨµ = 0 ,

γµν∂µΨν = 0 , → ∂µΨµ = 0 ,[/∂ +m

]Ψµ = 0 . (5.27)

Let’s count the degrees of freedom by examining the constraints among initial data. Thetime derivatives are no independent Cauchy data as the field equation is of first order. One

34

can e.g. obtain ∂0Ψµ from the last equation of (5.27). The first equation of (5.27) is certainlya constraint, and can e.g. be used to determine Ψ0 in terms of the spacelike components.Another constraint on just the spacelike components and their space derivatives is the µ = 0part of the field equation (5.25).

Ex. 5.3 Show that this latter equation, which is

γij∂iΨj −mγiΨi = 0 , (5.28)

can be obtained also from using the solution for Ψ0 following from the first line in (5.27)into the second line.

Thus there are 2 × 2[D/2] constraints on the initial data for the complex field Ψµ(x), so thesystem contains (D − 2)2[D/2] independent classical degrees of freedom. These thus lead to12(D−2)2[D/2] on-shell degrees of freedom. For D = 4 there are 4 helicity states in agreement

with the 2s + 1 required for a particle with s = 3/2. In the Kaluza-Klein situation, thereis a massive gravitino with m = |k|/L for every Fourier mode k, each with 1

2(D − 2)2[D/2]

states. Note that this is the same as the number of states of a massless gravitino in D + 1dimensions.

Ex. 5.4 Study the Kaluza-Klein reduction for the Rarita-Schwinger field assuming periodic-ity Ψµ(x, y + 2π) = Ψµ(x, y) in y. Show that the spectrum seen in MinkowskiD consists of amassive gravitino for each Fourier mode k 6= 0 plus a massless gravitino and massless Diracparticle for the zero mode.

35

6 Differential Geometry

In this chapter we collect the ideas of differential geometry that are required to formulategeneral relativity and supergravity.

In general relativity spacetime is viewed as a differentiable manifold of dimension D ≥ 4with a metric of Lorentzian signature (−,+,+, · · ·+) indicating one time dimension andD − 1 space dimensions. We assume that readers of this book are not intimidated by theidea of D−4 hidden dimensions that are not directly observed. We will also need to considermanifolds of purely Euclidean signature (+,+, . . . ,+), which may appear in the hidden extradimensions and as the target space of nonlinear σ-models.

6.1 Scalars, vectors, tensors, etc.

Scalars, contravariant and covariant vector fields, and (mixed) tensors are defined by theirbehavior under coordinate transformations, namely

f ′(x′) = f(x) ,

V ′µ(x′) =∂x′µ

∂xνV ν(x) ,

ω′(x′)µ =∂xν

∂x′µων(x) ,

T ′µν (x′) =∂x′µ

∂xσ∂xρ

∂x′νT σρ (x) . (6.1)

Lie derivatives with respect to a vector field are defined as

LV f(x) = V µ(x)∂f

∂xµ,

LVUµ = V ρ∂ρUµ − (∂ρV

µ)Uρ ,

LV ωµ = V ρ∂ρωµ + (∂µVρ)ωρ ,

LV T µν = V ρ∂ρTµν − (∂ρV

µ)T ρν + (∂νVρ)T µρ . (6.2)

Ex. 6.1 Show explicitly that LVUµ, LV ωµ, and LV T µν defined in (6.2) do transform undercoordinate transformations as required by (6.1).

Let us consider the transformation properties of (6.1) for infinitesimal coordinate transfor-mations, namely those for which x′µ = xµ − ξµ(x). We work to first order in ξµ(x).

Ex. 6.2 Show to first order in ξµ(x) that the previous transformation rules can be expressedin terms of Lie derivatives as

δφ(x) ≡ φ′(x)− φ(x) = Lξφ ,δV µ(x) ≡ V ′µ(x)− V µ(x) = LξV µ ,

δωµ(x) ≡ ω′µ(x)− ωµ(x) = Lξωµ ,δT µν (x) ≡ T ′µν (x)− T µν (x) = LξT µν . (6.3)

36

6.2 The algebra and calculus of differential forms

We consider p-forms and their coordinates, according to the notation

ω(1) = ωµ(x)dxµ ,

ω(2) =1

2ωµν(x)dxµ ∧ dxν ,

...

ω(p) =1

p!ωµ1µ2···µpdxµ1 ∧ dxµ2 ∧ . . . dxµp . (6.4)

The exterior calculus is based on the exterior derivative, which maps p-forms into (p + 1)-forms as follows

dω(p) =1

p!∂µωµ1µ2···µpdxµ ∧ dxµ1 ∧ dxµ2 ∧ . . . dxµp . (6.5)

There is also an interior derivative, which maps p forms into (p − 1)-forms. The latterdepends on a vector V and is denoted as iV . It is defined as follows:

(iV ω(p))(V1, . . . , Vp−1) = ω(p)(V, V1, . . . , Vp−1) ,

(iV ω(p)) =

1

(p− 1)!V µωµµ1...µp−1dx

µ1 ∧ dxµ2 ∧ . . . dxµp−1 ,

iV (dxµ1 ∧ dxµ2 ∧ . . . dxµp) = V µ1dxµ2 ∧ . . . dxµp − V µ2dxµ1 ∧ dxµ3 . . . dxµp + . . . .(6.6)

Ex. 6.3 Show that these are equivalent definitions.

Ex. 6.4 Prove that also the internal derivative is nilpotent, i.e. iV iV = 0.

With the internal and external derivative, the Lie derivative, which we introduced in Sec.(6.1), has a simple expression on p-forms:

LV = diV + iV d . (6.7)

The formulas that we gave in (6.2) for e.g. LV ωµ are in this context the values of thecomponents7 of LV (ωµdxµ), i.e. they are (Lω)µ and not L(ωµ).

6.3 The metric and frame field on a manifold

The metric is described by the line element

ds2 = gµν(x)dxµdxν . (6.8)

7Hence we warn the reader that one should not use them in LV (ωµdxµ) = (LV ωµ)dxµ + ωµLV dxµ

and then use (6.2) for the first term. When one uses the definitions with forms ωµ(x) is a 0-form andLV ωµ = iV dωµ = V ν∂νωµ.

37

It is a non-degenerate metric, which means that det gµν 6= 0, so the inverse metric gµν(x)exists as a rank 2 symmetric contravariant tensor, which satisfies

gµρgρν = gνρgρµ = δµν . (6.9)

In order to describe fermions in general relativity, we use the frame field eaµ. In 4 di-mensions this quantity is commonly called the tetrad or vierbein. In general dimension theterm vielbein is frequently used, but we prefer the term frame field for reasons that shouldbecome clear as we discuss its properties. These fields are defined as the fields that locallytrivialize the metric. This means that

gµν(x) = eaµ(x)ηabebν(x) , (6.10)

where ηab = diag(−1, 1, . . . , 1) is the metric of flat D-dimensional Minkowski spacetime.Further, eaµ is a non-singular D×D matrix, with det eaµ =

√− det g. The inverse frame field

eµa(x) satisfies eaµeµb = δab and eµae

aν = δµν .

Ex. 6.5 Show thateµa = gµνηabe

bν , eνagµνe

νb = ηab . (6.11)

The last relation shows that the (inverse) frame field can be used to relate a general metricof signature −+ +..+ to to the Minkowski metric.

Given the metric gµν(x), the frame field eaµ(x) is not uniquely determined. Any local Lorentztransformation Λa

b(x), which leaves ηab invariant, produces an equally good frame field

e′aµ (x) = Λ−1 ab(x)ebµ(x) . (6.12)

We require that there is no preferred Lorentz frame, which means that the frame field andgeometrical quantities derived from it must be used in a way that is covariant with respect tothe local Lorentz transformation (6.12). The results (6.10),(6.11) indicate that the frame fieldeaµ transforms as a covariant vector under coordinate transformations, while eµa transformsas a contravariant vector, viz.

e′aµ (x′) =∂xρ

∂x′µeaρ(x) , e′µa (x′) =

∂x′µ

∂xρeρa(x) . (6.13)

The equations (6.12), (6.13) give the basic transformations associated with the frame field.Note that the position of Lorentz indices a, b, . . . has no special significance since they canbe freely raised and lowered using ηab and ηab. Thus we may use eaµ(x) = ηabe

bµ if needed.

38

6.4 Volume forms and integration

On a D-dimensional manifold, one may choose any top degree D-form ω(D) as a volume formand define the integral

I =

∫ω(D)

=1

D!

∫ωµ1···µD

(x)dxµ1 ∧ . . . ∧ dxµD

=

∫ω01···D−1dx

0dx1 . . . dxD−1 . (6.14)

Although there are many possible volume forms, there are two types that usually appear inthe context of physics. The first, which is the more specialized, occurs when the physicaltheory contains form fields. A D-form is formed by exterior products of these forms, andthe integration of that one defines the action.

The second type of volume form is far more common in physics and we call it the canonicalvolume form. There are several ways to introduce it, and we will use the frame field eaµ(x)and the basis of frame 1-forms ea for this purpose. As a preliminary we define the Levi-Civitaalternating symbol in local frame components

εa1a2···aD=

+1 a1a2 · · · aD an even permutation of 01 . . . (D − 1)−1 a1a2 · · · aD an odd permutation of 01 . . . (D − 1)

0 otherwise(6.15)

Under (proper) Lorentz transformations, i.e., det Λab = 1, this is an invariant tensor that

takes the same form in any Lorentz frame. As usual Lorentz indices are raised with ηab.Note that ε01···(D−1) = −1.

Note that the Levi-Civita symbol provides a useful formula for the determinant of anyD ×D matrix Aab, namely

detAεb1b2···bD = εa1a2···aDAa1

b1Aa2b2 . . . A

aDbD , (6.16)

and that there are systematic identities for the contraction of p of the D = p+ q indices, aswe saw in (3.11).

We use the frame fields to transform this to the coordinate basis, but insert factors e −1

and e = det eaµ, i.e.

εµ1µ2···µD= e −1εa1a2···aD

ea1µ1ea2µ2· · · eaD

µD,

εµ1µ2···µD = e εa1a2···aDeµ1a1eµ2a2· · · eµD

aD. (6.17)

Note that these definitions ensure that εµ1···µD and εµ1···µDtake the constant values on the

right side of (6.15), as can be seen using (6.16). The tensor εµ1µ2···µD is as such a tensordensity. However, it is important to recognize that εµ1µ2···µD can not be obtained from raisingthe indices of εµ1µ2···µD

in the usual way using the inverse of the metric. Therefore expressionslike εµ1···µp

µp+1···µDare not well defined. There is no such problem for εa1···ap

bp+1···bD .

39

Ex. 6.6 Prove, using (6.16) that both εµ1µ2···µD and εµ1µ2···µDtakes values ±1. This should

guarantee that they are invariant under infinitesimal changes of the frame field. Show alsodirectly that δεµ1µ2···µD = 0 for any δeµa using the formula for matrices that

δ detM = (detM) Tr(M−1δM) , (6.18)

and the Schouten identity, see (3.9).

With these preliminaries, the canonical volume form is defined as

dV ≡ e0 ∧ e1 ∧ . . . ∧ eD−1

=1

D!εa1···aD

ea1 ∧ · · · ∧ eaD

=1

D!e εµ1···µD

dxµ1 ∧ . . . ∧ dxµD

= e dx0 . . . dxD−1

= dDx√− det g . (6.19)

Note that the determinant of the frame field eaµ appears in a natural fashion. In the last linewe give the abbreviated notation we will use in most applications. For example, given theLagrangian of a system of fields, such as the kinetic Lagrangian L = 1

2gµν∂µφ∂νφ of a scalar

field, the action integral is written as

S =

∫dV L =

∫dDx

√− det gL . (6.20)

6.5 Hodge duality of forms

The dimension of the component spaces of p- and q-forms with p + q = D is the same, sothat it is possible to define a 1 : 1 map between them. This map is the Hodge duality mapfrom Λp(M) → Λq(M), which is quite useful in the physics of supergravity. The map isdenoted by Ω(q) =∗ ω(p).

Since the map is linear we can define it on a basis of p-forms and then extend to a generalform. It is convenient to use the local frame basis initially and define

∗ea1 ∧ . . . eap =1

q!eb1 ∧ . . . ebqεb1···bqa1···ap . (6.21)

A general p-form can be expressed in this basis, and we can proceed to define its dual via

Ω(q) =∗ ω(p) = ∗(1

p!ωa1···ape

a1 ∧ . . . eap)

=1

p!ωa1···ap

∗ea1 ∧ . . . eap . (6.22)

40

Ex. 6.7 Show that the frame components of Ω(q) are given by

Ωb1···bq = (∗ω)b1···bq =1

p!εb1···bq

a1···apωa1···ap . (6.23)

These formulas are far less complicated than they look since there is only one independentterm in each sum. For example, for D = 4 the dual of a 3-form is a 1-form. For basiselements we have ∗e1 ∧ e2 ∧ e3 = e0 and ∗e0 ∧ e1 ∧ e2 = e3. For components, (∗ω)0 = ω123 and(∗ω)3 = ω012.

The duality has an important involutive property, which can be inferred from the follow-ing sequence of operations on basis elements:

∗(∗ea1 ∧ . . . eap) =1

q!

∗eb1 ∧ . . . ebq εb1···bqa1···ap

=1

p!q!ec1 ∧ . . . ecpεc1···cpb1···bqεb1···bqa1···ap

= −(−)pqec1 ∧ . . . ecpδa1···apc1···cp

= −(−)pqea1 ∧ . . . eap . (6.24)

This leads to the general relation ∗(∗ω(p)) = −(−)pqω(p). This is the correct relation for aLorentzian signature manifold. For Euclidean signature the involution property is ∗(∗ω(p)) =(−)pqω(p).

For even dimension D = 2m, it is possible to impose the constraint of self-duality (oranti-self-duality) on forms of degree m, i.e. Ω(m) = ±∗Ω(m). In a given dimension thiscondition is consistent only if duality is a strict involution, i.e. −(−)m

2= −(−)m = +1

for Lorentzian signature and (−)m = +1 for Euclidean signature. Thus we have self-dualYang-Mills instantons in 4 Euclidean dimensions and a self-dual F (5) is possible in D = 10and indeed appears in Type IIB Lorentzian signature supergravity as we will see later.

The duality relations defined above in a frame basis are easily transformed to a coordinatebasis using the relations ea = eaµ(x)dxµ and dxµ = eµa(x)ea. For coordinate basis elementsthe duality map is

∗(dxµ1 ∧ . . . dxµp) =1

q!e gµ1ρ1 . . . gµpρpdxν1 ∧ . . . dxνpεν1···νqρ1···ρp . (6.25)

For anti-symmetric tensor components, we have

(∗ω)µ1···µq =1

p!e εµ1···µqρ1···ρpg

ν1ρ1 . . . gνpρpων1···νp . (6.26)

Following the discussion in Sec. 6.4, we may take as a volume form on M the wedgeproduct ∗ω(p) ∧ ω(p) of any p form and its Hodge dual. The integral of this volume form issimply the standard invariant norm of the tensor components of ω(p), i.e.∫

∗ω(p) ∧ ω(p) =1

p!

∫dDx

√−g ωµ1···µpωµ1···µp . (6.27)

41

Ex. 6.8 Prove (6.27). Use the definitions above and those in Sec. 6.4 and the fact

ea1 ∧ . . . ∧ eaq ∧ eb1 ∧ . . . ∧ ebp = −εa1···aqb1···bpdV , (6.28)

where dV is the canonical volume element of (6.19).

Ex. 6.9 Compare these definitions with Sec. 4.1.2, to obtain

Fµν = −i (∗F )µν . (6.29)

See that the factor i ensures that the tilde operation squares to the identity. Self-duality isthen possible for complex 2-forms.

6.6 p-form gauge fields

With the terminology of the previous section, we can rewrite the simplest kinetic actions ofscalars and gauge vectors in the following way

S0 = −12

∫∗F (1) ∧ F (1) , F (1) = dφ ,

S1 = −12

∫∗F (2) ∧ F (2) , F (2) = dA(1) . (6.30)

In each case we have a Bianchi identity: dF (1) = 0 and dF (2) = 0, whose solution is that thesefield strengths can be written as the differential of a lower form. For the form A(1), whichdescribes the photon, there is a gauge transformation that can be written as δA(1) = dΛ(0).

This suggests that there is a generalization. When we call the previous actions thetheories for a 0-form and a 1-form, we can describe a p-form in terms of its field strength as

Sp = −12

∫∗F (p+1) ∧ F (p+1) , F (p+1) = dA(p) . (6.31)

Ex. 6.10 Translate this in components and find

Sp = − 1

2 (p+ 1)!

∫dDx

√−g F µ1···µp+1Fµ1···µp+1 , Fµ1···µp+1 = (p+ 1)∂[µ1Aµ2...µp+1] .

(6.32)

Again there is a gauge transformation δA(p) = dΛ(p−1), as such transformations of the p-form gauge field leave F (p+1) invariant. The number of independent components in Λ(p−1)

is(Dp−1

). However, not all the components of Λ(p−1) are independent symmetries, as all

transformations where Λ(p−1) = dΛ′ (p−2) are not effective transformations of the gauge field.But not all components of the latter are to be subtracted from the gauge symmetries. Indeed,

42

when Λ′ (p−2) = dΛ′′ (p−3), these are not zero modes. In this way, we can count the remaininggauge-invariant components of a p form as(

D

p

)−

(D

p− 1

)+

(D

p− 2

)− . . . =

(D − 1

p

), (6.33)

the number of components of an p-form in D− 1 dimensions. We thus find that the off-shelldegrees of freedom (field variables minus symmetries) form a representation of SO(D − 1),as we saw before for other fields. If furthermore the field equations of massless fields areused, the independent components form representations of SO(D − 2). This is the on-shellcounting of degrees of freedom. Thus the number of physical degrees of freedom is

(D−2p

).

This suggests that a p-form has the same degrees of freedom as a (D− 2− p)-form. Indeed,we will now show that not only the numbers match, but that the physical content is thesame.

One further indication in this direction is that the field equation of the action (6.31) is

d∗F (p+1) = 0 , (6.34)

while the Bianchi identity that says that F (p+1) is an exact form is

dF (p+1) = 0 . (6.35)

When one defines a dual form G(D−p−1 = ∗F (p+1), then these two equations have a similarform, and dG(D−p−1 = 0 can then be interpreted as the statement that it is the field strengthof a (D − p− 2)-form.

We can easily prove this duality by the following steps. First, we rewrite the action (6.31)as

Sp = −∫ [

12∗F (p+1) ∧ F (p+1) + b(D−p−2) ∧ dF (p+1)

]. (6.36)

The latter term is a Lagrange multiplier term where the field equation of b(D−p−2) impliesthat F (p+1) is the field strength of a p-form. If we include that term, we can consider F (p+1)

and b(D−p−2) as the independent fields. Therefore, we can first consider the field equationfor F (p+1), which states that

∗F (p+1) = (−)D−p db(D−p−2) . (6.37)

As this is an algebraic field equation, it can be used in the action, which leads to the actionof the type (6.31), where b(D−p−2) takes the role of A(p). This proves the equivalence of ap-form with a (D − 2− p)-form.

The simplest case are the gauge antisymmetric tensors (2-forms) in 4 dimensions. Thestatement is thus that the simplest kinetic actions for such fields are equivalent to actions forscalars. However, the arguments above are only true for the simplest actions, i.e. with abeliangauge fields. In non-abelian field theories, antisymmetric tensors can lead to non-equivalenttheories.

The duality transformations for gauge vectors in 4-dimensions is a self-duality. A 1-formis dual to another 1-form. In fact, it transforms electric in magnetic components, and thisis the duality that we discussed in Sec. 4.1.3.

43

6.7 Connections and covariant derivatives

A covariant derivative on a manifold is a rule to differentiate a tensor of type8 (p, q) producinga tensor of type (p, q+ 1). It is well known that one needs to introduce the affine connectionΓρµν(x) to accomplish this. On vector fields the covariant derivative is defined by

∇µVρ = ∂µV

ρ + ΓρµνVν ,

∇µVν = ∂νVν − ΓρµνVρ , (6.38)

and it is straightforward to extend this definition to tensors. Most discussions of the affineconnection in the physics literature are based on the idea of parallel transport and use acoordinate basis.

Since supergravity requires the frame field eaµ(x), and we have used the frame 1-forms ea

extensively, it is natural to introduce the affine connection in this framework and then makecontact with the more common treatment. In frames the affine connection is specified by the1-forms ωab = ωµ

ab(x)dxµ. The guiding principle of the discussion is the covariance underlocal Lorentz transformations (6.12) of the frame. The ωµ

ab(x) are the gauge fields for thesetransformations. They are usually called the spin connection because they are essential inthe description of spinors on manifolds. We will use the terms spin connection or frameconnection for ωµ

ab(x) or ωab.

6.7.1 The spin connection ωµab

Vectors with such local indices V a, Ua and tensors Tab transform as

V ′a(x) = Λ−1 ab(x)V b(x) ,

U ′a(x) = Ub(x)Λba(x) ,

T ′ab(x) = Tcd(x)Λca(x)Λd

b(x) , (6.39)

respectively. The spin connection is a field that transforms as

ω′µab = Λ−1 a

c ∂µΛcb + Λ−1 a

c ωµcd Λd

b . (6.40)

These are exactly the gauge transformation properties of a Yang-Mills potential for the groupO(D − 1, 1). We use the spin connection to define local Lorentz covariant derivatives as

DµVa = ∂µV

a + ωµabV

b ,

DµUa = ∂µUa − Ubωµba ,

DµTab = ∂µTab − Tcbωµca − Tacωµ

cb . (6.41)

Let’s apply (6.41) to the Lorentz metric tensor ηab. We have

Dµηab = −ηcbωµca − ηacωµcb = −ωµba − ωµab = 0 . (6.42)

8Type (p, q) means p upper and q lower indices.

44

This shows that the spin-connection (with indices at equal level) should be antisymmetric,related to the fact that it gauges orthogonal transformations.

An important quantity is the torsion two-form that is the antisymmetric part of Dµeaν ,

or in form language:T a ≡ dea + ωab ∧ eb . (6.43)

In most applications of differential geometry to gravity, the torsion vanishes, and one dealswith a torsion-free, metric-preserving connection. This is also called the Levi-Civita con-nection, for which the structure equation reads dea + ωab ∧ eb = 0. However non-vanishingtorsion can arise from coupling of gravity to certain matter fields, and this does occur insupergravity.

If the torsion is required to vanish, we can solve that equation for the components of thespin-connection. This equation has indeed as many components as ωµ

ab one finds

ωµab(e) = 2eν[a∂[µeν]

b] − eν[aeb]σeµc∂νeσc . (6.44)

The difference in general is called the contorsion tensor Kµab, and we find

ωµab = ωµ

ab(e) +Kµab ,

Kµ[νρ] = eνaeρbKµab = −1

2(T[µν]ρ − T[νρ]µ + T[ρµ]ν) , T[µν]ρ = T[µν]

aeρa . (6.45)

We now bring our discussion of Lorentz covariant derivatives of fields in the frame ba-sis to a close by discussing a more general viewpoint. In general one can consider fieldsthat transform in any finite dimensional matrix representation of O(D − 1, 1). The matrixrepresentative of a proper Lorentz transformation takes the form

Λ(x) = exp

(1

2λab(x)mab

). (6.46)

The anti-symmetric array λab(x) contains the boost and rotation parameters of the transfor-mation. The 1

2D(D−1) matrices mab are a representation of the Lie algebra so(D−1, 1), as

we saw e.g. in (2.14), and therefore satisfy the commutation relations (2.16) (in both theseequations the µ, ν, . . . indices are now replaced by a, b, . . . indices). Row and column indicesof the representation are suppressed in the equations in this paragraph. If Φ(x) is a fieldwhose frame components transform in an n-dimensional representation, then it transformsunder Lorentz transformations as

Φ′(x) = Λ−1(x)Φ(x) , (6.47)

and its covariant derivative is defined as

DµΦ(x) =

(∂µ +

1

2ωµ

abmab

)Φ(x) . (6.48)

We are particularly interested in two representations, The first is the defining vectorrepresentation (D-dimensional) in which the generators are given by (2.14). In this case the

45

covariant derivative of a frame vector V (x) computed using (6.48) and (2.14) is the same asin (6.41).

The second representation is the Dirac (or Majorana) spinor representation, which isvery important for supergravity. As discussed in Ch. 3, it has dimension 2[D/2], and thegenerators are the second rank Clifford matrices 1

2γab. In a gravitational theory, spinors

must be described through their local frame components. The local Lorentz transformationrule and covariant derivative of a spinor field are

Ψ′(x) = exp(−1

4λab(x)γab

)Ψ(x) ,

DµΨ(x) =(∂µ + 1

4ωµ

ab(x)γab)

Ψ(x) . (6.49)

6.7.2 The affine connection Γρµν

Our next task is to transform Lorentz covariant derivatives of vector and tensor frame fieldsto the coordinate basis where they become covariant derivatives with respect to generalcoordinate transformations and take the familiar form in (6.38), but include torsion. Itshould be emphasized that no new structure on the manifold is required. We simply reexpressthe information in the spin connection ωµ

ab in a coordinate basis where it is contained in

the affine connection Γρµν .Combining the connection formula (6.38) for a coordinate vector and (6.41) for a frame

vector, we define the full covariant derivative on the vierbein as

∇µeaν = ∂µe

aν + ωµ

abebν − Γσµνe

aσ . (6.50)

The vielbein postulate states that the vielbein is constant under this complete connection:

∇µeaν = ∂µe

aν + ωµ

abebν − Γσµνe

aσ = 0 . (6.51)

This allows then to obtain the covariant derivatives of coordinate vectors in terms of theLorentz covariant derivative:

∇µVρ ≡ ∇µ (eρaV

a) = eρaDµVa

= ∂µVρ + ΓρµνV

ν . (6.52)

In fact, the vielbein postulate can be seen as a definition of the affine connection in termsof the spin connection:

Γρµν = eρa(∂µeaν + ωµ

abebν) . (6.53)

The vielbein postulate leads to the more familiar metric postulate

∇µgνρ ≡ ∂µgνρ − Γσµνgσρ − Γσµρgνσ = 0 , (6.54)

The metric postulate means that the metric tensor is covariantly constant. Hence lengths andscalar products of vectors are preserved under parallel transport, and covariant differentiationcommutes with index raising, e.g. ∇µV

ρ = gρν∇µVν .

46

Last, but hardly least, we substitute the explicit form (6.45) of the spin connection inthe definition (6.53). After some calculation we obtain the explicit formula for Γρµν

Γρµν = Γρµν(g)−Kµνρ , (6.55)

Γρµν(g) =1

2gρσ(∂µgσν + ∂νgµσ − ∂σgµν) . (6.56)

(Remember that K is antisymmetric in the last two indices, i.e. Kµνρ = −Kµ

ρν). The

first term, written in detail in (6.56), is the torsion-free connection, frequently called theChristoffel symbol and denoted by ρµν instead of our Γρµν(g). It is well known that it is theunique symmetric affine connection that satisfies (6.54). When torsion is present the affineconnection is not symmetric, rather

Γρµν − Γρνµ = −Kµνρ +Kνµ

ρ = Tµνρ . (6.57)

For ‘mixed’ quantities with both coordinate and frame indices, it is useful to distinguishbetween local Lorentz and coordinate covariant derivatives. Thus, for a vector-spinor fieldΨµ(x), we define both

DµΨν ≡(∂µ + 1

4ωµabγ

ab)

Ψν ,

∇µΨν = DµΨν − ΓρµνΨρ . (6.58)

The first transforms as a local Lorentz spinor, but not a tensor, and the second transformsas a spinor and type (0, 2) tensor. After anti-symmetrization in µν, both derivatives yield(0, 2) tensors, but they differ by a torsion term, viz.

∇µΨν −∇νΨµ = DµΨν −DνΨµ − TµνρΨρ . (6.59)

6.7.3 Partial integration

The covariant derivatives without torsion are convenient to make integration by parts. Thekey relation on the Christoffel connection is

∂µ√−g =

√−g Γρρµ(g) . (6.60)

It implies, using the definitions (6.38) and (6.55),∫dDx

√−g∇µV

µ =

∫dDx ∂µ

(√−g V µ

)−

∫dDx

√−g Kνµ

νV µ . (6.61)

The first term is the total derivative that we often neglect in field theory due to the assump-tion of vanishing fields at infinity. The second term shows the violation of the manipulationsof integration by parts in case of torsion. It is proportional to

Kνµν = −Tνµν . (6.62)

47

6.8 The curvature tensor

In Sec. 6.7.1 we discussed the fact that the spin connection ωµab transforms as a Yang-Millsgauge potential for the group O(D − 1, 1), see (6.40). It then follows that the quantity

Rµνab ≡ ∂µωνab − ∂νωµab + ωµacωνcb − ωνacωµ

cb (6.63)

has the properties of a Yang-Mills field strength; transforming as a type (0, 2) Lorentz tensorunder local Lorentz transformations. See (6.41). Because of anti-symmetry in µν, it is alsoa (0,2) tensor under general coordinate transformation, called the curvature tensor. Thuswe can define the curvature 2-form

ρab =1

2Rµν

ab(x)dxµ ∧ dxν . (6.64)

Using (6.63) it is easy to see that the curvature 2-form is related to the connection 1-formby

dωab + ωac ∧ ωcb = ρab . (6.65)

It is this equation that is known as the second Cartan structure equation. Needless to say,it is the metric, curvature , and (if present) the torsion tensor, which carry the basic localinformation about a space-time manifold that are needed for gravitational physics.

Ex. 6.11 Consider the effect on Rµνab of an infinitesimal variation of the spin connection.Show that

δRµνab = Dµδωνab −Dνδωµab . (6.66)

One immediate application of the Cartan structure equations (6.43),(6.65) is to derivethe Bianchi identities for the curvature tensor.

Ex. 6.12 Apply the exterior derivative to (6.43)and (6.65) and obtain, respectively the 3-form relations

ρab ∧ eb = dT a + ωab ∧ Tb ,dρab + ωac ∧ ρcb + ρac ∧ ωcb = 0 . (6.67)

Show that the components of these 3-form equations read, using Rµνρa = Rµνb

aebρ,

Rµνρa +Rνρµ

a +Rρµνa = −DµTνρ

a −DνTρµa −DρTµν

a ,

DµRνρab +DνRρµ

ab +DρRµνab = 0 . (6.68)

The derivatives Dµ etc. in these equations are Lorentz covariant derivatives and containonly the spin connection. However, each relation is a type (0, 3) coordinate tensor becauseof anti-symmetry. The first relation is the conventional first Bianchi identity for the curvaturetensor, but corrected by torsion terms on the right side. This has no analogue in Yang-Millstheory. The second relation is the usual Bianchi identity of non-abelian gauge theory, andis called the second Bianchi identity for the curvature.

48

The commutator of Lorentz covariant derivatives leads to important relations, in bothgauge theory and gravity, which we call Ricci identities. We write the identity for a field Φ(x)transforming in a representation of the proper Lorentz group with generators Mab togetherwith special cases of interest, namely frame vector fields V a(x) and spinor fields Ψ(x),

[Dµ, Dν ]Φ = 12RµνabM

abΦ ,

[Dµ, Dν ]Va = Rµν

abV

b ,

[Dµ, Dν ]Ψ = 14Rµνabγ

abΨ . (6.69)

There are also generalized Ricci identities for the commutator of coordinate covariantderivatives, but they contain torsion terms. For example, for vector fields V ρ(x), one canderive by direct computation

[∇µ,∇ν ]Vρ = Rµν

ρσV

σ − Tµνσ∇σV

ρ , (6.70)

in which the curvature tensor appears as

Rµνρσ = ∂µΓρνσ − ∂νΓ

ρµσ + ΓρµτΓ

τνσ − ΓρντΓ

τµσ . (6.71)

Ex. 6.13 Verify (6.70). Show that Rµνρσ = Rµνabe

aρebσ by evaluating [∇µ,∇ν ]eρa = 0. Prove

the generalized second Bianchi identity

∇µRνρστ +∇νRρµ

στ +∇ρRµνστ = Tµν

ξRξρστ + Tνρ

ξRξµστ + Tρµ

ξRξνστ . (6.72)

Ex. 6.14 DeriveδRµν

ρσ = ∇µδΓ

ρνσ −∇νδΓ

ρµσ , (6.73)

which is closely related to (6.66)

The Ricci tensor is defined as Rµν = Rµσνσ and the curvature scalar is R = gµνRµν .

Ex. 6.15 Show that Rµν = Rνµ if and only if there is no torsion. Show that Rµνρσ = Rρσµν

if and only if the torsion vanishes.

Below is an exercise designed to show that the Hilbert action for general relativity,S ∼

∫dDx

√−gR can be written as the integral of a volume form involving the frame and

curvature forms. It is this form of the action that reveals how the mathematical frameworkof connections with torsion is realized in the physical setting of gravity coupled to fermions.We will explore this in the next chapter, but the exercise brings together some of the ideasdiscussed in this chapter.

Ex. 6.16 Show that

1

(D − 2)!

∫εabc1...c(D−2)

ec1 ∧ · · · ∧ ec(D−2) ∧ ρab =

∫dDx

√−gR . (6.74)

Hint: express ρab = 12Rcd

abec ∧ ed and use (6.28) and (6.19).

49

6.9 The nonlinear σ-model

The principal application of differential geometry is to theories of gravity and supergravityin spacetime, viewed as a differentiable manifold. The matter couplings in these theories,including fermions and p-form gauge fields also require the ideas of differential geometryreviewed in this chapter. There is still another physical application of differential geometrythat is important in supergravity. This application involves the dynamics of scalar fieldsin flat spacetime. Physicists usually call it the nonlinear σ-model because it first appearedas a description of the low-energy interactions of the triplet of ~π mesons which involve thecomposite field σ = (f 2

π+~π2)1/2. The notation ~π indicates that the three pion fields transformin the fundamental representation of an SO(3) internal symmetry. In mathematics the sametype of field theory may be called the theory of harmonic maps.

To begin the discussion let us consider an n-dimensional Riemannian manifold Mn. Thename Riemannian means that Mn has a smooth Euclidean signature metric. In local coor-dinates called φi, i = 1, 2 . . . , n, the metric tensor is gij(φ). In the nonlinear σ-model thecoordinates are fields φ(x) in which the xµ, µ = 1, 2, . . . D are the Cartesian coordinatesof a flat spacetime MD. We assume that MD is of Minkowski signature, but our discussionrequires little or no change for Euclidean signature. Thus we deal with maps from spacetimeto the internal space or target space Mn, see Fig. 1.

MD

Mn

Figure 1: Scalar fields as maps from spacetime to the target manifold.

We postulate that the dynamics of these maps is governed by the action

S[φ] = −1

2

∫dDx gij(φ)ηµν∂µφ

i∂νφj . (6.75)

Spacetime indices are raised and lowered with the Minkowski metric ηµν .

Ex. 6.17 Compute the Euler variation of this action. Observe that the Christoffel connec-tion (6.56) is obtained in the process and express the equation of motion in the form:

2φi + Γijk(φ)∂µφi∂µφj = 0 . (6.76)

50

For Euclidean signature the d’Alembertian 2 is replaced by the Laplacian ∇2. This is whythe solutions are called harmonic maps; they are a nonlinear generalization of harmonicfunctions. Indeed the conventional Laplace equation is obtained in the special case when thetarget space is also flat.

Suppose that we use a different set of coordinates φ′i(φ) on the target space Mn. In thesecoordinates the metric tensor becomes

g′ij(φ′) =

∂φk

∂φ′igk`(φ)

∂φ`

∂φ′j, (6.77)

and the action changes to

S ′[φ′] = −1

2

∫dDx g′ij(φ

′)ηµν∂µφ′i∂νφ

′j . (6.78)

This means that the equation of motion for φ′i(x) has the same structure form as in (6.76),but with metric and connection expressed in the new coordinates. Our viewpoint is thatsignificant physical information may be expressed in either set of coordinates.

6.10 Symmetries and Killing vectors

It is frequently and correctly said that the equations of a gravitational theory are invariantunder general coordinate transformations, and the same is true for the equations of thenonlinear σ-model. What this really means, however, is that the equations are constructedfrom the same elements, vectors, tensors, covariant derivatives, etc. in any coordinate system.This has some content, but it is only a special class of coordinate transformations which leadto symmetries of a system. As discussed in Ch. 2, a symmetry is a transformation that takesone solution of the equations of motion into another. The equations themselves must takethe same form in two coordinate transformations related by a symmetry. For each globalsymmetry there is a conserved Noether current.

Although the situation is similar for both gravitation and the nonlinear σ-model, weprefer to discuss symmetries in the latter setting. The equations of motion (6.76) take thesame form in the two sets of coordinates φi and φ′i if the metric g′ij in (6.77) has the samefunctional form as the original metric, i.e. if

g′ij(φ) = gij(φ) . (6.79)

A symmetry transformation of the metric is frequently called an isometry.We now assume that the coordinate transformation is continuous. This means that the

functional relation φ′i(φ) depends continuously on parameters θA. In the limit of small θA

it is assumed that the transformation is close to the identity and thus approximated byφ′i = φi + θAkiA(φ). The kiA(φ) are vector fields, which are the generators of the coordinatetransformation. Thus we have an infinitesimal coordinate transformation, whose effect on

51

the various tensor fields on the target space is given by Lie derivatives as defined in (6.3).In particular, the condition (6.79) for a symmetry holds to first order in θA if

δgij ≡ g′ij(φ)− gij(φ) = −θALkAgij = 0 . (6.80)

This means that each kiA is a vector field that satisfies

∇ikjA +∇jkiA = 0 , kiA = gijkjA , ∇ikjA = ∂ikjA − Γkij(g)kkA , (6.81)

where ∂i is the ordinary derivative with respect to φi and Γkij(g) is the Christoffel connectionon Mn (we are not dealing with torsion here). A vector field with this property is called aKilling vector, and each Killing vector is associated with a symmetry. To summarize; foreach continuous isometry of the target space metric there is a Killing vector kiA.

The Killing symmetries are infinitesimal variations of the fields φi(x) of the form

δ(θ)φi = θAkiA(φ) . (6.82)

As opposed to the symmetries discussed in Ch. 2, the Killing symmetries are frequentlynonlinear in the fields. It is easy to see that this transformation leaves the σ-model actioninvariant. To first order in θA, the transformation of (6.75) is9

δS[φ] = S[φ′]− S[φ]

=

∫dDx θA

[∂µgij∂φk

kkA ∂µφi ∂µφj + gij(φ)

∂kiA∂φk

∂µφk ∂µφj + gij(φ) ∂µφ

i ∂kjA

∂φk∂µφ

k

]=

∫dDx θA

[(∇ikjA +∇jkiA)∂µφi∂µφ

j]

= 0 . (6.83)

As described in Sec.6.1 it is frequently useful to encode a Killing vector in a differentialoperator

kA ≡ kjA∂

∂φj. (6.84)

One can then define the action of the symmetry transformation of any scalar function f(φ)on the target space as δf ≡ θAkAf . The elementary variation (6.82) is then the special casewhen f(φ) = φi.

The isometry group of a given metric on Mn is frequently non-abelian. Then there areseveral linearly independent Killing vectors. As in Sec.6.1, the Lie algebra is determinedfrom the commutators of the differential operators, i.e.

[kA , kB] = fABCkC = fAB

CkiC∂

∂φi. (6.85)

The vector field kiC is also a Killing vector. With the transformation parameters included,one finds that [

θA1 kA , θB2 kB

]= θC3 kC , θC3 = θB2 θ

A1 fAB

C . (6.86)

9The symmetry acts on the dynamical variables φi of the nonlinear σ-model, and the transformation ofgij(φ) is induced by the variations δφi.

52

Thus nonlinear Killing symmetries compose in the same way as the linear internal symmetriesin (2.9). Although Killing vectors kiA(φ) are generically nonlinear functions of the φk, theisometry group of a manifold Mn can include linearly realized matrix symmetries, which actin the same way as the internal symmetries of Sec. 2.1. In this case the Killing vectors kiAare related to the matrix generators tA by kiA(φ) = −(tA)ijφ

j and the differential operatorsby

kA = −(tA)ijφj ∂

∂φi(6.87)

Ex. 6.18 Check that for products of Killing vectors of the form (6.87) we have

kAkB = (tBtA)ijφj ∂

∂φi+ a term symmetric in A↔ B . (6.88)

This interchange of the matrices is the underlying reason for introducing − signs in thedefinitions of transformations defined by matrices already started in (2.4).

Finally we discuss how to recognize when a given metric has symmetries and how toobtain the Killing vectors. One simple situation occurs when the metric tensor gij does notdepend on a particular coordinate, say φk. Then ‘translation’ φk → φk + c is an isometry,and kk = ∂/∂φk is the differential operator which generates the symmetry. In general, itmay not be easy to examine a metric tensor and determine its isometries, Indeed there aretarget space metrics that have important applications in supergravity and string theory andhave no continuous isometries.10

As an example that will be relevant later in the book, we discuss the Poincare plane.This is the non-compact two-dimensional manifold with coordinates X, Y with Y > 0. TheRiemannian metric is defined by the line element

ds2 =dX2 + dY 2

Y 2. (6.89)

The isometry group of this metric is the Lie group S`(2,R). Its applications in supergravityand string theory are fundamental, so it is worthwhile to study. The group may be definedas the group of 2× 2 real matrices of the form(

a bc d

), ad− bc = 1 . (6.90)

First we will study the action of finite group transformations and then identify three Killingvectors as infinitesimal symmetries. For this purpose it is very convenient to use a complexcoordinate Z = X + iY on the upper half-plane. The line element (6.89) becomes

ds2 =dZdZ

Y 2. (6.91)

10For interested readers we note that compact Calabi-Yau metrics have no isometries.

53

S`(2,R) transformations act as nonlinear maps

Z → Z ′ =aZ + b

cZ + d= X ′ + iY ′ (6.92)

of the upper half-plane onto itself. Up to some trivial redefinitions these are the transforma-tions mentioned in (4.29) for a scalar field that can appear in the kinetic terms of a Maxwellgauge field. It is a matter of straightforward algebra to show that

X ′ =ac(X2 + Y 2) + (ad+ bc)X + bd

|cZ + d|2,

Y ′ =Y

|cZ + d|2,

dZ ′ =dZ

(cZ + d)2. (6.93)

Then, by direct substitution we find that the line element behaves as

ds2 =dZdZ

Y 2=

dZ ′dZ ′

Y ′2, (6.94)

corresponding to a Lagrangian

L =∂µZ∂

µZ

(Z − Z)2. (6.95)

It has the same form in both coordinate systems, so finite S`(2,R) transformations are indeedisometries.

Ex. 6.19 The straightforward algebra is highly recommend. Note that Y ′ is positive when-ever Y is positive. This shows that the transformation maps the upper half-plane ito itself.It would fail for complex a, b, c, d.

We could obtain the Killing vectors by expanding (6.92) around the unit transformations.However, as an illustrative example, in the following exercise the student is invited to startagain from the metric (i.e. the Lagrangian) and derive the Killing vectors systematically.

Ex. 6.20 Consider Z and Z as the independent fields, rather than X an d Y . The metriccomponents are then

gZZ = gZZ = 0 , gZZ = gZZ =1

(Z − Z)2. (6.96)

Prove that the non-vanishing components of the Christoffel connection are ΓZZZ and its com-plex conjugate ΓZ

ZZ. Calculate them and show that there are three Killing vectors (or sym-

metries):kZ1 = 1 , kZ2 = Z , kZ3 = Z2 , (6.97)

each with conjugate components kZA. Show that their Lie brackets give a Lie algebra whosenon-vanishing structure constants are

f121 = 1 , f13

2 = 2 , f233 = 1 . (6.98)

This is the algebra su(1, 1) = so(2, 1) = sl(2).

54

7 The first and second order formulations of general

relativity

In this chapter we discuss and compare the first and second order formulations of gravita-tional dynamics. The most familiar setup is the second order formalism in which the metrictensor or the frame field is the dynamical variable describing gravity. If fermions are presentone must use the frame field. The curvature tensor and covariant derivatives are constructedfrom the torsion-free connections Γρµν(g) and ωµab(e), see (6.44),(6.55). The name ‘secondorder’ refers to the fact that the gravitational field equation is second order in derivatives ofgµν or eaµ.

In the first order (or Palatini) formalism one starts with an action in which eaµ and ωµabare independent variables, and the Euler-Lagrange equations are first order in derivatives.Without matter, the solution of the ωµab field equation simply sets ωµab = ωµab(g). Whenthis result is substituted in the field equation for eaµ, one finds the conventional Einsteinequations, exactly as they emerge in the second order formulation for frame fields. Whengravity is coupled to spinor fields, the ωµab field equation contains terms bilinear in thespinors, and its solution is ωµab = ωµab(g) + Kµab, with contortion tensor determined as abilinear expression in the spinor fields. It is in this way that the mathematical formalism ofconnections with torsion is realized in physics.

When the result ωµab = ωµab(e)+Kµab is substituted in the other field equations, one findsthat the full effects of the torsion could have been obtained in the second order formalismwith an added set of quartic fermion terms in the action. One may then ask why bother?Why introduce the complication of torsion when its physical effects can be described moreconventionally? The answer is that the proof that a supergravity theory is invariant underlocal supersymmetry transformations is greatly simplified by the fact that quartic terms inthe gravitino field can be organized within the connection ωµab = ωµab(e) +Kµab.

7.1 Second order formalism for gravity and bosonic matter

In this section we review the conventional treatment of gravity coupled to bosonic matter. Tobe definite we consider the simplest and most common matter fields, a real scalar field φ(x)and an abelian gauge field with gauge potential Aµ(x) and field strength Fµν = ∂µAν−∂νAµ.The action functional of the coupled system is the sum of the Hilbert action for gravity plusthe action for matter fields:

S =

∫dDx

√− det g

(1

2κ2gµνRµν(g) + L

),

L = −1

2gµν∂µφ∂νφ−

1

4gµρgνσFµνFρσ . (7.1)

The constant κ2 = 8πGN is the gravitational coupling constant. The Ricci tensor Rµν(g)contains the torsion-free connection.

Ex. 7.1 Prove that κ2 should have dimension of a length to the power (D− 2) in order thatthe action be dimensionless.

55

The matter field Lagrangian was obtained from the Minkowski space kinetic Lagrangiansdiscussed in Chapters 2 and 4 by the minimal coupling prescription. This consists of therules:i. replace the Minkowski metric ηµν by the spacetime metric tensor gµν(x).ii. replace each derivative ∂µ by the appropriate covariant derivative ∇µ with connectionΓρµν(g).iii. Use the canonical volume form (6.19).These rules incorporate the equivalence principle of general relativity and the principle ofgeneral covariance. The Lagrangian transforms as a scalar under coordinate transformations,so the matter action is invariant. The second rule is not really needed for the scalar and vectorgauge fields of our matter system. For scalars, no connection is needed since ∂µφ = ∇µφ.The same is true for the gauge field, since Fµν can be written as Fµν = ∇µAν −∇νAµ. TheChristoffel connection cancels by symmetry.

7.2 Gravitational fluctuations of flat spacetime

In the absence of matter, the gravitational field satisfies the equation

Rµν = 0 . (7.2)

One solution is flat Minkowski spacetime with metric tensor gµν = ηµν . In this section westudy weak gravitational perturbations of Minkowski space, perturbations that are describedby metrics of the form

gµν(x) = ηµν + κhµν(x). (7.3)

We will work systematically to first order in the gravitational coupling constant κ. Thesymmetric tensor fluctuation hµν(x) is a gauge field. We will count the physical degrees offreedom and determine its propagator, as we have done for the free vector Aµ and vector-spinor ψµ fields in Chapters 4 and 5, respectively. The free equation of motion for hµν is thelinearization of the exact Ricci tensor, obtained by substituting the metric ansatz (7.3) andretaining terms of order κ. We ask readers to do this as an exercise.

Ex. 7.2 Obtain the linearized Christoffel connection

ΓρLinµν = 1

2κ ηρσ(∂µhσν + ∂νhµσ − ∂σhµν) . (7.4)

Indices are raised using ηµν. Then use the variational formula δRµν = ∇ρδΓρµν −∇µδΓ

ρνρ to

determine the linearized Ricci tensor

RLinµν = −1

2κ [2hµν − ∂ρ(∂µhρν + ∂νhµρ) + ∂µ∂νh

ρρ] . (7.5)

The gauge properties of the free field system are obtained by linearizing the exact trans-formation rules under infinitesimal diffeomorphisms (See Sec. 6.1)

δgµν = κ (∇µξν +∇νξµ) ,

δRµν = κ (ξρ∇ρRµν +∇µξρRρν +∇νξ

ρRµρ) , (7.6)

56

in which we have specified that the arbitrary vector ξ(x) is of order κ. Since the right-handsides of these transformation rules are of order κ or higher, the linearization is simply

δhµν = ∂µξν + ∂νξν , (7.7)

δRLinµν = 0 . (7.8)

Thus we learn that the gravitational fluctuation field is a gauge field with gauge transfor-mation (7.7), which obeys the gauge-invariant wave equation (7.8).

Ex. 7.3 Show explicitly that RLinµν in (7.5) is invariant under the gauge transformation (7.7).

For most applications of the linearized equation (7.8), it is desirable to fix the gauge.The Lorentz covariant de Donder gauge condition ∂ρhρν − ∂νh

ρρ/2 = 0 is commonly used.

It leads to the simple wave equation 2hµν = 0, which shows convincingly that the fielddescribes massless particles. However, the de Donder gauge condition does not fix the gaugecompletely since δ(∂ρhρν − ∂νh

ρρ/2) = 2ξν . This makes the argument for counting physical

degrees of freedom somewhat indirect.Instead, we will choose a non-covariant gauge condition, namely the D conditions

∂ihiµ = 0 , (7.9)

where the sum includes only the space coordinates i = 1, . . . , D− 1. We will show that thisfixes the gauge completely and then proceed to count the physical modes of the field.

First note that the variation of the gauge condition is

δ(∂ihiµ) = ∇2ξµ + ∂i∂µξi = 0 . (7.10)

Let µ→ j and contract with ∂j to learn that ∇2∂iξi = 0. Thus ∂iξi does not contain degreesof freedom. With this information incorporated, (7.10) tells us that ξµ(x) ≡ 0.

We now turn to the wave equation specified by (7.5),(7.8). To count the degrees offreedom, we use the gauge condition (7.9) to write it as

2hµν − ∂0(∂µh0ν + ∂νhµ0) + ∂µ∂ν(hii − h00) = 0 . (7.11)

We now distinguish the specific components

µ = ν = 0 : ∇2h00 + ∂20hii = 0 ,

µ = ν = i : 2∇2hii − ∂20hii −∇2h00 = 0 , (7.12)

in which the indices i are summed. The sum of the two equations tells us that ∇2hii = 0, sothat hii = 0. Going back to the 00 equation, we learn that h00 ≡ 0. With the informationjust learned incorporated, the µ = 0, ν = i components of (7.11) becomes ∇2h0i = 0, sothat h0i ≡ 0.

Only components hij remain, and they satisfy the simple wave equation

2hij = 0 . (7.13)

57

There are D(D − 1)/2 distinct components hij, but they satisfy the constraints ∂ihij = 0from the gauge condition (7.9) and the trace condition hii = 0, which was found above. Thusthe number of independent functions that must be supplied as initial data are D(D−1)/2−(D − 1)− 1 = D(D − 3)/2 together with their time derivatives. A graviton in D spacetimedimensions thus has D(D − 3)/2 degrees of freedom. As discussed in Ch. 4, the states ofa massless particle in D dimensions carry an irreducible representation of the orthogonalgroup SO(D − 2). For the graviton, this is the traceless symmetric tensor representation,which indeed has dimension D(D − 3)/2.

7.3 Second order formalism for gravity and fermions

In this section we explore the most commonly used framework for the coupling of spinor fieldsto gravity. The frame field eaµ(x) and covariant derivatives constructed with the torsion-freeconnection play an essential role. It is instructive to consider the simplest case of a masslessDirac field Ψ(x). The action functional is

S = S2 + S1/2 =

∫dDx e

[1

2κ2eµae

νbR

abµν(e) +

1

2Ψγµ∇µΨ− 1

2Ψ←∇µγ

µΨ

]. (7.14)

The first term contains the curvature tensor (6.63)

Rµνab = ∂µωνab − ∂νωµab + ωµacωνcb − ωνacωµ

cb , (7.15)

with ω → ω(e), see (6.44).

Ex. 7.4 Check that ω(e) is the solution of the field equation of S2 for the spin connection,i.e.

δS2

δωµab

∣∣∣∣ω=ω(e)

= 0 . (7.16)

The fermion action contains the curved space γµ matrix, whose construction and prop-erties we discuss below, and the covariant derivatives

∇µΨ = DµΨ = (∂µ + 14ωabµ γab)Ψ ,

Ψ←∇µ = Ψ

←Dµ = Ψ(

←∂µ − 1

4ωabµ γab) . (7.17)

For S1/2 we could have written one term Ψγµ∇µΨ, which differs from the two terms in (7.14)by a total derivative. The symmetric expression (7.14) is easier to handle.

The procedure to covariantize Dirac-Clifford matrices begins with the observation thatthe Clifford algebra is closely linked to the properties of spinors on the space-time manifoldand is therefore defined in local frames. The generators are the constant matrices γa, whichsatisfy γa, γb = 2ηab, and higher rank elements γab, γabc, . . . are defined as anti-symmetricproducts of the generators as in Ch. 3.

Frame fields are used to transform frame vector indices to a coordinate basis. For example,γµ = eaµγ

a or γµ = eµaγa = gµνγν . Thus γµ transforms as a covariant vector under coordinate

58

transformations. But it also has (suppressed) row and column spinor indices and is thereforea Lorentz bi-spinor.11 The covariant derivative of γµ is therefore

∇µγν = ∂µγν + 14ωabµ [γab, γν ]− Γρµνγρ . (7.18)

The spin connection appears with the commutator as required for a bi-spinor.

Ex. 7.5 Derive the very useful result that the covariant derivative of γν vanishes. Specificallyshow that

∇µγν = γa(∂µeaν + ωµabebν − Γρµνeaρ) = 0 . (7.19)

In the last step use (6.51). Note that ∇µγν = 0 holds for any affine connection, with orwithout torsion, provided that one uses the total covariant derivative.

The result (7.19) implies that covariant derivatives commute with multiplication by gammamatrices. For example, if Ψ(x) is a Dirac spinor field, then ∇µ(γνΨ) = γν∇µΨ.

Ex. 7.6 Show that Ψγµ∇µΨ is invariant under infinitesimal local Lorentz transformationsδΨ = −1

4λabγ

abΨ, δΨ = 14λabΨγ

ab, δeµa = −λabeµb . Show that it transforms as a scalar underinfinitesimal coordinate transformations δΨ = ξρ∂ρΨ, δeµa = ξρ∂ρe

µa − ∂ρξ

µeρa, etc. Thesetransformations are Lie derivatives, as in (6.2), under which local frame indices are inert.

Let us consider how Einstein equation follows from S2. As we have (7.16), the variation ofthe action is

δS2 =1

κ2

∫dDx e

(ebνRµνab −

1

2eaµR

)δeaµ . (7.20)

The object in brackets thus enters in the field equation. Changing indices by multiplyingwith eνa gives the Einstein tensor, and we get a field equation of the form

Rµν − 12gµνR = κ2Tµν , (7.21)

where the right-hand side is determined by the appearance of the frame field in the mattercouplings.

7.4 The first order formalism for gravity and fermions

Let us now describe the first order formalism for the coupling of fermions to the gravitationalfield. As discussed at the beginning of this chapter, the key points are that the frame field eaµand spin connection ωµab appear as independent variables in the first order action. The fieldequation for ωµab may then be solved, giving a connection with torsion, ωµab = ωµab(g)+Kµab.The contortion tensor is bilinear in the fermions. It is interesting to see how connectionswith torsion arise in a physical setting.

11The archetypical example of a bi-spinor is the product ΨαΨβ of a Dirac spinor and its adjoint, whereα, β are spinor indices. Under infinitesimal local Lorentz transformations, this bi-spinor transforms asδ(Ψ Ψ) = λab(x)[γab,ΨΨ].

59

We can still use (7.14) and (7.17) for the first order action and covariant derivativeswith the understanding that the connection ωabµ is an independent quantity, and that thecurvature tensor Rab

µν is constructed from this connection. See (7.15). Note that the firstorder spinor actions with anti-symmetric and right acting Dµ are not equivalent, since atorsion term appears upon partial integration. See Sec. 6.7.3.

Our goal is to solve the field equations for the connection. We will obtain δS/δωµab,

which is the change in the action due to a small variation δωµab of the connection. The

variation of the gravitational action, using (6.66), is

δS2 =1

2κ2

∫dDx e eµae

νb

(Dµδων

ab −Dνδωµab

). (7.22)

Due to the anti-symmetry in µν, we can replace the Lorentz covariant derivatives with fullycovariant derivatives up to torsion as in (6.59), and with the anti-symmetry in ab, we havethen

δS2 =1

2κ2

∫dDx e eµae

νb

(2∇µδων

ab + Tµνρδωρ

ab). (7.23)

Using (6.50), the covariant derivatives commute with the vielbeins, and the first term is atotal covariant derivative. Comparing with (6.61), and omitting the boundary terms, weobtain the connection variation

δS2 =1

2κ2

∫dDx e

(−2Kρµ

ρeµaeνbδων

ab + Tabρδωρ

ab)

=1

2κ2

∫dDx e (Tρa

ρeνb − Tρbρeνa + Tab

ν) δωνab , (7.24)

where we used (6.62) and made the [ab] anti-symmetry explicit.The connection variation of the spinor action is simpler. Using (7.17) we find

δS1/2 =1

8

∫dDx e Ψγν , γabΨ δωabν

=1

4

∫dDx e Ψ γνab Ψ δωabν . (7.25)

The connection field equation can now be identified as the coefficient of δωνab in δS2+δS1/2 =

0. We find directly an equation for the torsion tensor

Tabν − Taρ

ρ eνb + Tbρρ eνa = −1

2κ2Ψ γab

ν Ψ . (7.26)

Since the trace of the right side vanishes, the trace of the torsion vanishes too, and the

torsion of the spinor field with↔Dµ kinetic term is simply given by the totally anti-symmetric

tensorTab

ν = −12κ2Ψ γab

ν Ψ = −2Kνab . (7.27)

60

The physical effects of torsion are rather unexciting, but they are worth discussing forpedagogical reasons. We substitute ω = ω(e) +K in the first order action (7.14).

S =1

2κ2

∫dDx e

[R(g) + κ2Ψγµ

↔∇µΨ

−2∇µKννµ +KµνρK

νµρ −KρρµKσ

σµ +1

2ΨγµνρΨKµνρ

], (7.28)

The connection and curvature that appear in (7.28) are torsion-free, so the ∇K term is atotal derivative, which can be dropped. We substitute the specific form of the torsion tensorfrom (7.27) and obtain the physically equivalent second order action

S =1

2

∫dDx e

[1

κ2R(g) + Ψγµ

↔∇µΨ +

1

16κ2(ΨγµνρΨ)(ΨγµνρΨ)

]. (7.29)

Note that the (Kρρµ)2 term vanishes in this theory because the torsion is totally anti-

symmetric. Physical effects in the fermion theories with and without torsion differ only bythe presence of the quartic Ψ4 term, which is a dimension 6 operator suppressed by the Planckscale. This term generates 4-point contact Feynman diagrams in fermion-fermion scatteringamplitudes. The contribution of these diagrams could thus, in principle, be detected inexperiments, so the fermion theories with and without torsion are physically inequivalent.

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8 N = 1 Global Supersymmetry in D = 4

In global SUSY the scope of symmetries included in quantum field theory is extended fromPoincare and internal symmetry transformations, with charges Mµν, Pµ, and TA, to includespinor charges Qi

α, where α is a spacetime spinor index, and i = 1, . . .N is the index of thedefining representation of an SO(N) global internal symmetry. We will assume that the Qi

α

are 4-component Majorana spinors, although an equivalent formulation using 2-componentWeyl spinors is also commonly used. In this chapter we will repeat some formulae for thesimplest case where N = 1 and there is a single spinor charge Qα. This case is called N = 1SUSY or simple SUSY. Theories with N > 1 spinor charges are called extended SUSYtheories. We give only a few formulae as the students have studied global N = 1 theories inthe lectures of P. Fayet and A. Bilal. But there will be some other notations and viewpointsuseful for the supergravity theories.

In N = 1 global SUSY the Poincare generators and Qα join in a new algebraic structure,that of a superalgebra. the subalgebra of the bosonic charges Mµν and Pµ is the Lie algebraof the Poincare group discussed in Ch. 2, while the new structure relations involving Qα are

Qα, Qβ

= −12(γµ)α

βP µ ,[Mµν, Qα

]= −1

2(γµν)α

βQβ ,

[Pµ, Qα] = 0 . (8.1)

8.1 Basic SUSY Field Theory

The simplest representations of the algebra that lead to the most basic SUSY field theoriesarei) the chiral multiplet, which contains a self-conjugate spin 1/2 fermion described by theMajorana field χ(x) plus a complex spin 0 boson described by the scalar field Z(x). Alter-natively, χ(x) may be replaced by the Weyl spinor PLχ and/or Z(x) by the combinationZ(x) = (A(x) + iB(x))/

√2 where A and B are a real scalar and pseudoscalar, respectively.

A chiral multiplet can be either massless or massive.ii) the gauge multiplet consisting of a massless spin 1 particle, described by a vector gaugefield Aµ(x), plus its spin 1/2 fermionic partner, the gaugino, described by a Majorana spinorλ(x) (or the corresponding Weyl field PLλ).

8.2 SUSY field theories of the chiral multiplet

The physical fields of the chiral multiplet are a complex scalar Z(x) and the Majorana spinorχ(x). It simplifies the structure in several ways to bring in a complex scalar auxiliary fieldF (x). The field equations of F are algebraic, so F can be eliminated from the system at alater stage. The set of fields Z, PLχ, F constitute a chiral multiplet, and their conjugatesZ, PRχ, F are an anti-chiral multiplet. The treatment is streamlined because we use thechiral projections PLχ and PRχ, but can still regard χ as a Majorana spinor.

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The transformation rules of the chiral multiplet are:

δZ =1√2εPLχ ,

δPLχ =1√2PL(/∂Z + F )ε ,

δF =1√2ε /∂PLχ . (8.2)

The anti-chiral multiplet transformation rules are:

δZ =1√2εPRχ ,

δPRχ =1√2PR(/∂Z + F )ε ,

δF =1√2ε /∂PRχ . (8.3)

Ex. 8.1 Show that the variations δZ, δPRχ, δF are precisely the adjoints of δZ, δPLχ, δF .

8.2.1 Basic actions.

There are two basic actions, which are separately invariant under the transformation rulesabove. The first is the free kinetic action

Skin =

∫d4x

[−∂µZ∂µZ − χ/∂PLχ+ FF

], (8.4)

in which we have presented the spinor term in chiral form. The interaction is determinedby an arbitrary holomorphic function, the superpotential W (Z). Given this we define theaction

SF =

∫d4x

[FW ′(Z)− 1

2χPLW

′′(Z)χ]. (8.5)

(The reason for the apparent extra derivative will be explained shortly.) Note that SFinvolves only the fields of the chiral multiplet and no anti-chiral components. Thus theaction SF is not Hermitian, so we must also consider the conjugate action SF = (SF )†. Thecomplete action of the chiral multiplet is the sum

S = Skin + SF + SF . (8.6)

8.2.2 The SUSY algebra

In this section we will study the realization of the SUSY algebra on the components of achiral multiplet. It is convenient and interesting that the Q, Q anti-commutator in (8.1)

63

is realized in classical manipulations as the commutator of two successive variations of thefields with distinct (anti-commuting) parameters ε1, ε2.

Let’s carry out the computation of [δ1, δ2]Z(x) on the scalar field of a chiral multiplet.Using (8.2) we write

[δ1, δ2]Z =1√2δ1(ε2PLχ)− [1 ↔ 2]

=1

2ε2PL(/∂Z + F )ε1 − [1 ↔ 2]

= −1

2ε1γ

µε2∂µZ . (8.7)

The symmetry properties of Majorana spinor bilinears (see (3.39)) have been used to reachthe final result, which clearly shows the promised infinitesimal translation.

The analogous computation of [δ1, δ2]PLχ(x) is more complex because a Fierz rearrange-ment is required. We outline it here:

[δ1, δ2]PLχ =1√2PL(/∂δ1Z + δ1F )ε2 − [1 ↔ 2]

=1

2PLγ

µε2ε1PL∂µχ+1

2PLε2ε1/∂PLχ− [1 ↔ 2] . (8.8)

To continue, you have to make a Fierz transformation, see Ex. 3.10. Because of the anti-symmetrization in ε1 ↔ ε2 the only non-vanishing bilinears are γν or γνρ. However, only γν

survives the chiral projection in the last factor. Thus we find the expected result

[δ1, δ2]PLχ = −12ε1γ

µε2PL∂µχ (8.9)

as the just reward for our labor.

Ex. 8.2 It is quite simple to demonstrate that

[δ1, δ2]F = −12ε1γ

µε2∂µF , (8.10)

Zealous readers should do it.

The auxiliary field F can be eliminated from the action by substituting the value F =W ′(Z), which is the solution of its equation of motion from (8.6), without affecting theclassical (or quantum) dynamics. Here is an exercise to show that SUSY is also maintainedafter elimination.

Ex. 8.3 Consider the theory after elimination of F and F . Show that the action

S =

∫d4x

[−∂µZ∂µZ − χ/∂PLχ− W ′W ′ + 1

2χ(PLW

′′ + PRW′′)χ

], (8.11)

is invariant under the transformations rules (8.2) and their conjugates (8.3). Show that[δ1, δ2]Z is exactly the same as in (8.7), but [δ1, δ2]PLχ is modified as follows

[δ1, δ2]PLχ = −14ε1γ

µε2PL[∂µχ− γµ(/∂ − W ′′)χ

]. (8.12)

We find the spacetime translation plus an extra term that vanishes for any solutions of theEOM’s.

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Since the commutator of symmetries must give a symmetry of the action and translationsare a known symmetry, the remaining transformation, namely

δZ = 0 ,

δχ = vµγµ(/∂ − PLW′′ − PRW

′′)χ , (8.13)

is itself a symmetry for any constant vector vµ. However, its Noether charge vanishes whenthe fermion equation of motion is satisfied, so it has no physical effect. Such a symmetry issometimes called a ‘zilch symmetry’.

Although nothing physically essential is changed by eliminating auxiliary fields, we cannevertheless see that they play a useful role:i) it is only with F , F included that the form of the SUSY transformation rules (8.2),(8.3)is independent of the superpotential W (Z).ii) the SUSY algebra is also universal on all components of the chiral multiplet when F isincluded. The phrase used in the literature is that the SUSY algebra is “closed off-shell”when auxiliary fields are included and “closed only on-shell” when they are eliminated.iii) auxiliary fields are very useful in determining the terms in a SUSY Lagrangian describinginteractions between different multiplets such as the SUSY gauge theories described in thenext section. It is also the case that auxiliary fields are known only for a few extended SUSYtheories in 4-dimensional spacetime and also unavailable for dimension D > 4. Indeed manyof the most interesting SUSY theories have no known auxiliary fields.

Although we hope that some readers enjoy the detailed manipulations needed to studySUSY theories, we suspect that many are fed up with Fierz rearrangement and would likea more systematic approach. To a large extent the superspace formalism does exactly thatand has many advantages. Unfortunately, complete superspace methods are also unavailablewhen auxiliary fields are not known.

8.3 SUSY Gauge Theories

The basic SUSY gauge theory is the N = 1 SYM theory containing the gauge multiplet AAµ ,λA where A is the index of the adjoint representation of a compact, non-abelian gauge groupG. We assume that the group has an invariant metric that can be chosen as δAB. This is thecase for ‘reductive groups’, i.e. products of compact semisimple groups and Abelian factors,i.e. G = G1 ⊗ G2 ⊗ . . ., where each factor is a simple group or U(1). The normalization ofthe generators is fixed in each factor, which can lead to different gauge coupling constantsg1, g2, . . . for each of these factors. We have taken here the normalizations of the generatorswhere these coupling constants are not explicitly appearing. One can replace everywheretA with gi tA and fAB

C with gi fABC , where gi can be chosen independently in each simple

factor, to re-install these coupling constants. Usually one also redefines then the parametersθA to 1

giθA. This leads to the formulae with coupling constant g in Sec. 4.2. Further note

that for these compact algebras, the structure constants can be written as fABC , which arecompletely antisymmetric.

The action of the general theory is the sum of several terms

S = Sgauge + Smatter + Scoupling + SW + SW . (8.14)

65

We suppress representation indices when this causes no ambiguity.

Sgauge =

∫d4x

[−1

4F µνAFA

µν − 12λAγµDµλ

A + 12DADA

], (8.15)

Smatter =

∫d4x

[−DµZDµZ − χγµPLDµχ+ FF

], (8.16)

Scoupling =

∫d4x

[−√

2(λAZtAPLχ− χPRtAZλA) + iDAZtAZ

], (8.17)

SF =

∫d4x

[FαW,α +1

2χαPLW,αβ χ

β], (8.18)

SF =

∫d4x

[FαW

α + 12χαPRW ,αβ χβ

]. (8.19)

We can write covariant derivatives of the various fields as

DµλA = ∂µλ

A + fBCAABµ λ

C ,

DµZ = ∂µZ + tAAAµZ ,

DµPLχ = ∂µPLχ+ tAAAµPLχ ,

DµPRχ = ∂µPRχ+ t∗AAAµPRχ . (8.20)

The system need not contain a superpotential, but superpotentials W (Zα), which should beboth holomorphic and gauge invariant, are optional. It is useful to express the condition ofgauge invariance of W (Zα) as

δgaugeW = W,α δgaugeZα = −W,α θA(tA)αβZ

β . (8.21)

The full action is invariant under SUSY transformation rules, which are for the gaugemultiplet

δAAµ = −12εγµλ

A ,

δλA =[

14γρσFA

ρσ + 12iγ∗D

A]ε ,

δDA = 12i εγ∗γ

µDµλA , Dµλ

A ≡ ∂µλA + λCAµ

BfBCA . (8.22)

The fields of the gauge multiplet transform also under an internal gauge symmetry:

δ(θ)AAµ = ∂µθA + θCAµ

BfBCA ,

δ(θ)λA = θCλBfBCA ,

δ(θ)DA = θCDBfBCA . (8.23)

Let us first remark that the commutator of these internal gauge transformations and super-symmetry vanishes.

66

Ex. 8.4 Use the transformation rules above to derive the commutator algebra of the super-symmetries for the gauge multiplet

[δ1, δ2]AAµ = −1

2ε1γ

νε2FAνµ ,

[δ1, δ2]λA = −1

2ε1γ

νε2DνλA ,

[δ1, δ2]DA = −1

2ε1γ

νε2DνDA . (8.24)

It is no surprise that the commutator of two gauge-covariant variations from (8.22) is gaugecovariant, but at first glance the result seems to disagree with the space-time translationas found in Sec. 8.2.2. Note that in all 3 cases in ex. 8.4 the difference between thecovariant result in (8.24) and a translation is a gauge transformation by the field-dependentgauge parameter θA = 1

2ε1γ

νε2AAν . The covariant forms that occur in (8.24) are called

gauge-covariant translations. The conclusion is that on the fields of a gauge theory theSUSY algebra closes on gauge-covariant translations. These are the transformations that weencountered in Ex. 4.2.

In order that the modified supersymmetry algebra is also satisfied on the chiral multiplets,their transformations rules are modified to

δZ =1√2εPLχ ,

δPLχ =1√2PL(γµDµZ + F )ε ,

δF =1√2εPRγ

µDµχ− εPRλAtAZ . (8.25)

Some modifications in the action and transformation rules above, notably the introduc-tion of gauge covariant derivatives, are clearly required in a gauge theory, but other additionssuch as the form of the action Scoupling are surely not obvious. The best explanation is thatthey are dictated by the superspace formalism. However all features can be explained fromthe component viewpoint by checking the algebra and the invariance of the action.

8.4 Massless supersymmetry multiplets

We will give a brief introduction to the theory of massless unitary representations of thesupersymmetry algebras in 4 dimensions. You can consult e.g. Sec. 3 of the Phys. Rep. ofM. Sohnius for a more complete treatment.

Wigner defined a particle as a unitary representation of the Poincare group. To classifythese, he made a distinction between massive and massless states. For the massless case,he obtained that a particle consists of 2 helicity states h = ±s, where s is the spin of theparticle. Spin 0 particles have only one helicity state.

The theory of representations of the supersymmetry algebra has been investigated bySalam and Strathdee. They construct the physical states of a unitary representation of theN -extended algebra as representations of a Clifford algebra of dimension 2N . The minimalrepresentation has thus dimension 2N . In fact, one can divide the relevant parts of the

67

supersymmetry generators in N helicity-creation operators, say Ci+ where i = 1, . . . ,N and

N helicity-annihilation operators, say C−− . As these are supersymmetry operators, theychange the helicity by ±1/2. The states are then built from a Clifford vacuum that has itselfan helicity κ, which we may denote as |κ >. With the N creation operators, one obtains inthe supersymmetry multiplets states

|κ >, Ci+|κ >, Ci

+Cj+|κ >, , . . . , (8.26)

of helicity (κ, κ+ 1/2, . . . , κ+N/2). If this set of helicity states is not self-CPT conjugate,that means that states appear with helicity h and there is no corresponding state with helicity−h, then one has to introduce also the conjugate states (−κ,−κ− 1/2, . . . ,−κ−N /2), i.e.the multiplet with κ′ = −κ−N /2. Hence, it is clear that our enumeration should only startfrom κ = −N/4 or the higher half-integer from this number.12

For N = 1 we thus find for for κ = 0 the multiplet (0, 1/2) to be complemented with(0,−1/2) leading to the fields of the chiral multiplet. For κ = 1/2, we find fields with spin(1/2, 1), i.e. the gauge multiplet. In principle there is then the multiplet (1, 3/2). However,there is no good field theory known for this multiplet. This is due to the fact that we sawthat field theories for spin 3/2 fields involve a local supersymmetry. Hence, they shouldby the supersymmetry algebra also involve local translations, and hence general relativity.Therefore, we find the spin 3/2 particle only in the multiplet (3/2, 2). This is the supergravitytheory that we will consider in Ch. 9.

The problem that we mentioned above for field theories of spin 3/2 particles impliesthat global supersymmetry multiplets should be restricted to the spins ≤ 1, i.e. helicities−1 ≤ h ≤ 1. With the above rules, that means that we are restricted to N ≤ 4.

Despite a lot of efforts, it is still difficult if not impossible to construct interacting fieldtheories in Minkowski space with particles of spin ≥ 5/2. With the same principle as above,this implies that supergravity theories are restricted to N ≤ 8, which is a very importantfact of life for supergravity.

Ex. 8.5 With this information, the student should check the tables 3 and 4 of number offields with different spins in multiplets for different N .

12There is still a more delicate hermiticity requirement for N = 2 such that the multiplet (−1/2, 0, 0, 1/2)also has to be doubled.

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Table 3: Pure supergravity multiplets in 4 dimensions according to spin s

s N = 1 N = 2 N = 3 N = 4 N = 5 N = 6 N = 82 1 1 1 1 1 1 132

1 2 3 4 5 6 81 1 3 6 10 16 2812

1 4 11 26 560 2 10 30 70

Table 4: Matter multiplets in 4 dimensions

s N = 1 N = 2 N = 3, 41 1 1 112

1 2 40 2 6

s N = 1 N = 212

1 20 2 4

9 Supergravity, I

In earlier chapters we have reviewed the ideas and implementation of Lorentz invariance,relativistic spin, gauge principles, global supersymmetry, and spacetime geometry. We willnow begin to study how these elements combine in supergravity. The key idea is thatsupersymmetry holds locally in a supergravity theory. The action is invariant under SUSYtransformations in which the spinor parameters ε(x) are arbitrary functions of the spacetimecoordinates. The SUSY algebra (see (8.1)) will then involve local translation parametersε1γ

µε2 which must be viewed as diffeomorphisms. Thus local supersymmetry requires gravity.A supergravity theory is a nonlinear, and thus interacting, field theory that necessarily

contains the gauge or gravity multiplet plus, optionally, other matter multiplets of the un-derlying global supersymmetry algebra. The gauge multiplet consists of the frame field eaµ(x)describing the graviton, plus a specific number N of vector-spinor fields ΨI

µ(x), I = 1, . . . ,N ,whose quanta are the gravitinos, the supersymmetric partners of the graviton. In the basiccase of N = 1 supergravity in D = 4 spacetime dimensions, the gauge multiplet consistsentirely of the graviton and one Majorana spinor gravitino. In all other cases, both N ≥ 2in D = 4 dimensions and N ≥ 1 for D ≥ 5, additional fields are required in the gaugemultiplet.

Supergravity theories exist for spacetime dimensions D ≤ 11. For each dimension D, aspecific type of spinor is required, e.g. Majorana or Weyl. For D = 4, theories exist forN = 1, 2, . . . , 8. Beyond these limits local supersymmetry fails and the classical equationsof motion are inconsistent.

There are four major applications of supergravity.1. If some form of broken global supersymmetry describes the physics of elementary particles

69

beyond the standard model, it is quite clear that the accompanying theory of gravity mustalso be supersymmetric. It is then plausible that the right model of elementary particlephysics is a form of N = 1, D = 4 supergravity coupled to chiral and gauge multiplets ofglobal SUSY. Direct gravitational effects on elementary particle phenomena are probablytiny, but matter-coupled supergravity offers new ways to break SUSY.2. It is often said that D = 10 supergravity is the low energy limit of superstring theory.This means that the dynamics of the lowest energy modes of a superstring are describedby a supergravity action in ten spacetime dimensions. The form of such actions is quiterestrictive, but a surprisingly large array of solutions is known. Many of them exhibit whatis called spacetime compactification. The solutions describe a direct or warped product of amacroscopic universe of four dimensions with a compact Euclidean signature internal spaceof six dimensions.3. The several D = 10 superstring theories are related to each other by dualities, and thereare good arguments that they are all related to an 11 dimensional theory called M-theory.The framework of M-theory is not yet well understood, but it is clear that any reasonabledescription must involve the unique D=11 supergravity theory in an essential way.4. There are also compactifications of D = 10 string theory to a product of a five dimensionalspacetime which looks like anti-de Sitter space at large distances times a compact factorsuch as the 5-sphere. The AdS5 spacetime should be regarded as a manifold with a four-dimensional boundary at large distance. One surprising and profound development of thelast decade is the AdS/CFT correspondence which asserts that string theory in a productgeometry of this type is dual to a four-dimensional gauge theory whose spacetime domain isthe AdS boundary. The most precise results come from applying AdS/CFT in the limit inwhich string theory is approximated by supergravity. Correlation functions of the boundarygauge theory can be obtained in a strong coupling limit in which field theory methods are notavailable from weak coupling classical calculations in five and ten dimensional supergravity.

In this chapter we will concentrate on the basic N = 1, D = 4 supergravity theory. Wewill discuss the form of the action and transformation rules and prove local supersymmetry.However, the principal terms of the N = 1, D = 4 action are an important part of thestructure of all supergravity theories, and the initial steps in the proof of local SUSY areuniversal, that is, applicable in any dimension. We discuss these universal steps in the nextsection, and then refocus and complete the proof for N = 1, D = 4 supergravity.

It will be important to understand the structure of the proof, to gain familiarity withthe structure of a supergravity theory. However, despite the fact that we are just discussingsimple supergravity without matter couplings, the proof of invariance needs many ingredientsand complicated calculations. Later we will discuss another method, for which one needs theformalism of superconformal tensor calculus. Also that one is based on many technical tools,But these ones can be re-used for the matter-coupled supergravities, N = 2 supergravitiesand similar theories, and lead to more insights. Therefore the details of this proof (Sec. 9.4)are not very important for the general understanding of supergravity.

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9.1 The universal part of supergravity

The part of the supergravity action we will study in this section consists of the sum of theHilbert action for gravity in second order formalism plus a local Lorentz and diffeomorphisminvariant extension of the free gravitino action of (5.2) multiplied by 1/2 because we dealwith a Majorana gravitino and rescaled by the factor 1/κ2 for convenience.13 The action is

S = S2 + S3/2 , (9.1)

S2 =1

2κ2

∫dDx e eaµebνRµνab(ω) , (9.2)

S3/2 = − 1

2κ2

∫dDx e ψµγ

µνρDνψρ , (9.3)

where e stands for the determinant of eaµ. The gravitino covariant derivative is given by

Dνψρ ≡ ∂νψρ + 14ωνabγ

abψρ . (9.4)

It is the torsion-free spin connection ωνab(e), given in (6.44), that is used exclusively in thissection. We need not include the Christoffel connection term Γσνρ(g)ψσ in (9.4) because theconnection (6.56) is symmetric, and the term vanishes in the action S3/2 since the Lagrangianis antisymmetric in νρ.

We also need transformation rules, and we postulate the rules

δeaµ = 12εγaψµ , (9.5)

δψµ = Dµε(x) ≡ ∂µε+ 14ωµabγ

abε . (9.6)

The gravitino is the gauge field of local supersymmetry, so it is natural to postulate (9.6) asthe curved space generalization of (5.1). For the frame field, (9.5) is the simplest form con-sistent with the tensor structure required and the Bose-Fermi character of supersymmetry.14

Ex. 9.1 Deduce from (9.5) that

δeµa = −12εγµψa , δe = 1

2e (εγρψρ). (9.7)

The action and transformation rules above are not quite the whole story even for theN = 1, D = 4 theory, but they can be completed by incorporating the first order formalismwith torsion. This is done in the next section. For all other theories one must also addterms describing the other fields in the gauge multiplet. The variation of the action (9.1)consists of linear terms in ψµ from the frame field variation of S2 and the gravitino variationof S3/2 and cubic terms from the frame field variation of S3/2. It are the linear terms thatare universal, and we now proceed to calculate them.

13The derivation that we present depends on symmetry properties of Majorana bilinears and is thereforestrictly speaking only valid in dimensions D = 4, 10, 11, mod 8. However, using appropriate reality spinors,the other dimensions can be treated analogously.

14The possible form δeaµ ∼ εγµψ

a may be considered explicitly and shown to be incompatible with localsupersymmetry.

71

The variation of the gravitational action under (9.5) is obtained as in Sec. 7.3, see thefirst part of (7.20)

δS2 =1

2κ2

∫dDx e

(Rµν −

1

2gµνR

)(−εγµψν) . (9.8)

We now study the gravitino variation. For Majorana spinors in the second order formalism,partial integration is valid, so it is sufficient to vary δψµ and multiply by 2, obtaining

δS3/2 = − 1

κ2

∫dDx e ε

←Dµγ

µνρDνψρ (9.9)

=1

κ2

∫dDx e εγµνρDµDνψρ =

1

8κ2

∫dDx e εγµνρRµνabγ

abψρ .

We integrated by parts to move to the second line and then used the Ricci identity to obtainthe last expression.

We now need some Dirac algebra to evaluate the product γµνργab. This product wasalready discussed as an example in (3.21). We thus find

γµνργabRµνab = γµνρabRµνab + 6Rµν[ρbγµν]b + 6γ[µRµν

ρν]

= γµνρabRµνab + 2Rµνρbγµνb + 4Rµν

µbγνρb + 4γµRµν

ρν + 2γρRµννµ ,(9.10)

where in the second line we split the symmetry of [µνρ] according to the position of ρ. Thesymmetry of [µν] remains present due to the multiplication with Rµνab. The first term andsecond term vanish because of the above-mentioned first Bianchi identity. The third termvanishes because of the symmetry of the Ricci tensor, Rνb = Rµν

µb being contracted with

the antisymmetric γνρb. We are thus left with

δS3/2 =1

2κ2

∫dDx e (Rµν −

1

2gµνR)(εγµψν) . (9.11)

We now observe the exact cancellation between (9.8) and (9.11) which shows that localsupersymmetry holds to linear order in ψµ for any spacetime dimension D.15 Notice that thecalculation involves an intimate mix of key properties of Riemannian geometry and Diracalgebra!

9.2 Supergravity in the First Order Formalism

Beyond linear order it becomes complicated to establish local supersymmetry for any super-gravity theory. It is therefore very useful to recognize simplifications which lead to convenientorganization of terms. One important simplification is to express the action and transfor-mation rules in the first order formalism. We continue to work with the D-dimensionalaction (9.1) and transformation rules (9.5), (9.6), but we regard the spin connection ωµab

15Restrictions on D appear if local SUSY is imposed beyond linear order.

72

as an independent variable. There is no Christoffel connection term in S3/2. It would beinconsistent with local supersymmetry to include it.16

As discussed in Ch. 7, in the first order formalism the equation of motion for the spinconnection can be solved to obtain a connection with torsion. This result can then besubstituted in the action to obtain the physically equivalent second order form of the theorywith torsion-free connection and explicit 4-fermion contact terms. We now carry out thisprocess for the supergravity action (9.2). We will use the connection variation (7.24) of S2

derived in Ch. 7.For S3/2 we easily obtain

δS3/2 = − 1

8κ2

∫dDx e (ψµγ

µνργabψρ)δωνab . (9.12)

The Clifford algebra relation needed to simplify the spinor bilinear in (9.12) is again theone given in (3.21). We restrict here to the dimensions in which we have (pseudo-)Majoranaspinors, D = 2, 3, 4, 8, 9, 10, 11. Then, according to table 1 spinor bilinears of rank 3 are sym-metric, which clashes with the antisymmetry in the indices µ, ρ on the gravitini. Therefore,we obtain

ψµγµνργabψρ = ψµ

(γµνρab + 6γ[µeν [be

ρ]a]

)ψρ . (9.13)

It is now very easy to solve the connection field equation

δS2 + δS3/2 = 0 , (9.14)

using (7.24). The trace structure in the two terms matches exactly, so the unique solutionfor the torsion is

Tabν = 1

2ψaγ

νψb + 14ψµγ

µνρabψρ . (9.15)

The fifth rank tensor term is one of the complications of supergravity for D ≥ 5, but itsimply vanishes when D = 4.

For gravity coupled to a spin 1/2 Dirac field, we showed in (7.28), (7.29) how to obtainthe physically equivalent second order form of the theory by substituting the value of thetorsion tensor in the first order action. Here is an exercise to do the same for supergravity.

Ex. 9.2 For D = 4 substitute the torsion tensor (9.15) in the action (9.2) to obtain thesecond order action of supergravity

S =1

2κ2

∫d4x e [R−ψµγµνρDνψρ+

1

16

((ψργµψν)(ψργµψν+2ψργνψµ)−4(ψµγ ·ψ)(ψµγ ·ψ)

)] ,

(9.16)in which the curvature R and the gravitino covariant derivative now contains the torsion-freeconnection

Dνψρ ≡ ∂νψρ + 14ωνab(e)γ

abψρ . (9.17)

16Note that a Christoffel term is permitted by diffeomorphism invariance since the antisymmetric part ofΓσ

νρ is proportional to the torsion tensor. See (6.57).

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The theory expressed by (9.16) contains 4-point contact diagrams for gravitino scattering.Their physical effects are certainly not dramatic, but they are necessary for the consistencyof the theory, which does not otherwise obey the requirement of local supersymmetry. Localsupersymmetry means that the action (9.16) is invariant under the transformation rules(9.5),(9.6), with ω = ω(e) +K, see (6.45).

9.3 The 1.5 order formalism

The second order action (9.16) for N = 1, D = 4 supergravity is complete, and it possessescomplete local supersymmetry under the transformation rules (9.5), (9.6) with the connection

δψµ = Dµε ≡ ∂µε+ 14(ωµab(e) +Kµab)γ

abε ,

Kµνρ = −14(ψµγρψν − ψνγµψρ + ψργνψµ) , (9.18)

which includes the gravitino torsion. We will prove this invariance property in the nextsection.

Let us now think schematically about the structure of the proof. The variation of theaction contains terms which are first, third, and fifth order in the gravitino field. They areindependent and must cancel separately. In the first construction of the theory, which usedthe second order formalism, the first and third order terms were treated analytically, but itrequired a computer calculation to show that the complicated order (ψµ)5 term.17

In the first order form of the theory, the fifth order variation is avoided, but one mustspecify a new transformation rule δωµab, since the connection is an independent variable. Thisapproach becomes quite complicated when matter multiplets are coupled to supergravity.

For the reasons above, the simplest treatment of supergravity uses a formalism interme-diate between the first and second order versions we have discussed. It is therefore calledthe 1.5 order formalism, and it is applicable to supergravity theories in any dimension.

One is really working in the second order formalism since there are only two independentfields, eaµ, ψµ. However, since the second order Lagrangian is obtained by substitutingω = ω(e) +K in the first order action, one can simplify the proof of invariance by retainingthe original grouping of terms. To see more concretely how this works, let’s consider anaction which is a functional of the three variables S[e, ω, ψ]. We use the chain rule tocalculate its variation in the second order formalism, viz.

δS[e, ω(e) +K, ψ] =

∫dDx

[δS

δeδe +

δS

δωδ(ω(e) +K) +

δS

δψδψ

]. (9.19)

In particular, the variation δ(ω(e) + K) is calculated using (9.5) and (9.18). However, nocalculation is needed since the second term vanishes because ω = ω(e) + K is obtained bysolving the algebraic field equationδS/δω = 0. Thus, in the 1.5 order formalism, we canneglect all δω variations as we proceed to establish local supersymmetry.

17Such terms come from the variation of the order (ψµ)4 contact terms in (9.16) with respect to the framefield and the torsion part of δψµ.

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Summary of the prescription for δS in the 1.5 order formalism:1. Use the first order form of the action S[e, ω, ψ] and the transformation rules δe, δψ withconnection ω unspecified.2. Ignore the connection variation and calculate

δS =

∫dDx

[δS

δeδe +

δS

δψδψ

]. (9.20)

3. Substitute ω = ω(e) + K from (9.18) in the result, which must vanish for a consistentsupergravity theory.

9.4 Local supersymmetry of N = 1, D = 4 supergravity

This section will finally proof that the action is supersymmetric. This step could of course notbe omitted in this chapter. However, nothing new will be learned here. All the ingredientsand ideas have been explained in the previous sections. Therefore, the student could skipthis section.

In this section we will use the 1.5 order formalism to prove that N = 1, D = 4 supergrav-ity is a consistent gauge theory, invariant under the transformation rules (9.5) and (9.18)with arbitrary ε(x). The proof is specific to D = 4 spacetime dimensions, so we first simplifythe gravitino action by introducing the highest rank Clifford element γ∗ = iγ0γ1γ2γ3. Thisis related to the third rank Clifford matrices by

γabc = −iεabcdγ∗γd ,

γµνρ = −ie −1εµνρσγ∗γσ . (9.21)

The first relation holds in local frames and the second, which we need, in the coordinate basisof spacetime. The Levi-Civita tensor density, with ε0123 = −1, appears in both relations.Using (9.21) and the fact that det(eaµe

νa) = 1, we can rewrite the gravitino action (9.3) as

S3/2 =i

2κ2

∫d4x εµνρσψµγ∗γσDνψρ . (9.22)

This is the way that supergravity was written before it was discovered that the universe has6 or 7 hidden dimensions. The advantage of this form is that the frame field variation δe isneeded only in γσ rather than in four positions in e γµνρ.

We now proceed to the proof of local supersymmetry. There are a number of subtletiesbecause of the connection with torsion. We need the δe variation of S2 (with the δω variationof Rµνab ignored in the 1.5 order formalism). The Ricci tensor with torsion is not symmetric,and we call it Rµν(ω) as a reminder. But it is easy to check that (9.8) is still valid, and canbe rewritten as

δS2 =1

2κ2

∫dDx e (Rµν(ω)− 1

2gµνR(ω))(−εγµψν) . (9.23)

The variation of S3/2 is more complex. For example we must obtain the ψ and ψ varia-tions separately because partial integration of the local Lorentz derivative Dµ must be done

75

carefully. The simplest term is the ψ variation

δSψ =i

2κ2

∫d4x εµνρσψµγ∗γσDνDρε

=i

16κ2

∫d4x ψµε

µνρσγ∗γσγabRµνab(ω)ε . (9.24)

The curvature tensor appears through the commutator of covariant derivatives.Next we write the ψµ variation and exchange the spinors Dµε and ψρ (see (3.39)) to

obtain

δSψ =−i

2κ2

∫d4x εµνρσψρ

←Dνγ∗γσDµε . (9.25)

With some thought one see that the left-acting derivative ψρ←Dν = ∂νψρ− 1

4ψρωνabγ

ab can bepartially integrated and acts distributively to give

δSψ =i

2κ2

∫d4x εµνρσψργ∗[(Dνγσ)Dµε+ γσDνDµε]

=−i

2κ2

∫d4x εµνρσψργ∗

[1

2Tνσ

aγaDµε−1

8γσγ

abRµνabε

]. (9.26)

As indicated in (7.19) the full covariant ∇νγσ = 0. But supergravity employs only the localLorentz covariant derivative Dν . When we add back the Christoffel connection and useantisymmetry in νσ, we obtain the torsion tensor in (9.26).

Note that the Rµνab terms in (9.24) and (9.26) are equal. To treat this term we use theγ-matrix manipulation using the methods that we learned in Sect. 3.1.2

γσγab = γσab + 2eσ[aγb] = iedσεabcdγ∗γc + 2eσ[aγb] . (9.27)

(The last expression is only valid for D = 4). In the first term we encounter the contractionof two Levi-Civita symbols and Riemann tensor. Using (3.11), we write

εµνρσεabcdedσRνρ

ab = −6e δµνρabc Rνρab

= −2e [eµaeνbeρc + eµb e

νceρa + eµc e

νaeρb ]Rνρ

ab

= 4e[Rc

µ − 12eµcR

]. (9.28)

When this relation is inserted in (9.24) and (9.26) and we use ψµγcε = −εγcψµ, the result

exactly cancels δS2 in (9.23). The work so far has brought us to the level of linear localsupersymmetry proven in Sec. 9.1, although the present cancellation includes cubic termsfrom the torsion contribution to (9.23).

The contribution of the commutator term in (9.27) to the integrand of (9.24) or (9.26)involves the factor

εµνρσRνρσb = −εµνρσDνTρσb , (9.29)

76

in which we have used the modified first Bianchi identity, derived in (6.68), where it isshown that the derivative Dν contains only the spin connection acting on the index b, Thiscontributes to the sum of (9.24) plus (9.26) as

−i

4κ2

∫d4x εµνρσDνTρσ

b ψµγ∗γbε . (9.30)

We are left with uncancelled torsion contributions from (9.26) and (9.30), plus the framefield variation of S3/2

δSe =i

4κ2

∫d4x εµνρσ(εγaψσ)(ψµγ∗γaDνψρ) . (9.31)

In the integrand we reorder the spinors using Fierz rearrangement technique. From thegeneral Fierzing rules and some gamma algebra, one can obtain

(γµ)αβ(γµ)γ

δ =1

2m

∑A

vA(ΓA)αδ(ΓA)γ

β . (9.32)

For even spacetime dimension D = 2m the sum extends over the rank rA independentelements of the Clifford algebra 0 ≤ rA ≤ D, and the coefficients vA = (−)rA(D − 2rA).When applied to the integrand of (9.31), this reordering identity reads:

(εγaψσ)(ψµγaγ∗Dνψρ) = −14

∑A

(−)rA(4− 2rA)(εΓAγ∗Dνψρ)(ψµΓAψσ)

= 12(εγaγ∗Dνψρ)(ψµγ

aψσ)

= (εγ∗γaDνψρ)Tµσa . (9.33)

The second line contains a remarkable simplification. Due to the antisymmetry in indices µσof (9.31) and the vanishing of v2 for D = 4, only the rank 1 ΓA → γa invariant survives. Wehave rewritten the last line using the D = 4 torsion tensor (9.15). We insert this in (9.31),reorder the (e . . . Dψ) bilinear, and exchange the indices µρ to obtain

δSe =−i

4κ2

∫d4x εµνρσTρσ

aψµ←Dνγ∗γaε . (9.34)

We have now reached the final step of the proof in which we combine the three uncancelledtorsion terms (in the order (9.24) +(9.26), (9.34), (9.30)) to obtain the sum:

δStorsion =−i

4κ2

∫d4x εµνρσ

[Tρσ

a(ψµγ∗γaDνε+ ψµ←Dνγ∗γbε) +DνTρσ

b ψµγ∗γbε

]=

−i

4κ2

∫d4x εµνρσ∂ν [Tρσ

bψµγ∗γbε] ≡ 0 . (9.35)

The integrand is a total derivative because the local Lorentz derivative works distributively.Indeed one may see, using (9.27), that the spin connection cancels among the three termsin the second line of (9.35). This proves that N = 1, D = 4 supergravity is locally su-persymmetric and thus consistent as a classical theory of the graviton and gravitino! Thebasic relations of differential geometry with torsion and the Clifford algebra and spinor anti-commutativity combine in the proof in a very striking way!

77

9.5 The algebra of local supersymmetry

The commutator of two local SUSY transformations should realize an algebra which is com-patible with that of global SUSY. On any component field Φ of a chiral multiplet we foundin Sec. 8.2.2

[δ1, δ2] Φ = −12ε1γ

µε2∂µΦ . (9.36)

It is natural to expect that the local extension of the global result (9.36) should be a generalcoordinate transformation with parameter ξµ(x) = −1

2ε1(x)γµε2(x). However, the general

formalism for symmetries requires only that the commutator closes on a sum of the gaugesymmetries of the theory, in this case a sum of general coordinate, local Lorentz, and localSUSY transformations. Furthermore, the gauge parameters that appear in the commutatorcan be field dependent, a phenomenon already encountered in SUSY gauge theories in Sec.8.3. Thus it is not immediately clear what we will find in supergravity, and it is well advisedto do the computation.

The computation is quite simple for the frame field. Here it is:

[δ1, δ2] eaµ = 1

2δ1ε2γ

aψµ − (1 ↔ 2) = 12ε2γ

aδ1ψµ − (1 ↔ 2)

= 12ε2γ

aDµε1 − (1 ↔ 2)

= 12(ε2γ

aDµε1 +Dµε2γaε1)

= Dµξa , ξa = 1

2ε2γ

aε1 = −12ε1γ

aε2 . (9.37)

Under the expected general coordinate transformation x′µ = xµ − ξµ(x), the frame fieldtransforms as a covariant vector, viz.

eaµ(x′) =∂xν

∂x′µeaν . (9.38)

The infinitesimal form isδξe

aµ = ξρ∂ρe

aµ + ∂µξ

ρeaρ . (9.39)

Let us ‘covariantize’ the derivatives by adding and subtracting the ω and Γ connectionterms. This must be done with some care since there is torsion. The symbol ∇ρ includes allappropriate connections.

δξeaµ = ξρ∇ρe

aµ − ξρωρ

abebρ + ξρΓσρµe

aσ +∇µξ

ρ eaρ − Γρµσξσeaρ

= ∇µξρeaρ − ξρωρ

abebρ + ξρTρµ

a . (9.40)

Since ∇ρeaµ = 0 (see (6.51) which is valid with torsion), we have dropped it in moving to the

second line. Since eaρ∇µξρ = Dµξ

a, the first term in (9.40) matches the supergravity result(9.37). The last two terms seem mysterious, but, in the light of the remarks above, let’s tryto interpret then as field dependent symmetry transformations. The second term in (9.40)is simply a local Lorentz transformation of the frame field with field dependent parameterλab = ξρωρab. To interpret the third term, we use the explicit form (9.15) of the torsiontensor to write

ξρTρµa = 1

2(ξρψρ)γ

aψµ . (9.41)

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This is just a local SUSY transformation of eaµ with field dependent ε = ξρψρ. Thus we havederived

[δ1, δ2] eaµ = (δξ − δλ − δε) e

aµ . (9.42)

The combination of symmetries that is in the right-hand side is reminiscent of the transfor-mations that we discussed in Ex. 4.2. This combination is a ‘covariant general coordinatetransformation’.

It is more difficult to calculate the SUSY commutator on the gravitino field largelybecause Fierz rearrangement is required. The result is

[δ1, δ2]ψµ = ξρ(Dρψµ −Dµψρ) + . . . . (9.43)

We will discuss the omitted terms . . . momentarily. First we note that, as done above, onecan manipulate the formula for a general coordinate transformation of a vector-spinor fieldto bring (9.43) to the same form as (9.42), namely

[δ1, δ2]ψµ = (δξ − δλ − δε)ψµ + . . . . (9.44)

The omitted terms . . . vanish when ψµ satisfies its equation of motion. Such terms do notaffect the commutator algebra on physical states, so they can be dropped for most purposes.Similar equation of motion terms also appear in the commutator algebra of global SUSY,after elimination of auxiliary fields, see Ex. 8.3 and the discussion which follows it. Theirpresence in (9.44) means that the local SUSY algebra also ‘closes only on-shell’. As in globalSUSY, the physical fields eaµ, ψµ are only an ‘on-shell multiplet’ which can be completed toan ‘off-shell multiplet’ by adding auxiliary fields (for N = 1, D = 4 only!). The auxiliaryfields are important in the formulation of systematic methods for the coupling of chiral andgauge multiplets to supergravity.

Finally we note that the field-dependent gauge transformations of this section can bereinterpreted as modifications of the algebra of local supersymmetry leading to the conceptof ‘soft algebras’, which will be discussed later.

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10 D = 11 supergravity

The basic N = 1, D = 4 supergravity theory discussed above, has been generalized in severalways. In four dimensions, one can couple the gravity multiplet (eaµ, ψµ) to gauge (AAµ , λ

A)and chiral (zα, PLχ

α) multiplets and promote the global SUSY theories of Ch. 8 to localSUSY.

One can also combine N − 1 gravitino multiplets consisting of (ψµ, Aµ) with the specificnumbers of gauge and chiral multiplets required for representations of N -extended super-symmetry algebras to construct extended supergravities for N ≤ 8.

Still another generalization of supergravity is to spacetime dimension D ≥ 5. The twodifferent classical D = 10 supergravities, called Type IIA and Type IIB respectively, are thelow energy limits of the superstring theories of the same name. Type IIB and gauged D = 5supergravities have important applications to the AdS/CFT correspondence. Most higherdimensional supergravities are quite complicated, but the maximum dimension D = 11theory has a relatively simple structure. It is an important theory for at least two reasons.First, many interesting lower dimensional cases, such as the maximal N = 8, D = 4 theory,can be obtained through dimensional reduction. Second, D = 11 supergravity, together withits M2- and M5-brane solutions is the basis of the extended object theory called M-theory,which is widely considered to be the master theory that contains the various string theories.Thus we devote the rest of this chapter to D = 11 supergravity.

10.1 D ≤ 11 from a dimensional reduction argument

Before we embark on a technical discussion of the theory, let’s review the argument whyD = 11 is the largest spacetime dimension allowed for supergravity. The argument is basedon a generalization of the dimensional reduction technique we discussed in Sec. 5.2. Therewe studied Kaluza-Klein compactifications of D+ 1-dimensional fields on MinkowskiD ×S1.That discussion did not include symmetric tensor fields, but it is clear that the Fouriermodes of a symmetric tensor hMN in D + 1 dimensions gives rise to symmetric tensor fieldshµνk, vector fields hµDk, and scalar fields hDDk in D dimensions. Here k is the Fourier modenumber.

More generally we can consider the compactification of a D′ dimensional theory on aproduct spacetime MD′ = MD × Xd, with D′ = D + d and Xd a compact d-dimensionalinternal space. Fields of the lower dimensional theory arise from harmonic expansion on Xd

of the various higher dimensional fields. In a Kaluza-Klein compactification one keeps theentire infinite set of harmonic modes, which describe both massless and massive fields in Ddimensions. In the related process called dimensional reduction one keeps only a finite setof modes which must be a consistent truncation of the full set. Usually the modes which arekept are the massless or light modes, and the omitted modes are heavy modes. A consistenttruncation is one in which the field equations of the omitted heavy modes are not sourcedby the light modes which are kept. Thus setting the heavy modes to zero is consistent withthe field equations.

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Ex. 10.1 Consider electrodynamics, i.e. the Lagrangian of electrons and photons

S[Aµ, Ψ,Ψ] =

∫dDx

[−1

4F µνFµν + Ψγµ(∂µ − (∂µ − iqAµ)Ψ

]. (10.1)

One may add or not a mass term for the fermions, but this is not relevant for the argument.Show that removing the fermions is a consistent truncation, while removing the Maxwellgauge field is not a consistent truncation. Remark that the action after consistent truncationis still gauge invariant, while the non-consistently truncated one is not. Thus in this example,the fermions are the ‘heavy’ fields in the terminology above, despite the fact that we kept themmassless.

To show that D = 11 is the maximal dimension, we consider the toroidal compactification ofa D′ dimensional theory to M4 × TD

′−4. Fourier modes on the torus are generalized Fouriermodes labelled by integers k1, . . . kD′−4. Only the lowest modes with ki = 0 are massless infour dimensions, and these are retained in the 4-dimensional theory. We don’t yet know thefull field content of the putative D′-dimensional supergravity, but we can anticipate that itmust contain the metric tensor and at least one gravitino of the simplest spinor type (e.g.Majorana) permitted in dimension D′.

We need to see what 4-dimensional fields arise from the gravitino in the truncation.18.Suppose that D′ = 11, in which the simplest spinor is a 32 component Majorana spinor.The 11 matrices ΓM which generate the Dirac-Clifford algebra in 11 dimensions can berepresented as products of 4× 4 γµ and 8× 8 γi as

Γµ = γµ × , µ = 0, 1, 2, 3 ,

Γi = γ∗ × γi , i = 4, 5, 6, 7, 8, 9, 10 . (10.2)

Here is the 8×8 unit matrix, and γ∗ = iγ0γ1γ2γ3, while the γi are the generating elementsof the Clifford algebra in 7 dimensional Euclidean space. In this basis the 11-dimensionalgravitino field is labelled as ΨM αa in which α = 1, 2, 3, 4 is a 4-dimensional spinor index anda = 1, . . . 8 is the index on which the γi (and their products) act.

From the 4-dimensional standpoint Ψµαa transforms under Lorentz transformations as aset of 8 gravitinos, while Ψi αa transforms as a set of 7 × 8 = 56 spin 1/2 fields. However,note that 8 Majorana gravitinos plus 56 Majorana is the complete fermion content of theκ = −2 (maximal spin 2) particle representation of the N = 8 SUSY algebra, see Sec. 8.4,table 3. If we started with a single gravitino in D′ ≥ 12 dimensions or more gravitinos in11 dimensions, then the dimensionally reduced theory would contain more than 8 gravitinoswhich can only be accommodated in a representation that involves spins ≥ 5/2 for which noconsistent interactions are known.

Ex. 10.2 Show that the product representation of the 11 Γµ matrices defined above doessatisfy ΓM , ΓN = 2ηMN . Compute commutators of the set of 28 matrices consisting of the

18In this section and the next, we use upper case M,N . . . to denote a vector index in D′ dimensions,i, j, . . . for a direction on the torus, and reserve µν, . . . for 4-dimensional vector indices.

81

21 independent γij = γ[iγj], plus the 7 matrices iγk and show that they are a basis for an 8dimensional representation of the Lie algebra of the SO(8) group that acts on the 8 gravitinosof the reduced theory.

10.2 The field content of D = 11 supergravity

The previous argument has taught us quite a bit about the theory. We know that it containsthe gravitational field whose quantum excitations (see Sec. 7.3) transform in the tracelesssymmetric tensor representation of SO(D−2) of dimension D(D−3)/2. In 11 dimensions thiscontains 44 bosonic states. There is also one Majorana spinor gravitino, whose excitations(see Sec. 5.1.2) transform in a vector-spinor representation of SO(D − 2) of dimension(D − 3) 2[(D−2)/2]. For D = 11 this contains 128 real fermion states.

The theory must contain an equal number of boson and fermion states, so we are missing84 bosons. Where are they? Recall the discussion in Sec. 6.6 that bosons in spacetimedimension D can be described by p-form gauge fields or equivalently anti-symmetric tensorAM1,M2,...,Mp potentials of rank p < D. The excitations of such a field transform in the rank p

anti-symmetric tensor representation of SO(D−2), which contains(D−2p

)states. For rank 3 in

11 dimensions, this contains exactly 84 quantum degrees of freedom. Thus, Cremmer, Julia,and Scherk, who first formulated supergravity in 11 dimensions, made the elegant hypothesisthat the theory should contain the metric tensor gMN , the 3-form potential AMNP , and theMajorana vector-spinor ΨM .

We already know that upon dimensional reduction on T 7, the vector-spinor producesthe 8 gravitinos and 56 spin 1/2 fermions of N = 8, D = 4 supergravity. The metrictensor components gµν give the 4-dimensional spacetime metric, while gµi produces 7 spinone particles, and gij contains 28 scalars. The N = 8 theory contains 28 vectors, and themissing 21 are supplied by the 3-form components Aµij. The form components Aijk contain35 scalars, and the components Aµνi give an additional 7 scalars. In this way the fieldassignment of Cremmer, Julia and Scherk accounts for the 35 scalars and 35 pseudoscalarsof the dimensionally reduced theory.19

10.3 Construction of the action and transformation rules

We now know the field content of the theory, and we need to be more precise and determinethe Lagrangian and transformation rules. To find them we start with an initial ansatz forthe action compatible with the expected symmetries and use some of the ideas of Ch. 8 andCh. 9 to finalize the construction. Several additional calculations are needed to demonstratelocal SUSY completely. We present some of these and refer readers to the literature for therest.

To start we note that a theory containing the 3-form potential Aµνρ must be invariantunder a gauge transformation involving a gauge parameter θνρ which is a 2-form. The theory

19The field components Aµνρ contain no degrees of freedom in D = 4.

82

thus involves a gauge invariant 4-form field strength Fµνρσ. The basic equations are

δAµνρ = 3∂[µθνρ] ≡ ∂µθνρ + ∂νθρµ + ∂ρθµν ,

Fµνρσ = 4∂[µAνρσ] ≡ ∂µAνρσ − ∂νAρσµ + ∂ρAσµν − ∂σAµνρ ,

∂[τFµνρσ] ≡ 0 . (10.3)

Another important ingredient we need is the exchange property of Majorana spinor bi-linears χΓAλ where ΓA is a general element of rank n of the Clifford algebra. As we saw in(3.39), we have the same properties as for 4 dimensions:

χγµ1µ2...µnλ = tnλγµ1µ2...µnχ , t0 = t3 = 1 , t1 = t2 = −1 , tn+4 = tn . (10.4)

We postulate that the action contains the universal graviton and gravitino terms of Sec.9.1 plus the covariant kinetic action for the 3-form plus additional terms (denoted by . . .)which we must find. Thus we write

S =1

2κ2

∫d11x e

[eaµebνRµνab − ψµγ

µνρDνψρ −1

24F µνρσFµνρσ + . . .

]. (10.5)

Initially we use the second order formalism with torsion-free spin connection ωµab(e). Wealso need transformation rules and make the ansatz

δeaµ = 12εγaψµ ,

δψµ = Dµε+(a γαβγδµ + b γβγδδαµ

)Fαβγδε ,

δAµνρ = −c εγ[µνψρ] = −13c ε(γµνψρ + γνρψµ + γρµψν) . (10.6)

For δAµνρ and the new terms of δψµ we have postulated general expressions consistent withcoordinate and gauge symmetries which contain the numerical constants a, b, c.

We will determine these constants and other useful information by temporarily treatingthe fields ψµ and Aµνρ as a free system with global SUSY in D-dimensional Minkowski space.The free action is (dropping here an irrelevant factor κ2)

S0 =1

2

∫d11x e

[−ψµγµνρ∂νψρ −

1

24F µνρσFµνρσ

]. (10.7)

Of course there is no true supermultiplet containing only ψµ and Aµνρ, but for a free theory,this need not be an obstacle. Using (3.41) we have

δψµ = ε(−aγαβγδµ + bγβγδδαµ

)Fαβγδ , (10.8)

we compute the variation

δS0 =

∫d11x ε

[(aγαβγδµ − bγβγδδαµ

)Fαβγδγ

µνρ∂νψρ − 16cγνρψσ∂µF

µνρσ]

=

∫d11x ε

[(−aγαβγδµ + bγβγδδαµ

)∂νFαβγδγ

µνρψρ − 16cγνρψσ∂µF

µνρσ]. (10.9)

83

We used partial integration to obtain the last line, assuming that ε is constant as in globalSUSY.

We need matrix identities to reduce the products of γ matrices in (10.9) to sums overrank 6, rank 4, and rank 2 elements ΓA of the Clifford algebra. In the spirit of the discussionat the end of section 3.1.2 (the first identity is (3.23)), we write

γαβγδµγµνρFαβγδ = (D − 6)γαβγδνρFαβγδ + 8(D − 5)γαβγ[νF ρ]

αβγ − 12(D − 4)γαβFαβνρ ,

γβγδγµνρFµβγδ = −γνραβγδFαβγδ − 6γαβγ[νF ρ]αβγ + 6γαβFαβ

νρ . (10.10)

Ex. 10.3 Conscientious readers should verify these identities.

When inserted in the first term of (10.9) both rank 6 terms vanish due to the Bianchi identity.The sum of the two rank 4 contributions must vanish while the rank 2 terms must cancel withthe second term of (10.9). These conditions lead to the following two numerical relationsamong a, b, c:

8(D − 5)a+ 6b = 0 ,

12(D − 4)a+ 6b = 16c . (10.11)

It is only the case D = 11 which is relevant here, and the solution of (10.11) in this case isa = c/216, b = −8a. Thus for the free theory we have found the transformation rules

δψµ = ∂µε+c

216

(γαβγδµ − 8γβγδδαµ

)Fαβγδε ,

δAµνρ = −cεγ[µνψρ] . (10.12)

To fix the remaining parameter c, we examine the commutator of two SUSY transforma-tions and require that this agree with the local supergravity algebra discussed in Sec. 9.5.It is simplest to work with the gauge potential, so we write (for constant ε)

[δ1, δ2]Aµνρ = − 1

216c2ε2γ[µν

(γαβγδρ] − 8γβγδδαρ]

)ε1Fαβγδ − (1 ↔ 2) . (10.13)

It would be good practice to work out the detailed identities for the products of γ-matricesin (10.13), which involve contributions from Clifford elements of rank 1, 3, 5, 7, but this taskcan be simplified by the following observations:i. since the spinor parameters ε1, ε2 are anti-symmetrized, only the rank 1 and rank 5 termscan contribute.ii. the first product has no rank 1 part. The rank 5 part should cancel between the twoterms.iii. except for index changes, the second product is already given in (3.21).

The SUSY algebra must contain a spacetime translation involving the rank 1 bilinearε1γ

σε2 from the product γµνγβγδ. Using (3.21) we obtain the rank 1 contribution to the

commutator

[δ1, δ2]Aµνρ = −4

9c2ε1γ

σε2Fσµνρ . (10.14)

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The rank 5 contribution actually cancels between the two terms of (10.13) as the followingexercise shows.

Ex. 10.4 Show that the rank 5 terms cancel in the expression γ[µν

(γρ]

αβγδ − 8γβγδδαρ]

)Fαβγδ.

The interpretation of the result (10.14) is straightforward if we refer to the detailed formof the field strength in (10.3). The first term ∂σAµνρ is the spacetime translation we arelooking for, while the remaining terms just add up to a gauge transformation of the 3-formpotential with field-dependent gauge parameter proportional to θµν = −ε1γσε2Aσµν . Suchfield-dependent gauge transformations were already found in SUSY gauge theories, see Ex.8.4, and were found also in the local algebra in Sec. 9.5. We must normalize the coefficientof the translation term to agree with these results, which were (and should be) uniform forall fields. Thus we fix the parameter c2 = 9/8, and we choose the positive root c = 3/2

√2.

Recall that we have been studying the global supersymmetry of the free system of ψµand Aµνρ in flat spacetime. There is another important piece of information from that study,namely the effective supercurrent of the system, obtained by allowing the spinor parameterε to depend on xµ. After partial integration we find that δS0 contains a term proportionalto Dνε whose coefficient is the supercurrent

J ν =

√2

192

(γαβγδνρFαβγδ + 12γαβFαβ

νρ)ψρ . (10.15)

As we will see below this provides a new term in the D = 11 supergravity action.Our results on the free ψµ, Aµνρ system are valid for the interacting supergravity theory

if we introduce a general frame field eaµ(x) and consider general ε(x). We then have thetransformation rules

δeaµ =1

2εγaψµ ,

δψµ = Dµε+

√2

288

(γαβγδµ − 8γβγδδαµ

)Fαβγδε ,

δAµνρ = −3√

2

4εγ[µνψρ] , (10.16)

and the action

S =1

2κ2

∫d11x e

[eaµebνRµνab − ψµγ

µνρDνψρ −1

24F µνρσFµνρσ

−√

2

96ψν

(γαβγδνρFαβγδ + 12γαβFαβ

νρ)ψρ + . . .

]. (10.17)

The previous discussion ensures that all terms in δS of the form εRµνabψρ and εFαβγδψρcancel. The curvature terms vanish by the universal manipulations of Sec. 9.1. For termslinear in Fαβγδ, the calculations done above in the free limit are essentially the same in ageneral background geometry. The only new feature is the term (Dν ε)J ν which is canceled

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by the δψν = Dν ε variation of the last term written in (10.17). We still leave . . . in (10.17)because the action is not yet complete.

The next step involves variations of the Lagrangian of order eF 2ψ which come from theframe field variation of the order F 2 term in L and the δψ ∼ Fε variation of the order ψFψterm. After calculations that we will omit here, one finds that a term with a rank 9 Cliffordmatrix,

δLFF + δLψFψ = − 1

16 · 144εγα

′β′γ′δ′αβγδρψρ Fα′β′γ′δ′Fαβγδ , (10.18)

survives, and we must find a way to cancel it.To get rid of the high-rank gamma matrix, recall the discussion of Sec. 3.1.4 of the Dirac-

Clifford algebra for spacetimes of odd dimension D = 2m + 1. For D = 11, the generatingmatrices are the 32 × 32 matrices γ0, γ1, . . . γ2m−1, γ2m = γ∗, with γ∗ = γ0γ1 . . . γ9 (i.e. wetake the + sign in (3.30). The rank 9 clifford element is related to rank 2 by (3.31):

γα′β′γ′δ′αβγδρ = − 1

2eεα

′β′γ′δ′αβγδρµν γνµ . (10.19)

This allows to rewrite (10.18) as

e(δLFF + δLψFψ

)=

1

32 · 144εα

′β′γ′δ′αβγδρµν εγνµψρ Fα′β′γ′δ′Fαβγδ ,

=4

3√

2 · 32 · 144εα

′β′γ′δ′αβγδρµν(δAµνρ)Fα′β′γ′δ′Fαβγδ . (10.20)

This suggests that one can add a term in the Lagrangian to cancel this variation:

SC−S = −√

2

(144κ)2

∫d11x εα

′β′γ′δ′αβγδµνρFα′β′γ′δ′FαβγδAµνρ ,

= −√

2

6κ2

∫F (4) ∧ F (4) ∧ A(3) , (10.21)

where we used the form notation in the last line, with F (4) = dA(3), which simplifies theformula. Such a term has been called a Chern-Simons term, and has some special properties.First of all, using integration by parts one sees that a variation of A(3) gives 3 similar termsas

δ

∫F (4) ∧ F (4) ∧ A(3) =

∫ [2dδA(3) ∧ F (4) ∧ A(3) + F (4) ∧ F (4) ∧ δA(3)

]= 3

∫F (4) ∧ F (4) ∧ δA(3) , (10.22)

where we used the Bianchi identity dF (4) = 0. This shows how such a variation of (10.21)cancels (10.20). Further, this term does not produce other variations, as there are no framefields in the expression.

One might also wonder whether such a term is invariant under gauge transformationsof the form (10.3) in view of the explicit presence of the gauge field. For that one notes

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that this transformation can be written as δA(3) = dθ(2). Using similar integrations by partsand the Bianchi identity as above, the gauge invariance is guaranteed. These are typicalproperties of such Chern-Simons actions.

Finally, there is still the issue of which spin-connection to use, but we will not continuewith that derivation in these lectures, but give the result. The final result is obtained by

• replacing in the covariant derivative of the gravitino ω by 12(ω + ω(e)), where ω is the

spin connection with torsion. It is the solution of

D[µ(ω)eaν] = 14ψµγ

aψν . (10.23)

• In the Noether term, we should replace the field strength Fµνρσ by 12(Fµνρσ + Fµνρσ)

where F is the covariant field strength

Fµνρσ = 4 ∂[µAνρσ] +3

2

√2 ψ[µγνρψσ] . (10.24)

It is this covariant field strength that enters in the final supersymmetry transformationlaw of the gravitino.

The commutators of the supersymmetry transformations lead to

[δQ(ε1), δQ(ε2)] = δgct(ξµ) + δL(λab) + δQ(ε3) + δA(θµν) , (10.25)

where the terms are respectively general coordinate transformations, local Lorentz transfor-mations, supersymmetry and the gauge transformation of the 3-index form in (10.3). Theparameters in (10.25) are

ξµ = 12ε2γ

µε1 ,

λab = −ξµωµab +1

288

√2ε1

(γabµνρσFµνρσ + 24γµνF

abµν)ε2 ,

ε3 = −ξµψµ ,θµν = −ξρAρµν + 1

4

√2ε1γµνε2 . (10.26)

This algebra is satisfied on the gravitino only modulo field equations. This is thus an examplewhere we have no auxiliary fields. Despite much research for a solution of this problem, aset of auxiliary fields such that the algebra (10.25) is satisfied on all fields has never beenfound. Such absence of auxiliary fields also implies that there are 4-ghost interactions in thequantum theory.

The first terms in (10.26) are by now familiar as a covariantization of general coordinatetransformation. But apart from these we have here some extra Lorentz transformation and agauge transformation of the 3-form. These extra terms are either terms with rank 6 gammamatrices (which is equivalent with rank 5 in 11 dimensions) and rank 2 gamma matrices.When a solution is such that these expressions are non-vanishing, these terms act as centralcharges in the algebra. These central charges are important to allow BPS solutions withM5 and M2 branes, respectively. We will leave this important aspect to later studies of thereader.

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11 General gauge theory

In the previous chapters we studied the construction of the simplest supergravity theories,namely D = 4 and D = 11, N = 1 supergravity, and we proved invariance under localsupersymmetry. But N = 1 supergravity in 4 dimensions contains much more. Chiralmultiplets and gauge multiplets of global supersymmetry (see chapter 8) can be coupled tosupergravity yielding the rich structure of matter-coupled N = 1 supergravities with manyapplications.

To investigate matter-coupled theories, we will use more advanced methods, which willbe developed in the chapters ahead. To prepare for this, we will sharpen our knives in thischapter and discuss a general formulation of gauge theory of the type that we will needfor supergravity. This consists largely of the refinement and extension of the notations andconcepts used for symmetries earlier in this book. We will formalize several manipulationsand concepts that we encountered in the previous sections. This will allow us to performcalculations with covariant derivatives more effectively. However, we will also see how thesupergravity transformation rules postulated in (9.5),(9.6) are actually determined by thestructure constants of the Poincare supersymmetry algebra. In Sec. 11.3, we will explainhow the general rules of gauge theory can be applied for gravity theories. We will see thatsome formulas have to be modified.

11.1 Symmetries

We defined symmetries as maps of the configuration space such that solutions of the equationsof motion are transformed into new solutions. As we did there, we restrict the discussionto continuous symmetries that leave the action invariant. We consider only infinitesimaltransformations, and hence are concerned with the Lie algebra rather than the Lie group.However, we need also extensions of these mathematical concepts, e.g. allowing structureconstants to depend on fields and thus becoming rather ‘structure functions’. We use thesame formalism to describe both spacetime and internal symmetries and also supersymme-tries.

11.1.1 Global symmetries

The algebra. An infinitesimal symmetry transformation is determined by a parameter,which we denote in general by εA and an operation δ(ε), which depends linearly on the pa-rameter, and acts on the fields of the dynamical system under study. For global symmetries,often called rigid symmetries, the parameters do not depend on the spacetime point wherethe symmetry operation is applied. Since the symmetry operation is linear in ε we can writeit in general as

δ(ε) = εATA , (11.1)

in which TA is an operator on the space of fields. It describes the symmetry transformationwith the parameter stripped. We call TA the field space generator of the transformation. Thenotation of (11.1) and the formalism we develop below apply to all the types of symmetries

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encountered in this book. Internal, spacetime, and supersymmetry can be viewed as specialcases. However, to be concrete, we begin by rephrasing the discussion of (linearly realized)internal symmetry of Sec. 2.1 in the present notation. (See also Sec. 4.2.1.) Then we willdiscuss how spacetime and supersymmetries fit the pattern.

Suppose that the matrices (tA)i j are the matrix generators of a representation of a Liealgebra with commutator

[tA, tB] = fABCtC , (11.2)

and suppose that the system contains a set of fields φi that transforms in this representation.Then the transformation rule (2.6) can be expressed as the action of the generator TA asfollows:

TAφi = −(tA)ijφ

j . (11.3)

The Lie algebra is determined by the commutator of two such transformations, so itis important to define the product of two symmetry operations carefully. By the productδ(ε1)δ(ε2)φ

i, we mean that we first make a transformation with parameter ε2 followed byanother with parameter ε1. In the notation of (11.1), the product operation reads

δ(ε1)δ(ε2)φi = εA1 TA

(εB2 TBφ

i). (11.4)

As discussed in section 2.1, a symmetry transformation acts on the fields, which are thedynamical variables of the system, and not on the matrices which are the result of a priortransformation. Therefore, using (11.3), the explicit form of the product operation becomes

δ(ε1)δ(ε2)φi = εA1 TAε

B2

[−(tB)ijφ

j]

= εA1 εB2 (−tB)ijTAφ

j

= εA1 εB2 (−tB)ij(−tA)jkφ

k . (11.5)

It is important to realize that in the second line, the transformation operator acts on thefield φj, and not on the ‘numbers’ (tB)ij. This is similar to the way in which we calculatecommutators as e.g. in (9.37) where δ1 is similar to the TA here and acts on ψµ and goesafter γa. The commutator is then

[δ(ε1), δ(ε2)]φi = εB2 ε

A1 [TA, TB]φi

= εB2 εA1 ([tB, tA]φ)i = −εB2 εA1 fABC(tCφ)i

= εB2 εA1 fAB

CTCφi . (11.6)

Therefore the commutator of the generators TA, namely

[TA, TB] = fABCTC , (11.7)

conforms to the matrix commutator (11.2). The minus sign in the transformation rule (11.3)is necessary for having the same result in (11.7) as in (11.2), and it is a general feature ofsymmetry transformations defined by left multiplication by a matrix.

Although every case of internal symmetry in which the symmetry transformations actlinearly on the fields (linearly realized internal symmetry) can be expressed by the matrix

89

transformations, it is sometimes convenient to use the more general notation with TA. Forexample the transformation rules for fields Ψα in a complex representation of a compactsymmetry group, their conjugates Ψα, and fields φB in the adjoint representation were definedin (4.33). In each case we obtain the generator TA by stripping the parameter εA and write

TAΨα = −(tA)αβΨβ ,

TAΨα = Ψβ(tA)βα ,

TAφB = −fACBφC . (11.8)

Here the matrix generator tA acts by left multiplication on Ψα, but by right multiplication onΨα.20 On adjoint fields we again have left action by the matrix generators (tA)BC = fAC

B.

Ex. 11.1 Show that (11.7) holds for repeated symmetry transformations of Ψα and φB.Recall that the second transformation always acts only on the fields.

Our conventions are easily extended to include spacetime symmetries. For Lorentz transfor-mations we obtain the field space generator Mµν for scalar fields and for spinor fields:

Mµνφ(x) = −Lµνφ(x) ,

MµνΨ(x) = −(Lµν + 12γµν)Ψ(x) . (11.9)

The differential operator Lµν is defined in (2.20). Its commutators realize the Lie algebra(2.16) of the Lorentz group. The same is true of the matrix generators Σµν = 1

2γµν for the

spinor representation. For global translations we extract from (2.25)

Pµ = ∂µ =∂

∂xµ, (11.10)

which acts by differentiation on fields belonging to any representation of the Lorentz group.The commutators of the field space generators Mµν and Pµ conform to the Lie algebra(2.26) of the Poincare group (provided they are calculated with the convention that thesecond transformation acts only on the fields resulting from the first).

Ex. 11.2 Readers should verify this.

Let’s return to the equations (11.5) and (11.6). The order in which the symmetry pa-rameters are written is irrelevant for internal symmetries since the parameters commute.However, the order written in (11.6) will also be valid for fermionic symmetries for whichthe parameters anti-commute. Indeed we now proceed to formulate global supersymmetryin the present framework.

Here we need the notation with spinor indices, explained in Sec. 3.4. Remember thatbarred spinors are indicated with an upper spinor indices and ordinary spinors with a lowerspinor index. Indices are raised and lowered with the charge conjugation matrix. Symmetries

20Because the matrices tA are anti-Hermitian for a compact group, this is equivalent to left action by−(tA)†. See Sec. 4.2.1.

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of gamma matrices are determined by (3.57), such that e.g. in 4 dimensions (γµ)αβ issymmetric, and flipping indices up-down gives a sign according to (3.58), giving e.g. a minussign in 4 dimensions.

For supersymmetry the generators TA are replaced by the four-component Majoranaspinor Qα and the parameters εA by the anti-commuting conjugate (Majorana) spinor εα. Asupersymmetry transformation is an operation on fields denoted by

δ(ε) = εαQα . (11.11)

As an explicit example of supersymmetry we rewrite the transformation rule (8.2) of thescalar field Z(x) of a chiral multiplet:

QαZ =1√2

(PLχ)α . (11.12)

In parallel with (11.4), we write the product of two SUSY transformations as

δ(ε1)δ(ε2) = (ε1)αQα(ε2)

βQβ = (ε2)β(ε1)

αQαQβ . (11.13)

Note that there is no sign change when (ε2)β is moved through the bosonic quantity (ε1)αQα.

It is then easy to check that the commutator of two transformations is

[δ(ε1), δ(ε2)] = (ε2)β(ε1)

α (QαQβ +QβQα) . (11.14)

When applied to any field of a supersymmetric theory this result has the same structure asthe first line of (11.6), except that the anticommutator appears. Of course, this is entirelyexpected for the fermionic elements of a superalgebra, whose structure relation reads

Qα, Qβ = fαβCTC , (11.15)

with structure constants fαβC , which are symmetric in α, β and connect fermionic elements

to a sum of bosonic elements TC .In Sec. 8.2.2, we computed the commutator of SUSY variations on the fields of a chiral

multiplet and found

[δ(ε1), δ(ε2)] = −12ε1γ

µε2Pµ = 12ε2γ

µε1Pµ = −12εβ2 (γµ)βαε

α1Pµ , (11.16)

where Pµ acts on the fields of the chiral multiplet as ∂µ. Thus the structure constants ofsupersymmetry are given by

Qα, Qβ = −12(γµ)αβPµ , → fαβ

µ = −12(γµ)αβ = fβα

µ . (11.17)

Finally we would like to show that the treatments of supersymmetry and internal sym-metry can be united in a common notation. Toward this end we insert the definition (11.15)in (11.14) and obtain

[δ(ε1), δ(ε2)] = (ε2)β(ε1)

αfαβCTC . (11.18)

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This relation has the same structure as the last line of (11.6). Thus the result

[δ(ε1), δ(ε2)] = δ(εC3 = εB2 εA1 fAB

C) , (11.19)

holds for both bosonic and fermionic symmetries. The presence of a fermionic symmetry issignalled only by the fact that its parameter (called εα for supersymmetry) is anticommuting.The left side of (11.19) then contains the necessary anticommutator of the generators as in(11.14).

A superalgebra also contains structure relations of bosonic and fermionic elements, andthey are realized by commutators of the generators. In supersymmetry the commutator ofa Lorentz generator and a supercharge component was defined in (8.1). Here is a relevantexercise for the reader.

Ex. 11.3 Use (11.12) and (11.9) to show that the commutator of the field space generatorsMµν and Qα conforms conforms to (8.1) when acting on Z(x), specifically[

Mµν, Qα

]Z = −1

2(γµν)α

βQβZ . (11.20)

The nonlinear σ-model and Killing symmetries. Internal symmetries do not alwaysact linearly. We now consider symmetries in the non-linear σ-model, which was discussedin Sec. 6.9. It contains scalar fields that transform as in (6.82), where the Killing vectorskiA(φ) should satisfy (6.81). Hence, in this case the symmetry generator is

TAφi = kiA(φ) . (11.21)

The commutators of Killing vector fields determine the Lie algebra of the isometry groupof Mn. The Lie bracket, which is the commutator of the differential operators kA, reads asin (6.85)

[kA, kB] = fABCkC , (11.22)

or, in components,kjA∂jk

iB − kjB∂jk

iA = fAB

CkiC . (11.23)

It is then clear that the commutator of the field space generators TA conforms to (11.7). Thefollowing exercise confirms this.

Ex. 11.4 Use (11.21) to obtain

δ(ε1)δ(ε2)φi = δ(ε1)ε

A2 k

iA = εA2 ε

B1 k

jB∂jk

iA . (11.24)

Use (11.23) to show that for any smooth scalar function f(φi) on Mn,

[TA, TB]f(φi) = fABCTCf(φi) . (11.25)

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11.1.2 Local symmetries

Gauge fields. We now consider gauge symmetries and use the notation of the previoussection in which symmetries are indexed by A,B,C . . . . For each symmetry there is a fieldspace generator TA, but the parameters εA(x) are arbitrary functions on spacetime. Torealize local symmetry in Lagrangian field theory, one needs a gauge field, which we willgenerically denote as BA

µ (x), for every gauged symmetry. The gauge fields transform as

δ(ε)BµA ≡ ∂µε

A + εCBµBfBC

A . (11.26)

It is easy to check that these transformations satisfy the algebra (11.19). This definition ismodeled on the transformation of Yang-Mills fields in (4.37), but the definition is valid forinternal and spacetime symmetries and for supersymmetry. The gauge fields for each casetransform in the same way, although each case has its own specific notation in which thegeneric index A is appropriately changed.

Almost exclusively, we will be applying (11.26) in theories of gravity. Therefore, wedistinguish between coordinate indices µ, ν, ρ . . . and local frame indices a, b, c, . . ., and wenow use Mab and Pa to denote the generators of local Lorentz transformations and transla-tions. There are some subtleties for local translations, which are actually general coordinatetransformations, which we will discuss in section 11.3, but the major ingredients of our treat-ment of gauge symmetries are developed in this chapter. Table 5 indicates both the genericnotation and the particular cases we are concerned with.21

Table 5: Gauge transformations, parameters and gauge fields

generic gauge symmetry parameter gauge fieldTA εA BA

µ

local translations Pa ξa eaµ

Lorentz transformations Mab λab ωµab

Supersymmetry Qα εα ψαµ

Internal symmetry TA θA AAµ

The structure constants fBCA are the important data in the formula fBC

A, and we extractthem from the symmetry algebras in each specific case. Table 6 gives examples for the mostcommon (anti)commutators (remember that the calculation takes into account factors of 2in summations over pairs of antisymmetric indices as in ex. 2.1)

21The notation for internal symmetry is very similar to the general case, but the distinction will be clearfrom the context of each application. For the other symmetries, the index A is replaced by either an indexa, or an antisymmetric pair ab or a spinor index. E.g. ξa = ea

µξµ is the parameter of translations. Our

convention for spinors is that barred spinors carry an upper index. Thus we have written barred spinors inthe table. But we consider only Majorana spinors in supersymmetry, so they are uniquely related to ε orψµ.

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Table 6: Useful commutators, structure constants and third parameter in the commutationrelation (11.19).

(anti)commutators structure constants third parameter[Mab,Mcd

]= 4η[a[cMd]b] fabcd

ef = 8η[c[bδ[ea]δ

f ]d] λab3 = −2λ1

[acλ2

b]c[Pa,Mbc

]= 2ηa[bPc] fa,bc

d = 2ηa[bδdc] ξa3 = −λab2 ξ1b + λab1 ξ2b

[Pa, Pb] = 0

Qα, Qβ = −12(γa)αβPa fαβ

a = −12(γa)αβ ξa3 = 1

2ε2γ

aε1[Mab, Q

]= −1

2γabQ fab,α

β = −12

(γab)αβ ε3 = 1

4λab1 γabε2 − 1

4λab2 γabε1

[Pa, Q] = 0

Ex. 11.5 The spin connection field ωµab transforms under Lorentz transformations22 as

δωµab = ∂µλ

ab + 2ωµc[aλb]c . (11.27)

Check that this equation corresponds to (11.26). The gauge field BAµ is then replaced by ωµ

ab.You must replace each of the indices A,B,C by antisymmetric pairs ab, etc., and insertfactors 1

2as in Ex. 2.1.

When we started with N = 1 supergravity, we made an ansatz for the transformationsof the frame field and the gravitino in (9.5),(9.6). However, now we can consider this againin the context of the super-Poincare algebra and it turns out that this ansatz is actuallydetermined by (11.26) as the following exercise shows.

Ex. 11.6 Use (11.26) to calculate the supersymmetry transform of the vierbein eaµ, usinginformation from table 6. As the exercise asks only for the supersymmetry transformation,the index on the parameter can be restricted to a spinor index α. On the other hand, the Aindex is a for translations. Therefore, the first term of (11.26) does not contribute. Fromthe second term you should obtain (9.5). Consider now in the same way the transformationof the gravitino and re-obtain (9.6).

Soft algebra. In supersymmetry we often encounter a generalization of standard Lie al-gebras, which we call ‘soft algebras’. This means that the usual structure constants fAB

C

can depend on fields and are thus called ‘structure functions’. We already encountered thisphenomena when we studied the commutator of SUSY transformations in supersymmetricgauge theories. Consider the result (8.24), which we can write as

[δQ(ε1), δQ(ε2)] = δP(ξ)− δgauge(θA = AAµ ξ

µ) , ξµ = 12ε2γ

µε1 . (11.28)

22The orbital part of Lorentz transformations is not included here. Why this can be omitted will beexplained in section 11.3.1.

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The first term is a translation, which acts as ξµ∂µ on the fields. The second term wascalled a field dependent gauge transformation in Ch. 8. However, we now interpret it as amodification of the SUSY algebra in which the commutator of supercharges contains a gaugetransformation giving the new ‘structure function’

fαβA = −1

2AAµ (γµ)αβ , (11.29)

which depends on the gauge field Aµ.The extension to soft algebras does not affect most of the formulas we have developed

for gauge symmetries. Jacobi identities are modified due to the variation of the structurefunctions, but we will not need this explicitly. For the interested reader we point out thatthis type of generalized gauge theory is quite naturally described in the framework of theBatalin-Vilkovisky or field-antifield formalism.

When one has a solution of the field equations of the full theory, one may plug in thevalues of the fields into this soft algebra. That leads to a new honest Lie algebra, whichencodes the remaining symmetries for that particular solution. This is e.g. the way in whichthe anti-de Sitter algebra turns up when one has certain solutions of supergravity.

Zilch symmetries. A zilch symmetry is one whose transformation rules vanish on solu-tions of the equations of motion. The corresponding Noether current then vanishes. Anyaction with at least two fields has zilch symmetries. Indeed consider an action S(φ) contain-ing fields φi. Consider the transformation

δφi = εhijδS

δφj, (11.30)

for any antisymmetric matrix hij = −hji. One finds that

δS =δS

δφiεhij

δS

δφj= 0 . (11.31)

We assumed bosonic fields here, but the idea extends to fermions if we change the symmetryproperties of h. This is in fact the way that we encountered such a zilch symmetry in (8.13).In that case the hij is represented by vµ(γµ)αβ, which is symmetric in the indices α, β thatrefer to the spinor fields. The transformation (8.13) is a zilch symmetry as it vanishes if thefield χ satisfies the equation of motion.

The existence of zilch symmetries implies that symmetries are not uniquely defined. Onecan add to any symmetry transformation TAφ

i a zilch symmetry hijAδSδφj , and it is still a

symmetry. We view this as a change of basis in the algebra of symmetries.

Open algebra As we saw already in Ch. 8, zilch symmetries sometimes occur in thecommutators of conventional symmetries such as SUSY. We found this in the algebra ofsupersymmetry transformations of a chiral multiplet after elimination of the auxiliary field.See Ex. 8.3 and the following discussion.

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From first principles it follows that the commutator of two symmetries of the action isalso a symmetry. The result of the commutator can thus be expanded in the set of allsymmetries, as indicated in (11.19). However, this expansion may also include these zilchsymmetries.

Thus, in a general gauge theory, a commutator of symmetries such as (11.19) will only bevalid after equations of motion are used. In this case algebra of field transformations is saidto be closed only ‘on-shell’. If one has a basis such that (11.19) are valid without use of theequations of motions, then the commutator closes for any configuration of fields, whetherthe equations of motion are satisfied or not. In this case we say that the algebra is closed‘off-shell’.

In the situation of Ex. 8.3, we could obtain off-shell closure by keeping the auxiliary fields.This is one of the advantages of using auxiliary fields. However, for extended supersymmetry,or for supersymmetry in higher dimensions, it is often not possible to find suitable auxiliaryfields to obtain off-shell closure. One must then be careful to check whether commutatorformulas hold on- or off-shell.

11.2 Covariant derivatives and curvatures

We now want to consider field theories in which the Lagrangian contains both gauge fieldsBAµ and other fields φi whose transformation rules,

δ(ε)φi(x) = εA(TAφi)(x) . (11.32)

do not involve derivatives of the gauge parameters. Readers are already familiar with theimportant case of gauged non-abelian internal symmetry discussed in Sec. 4.2.2. In thatsection we saw how to define and use covariant derivatives and field strengths as the buildingblocks of the theory which also transform without derivatives of the εA. The Yang-Millsaction (4.42) is directly constructed from such covariant building blocks.

In this section we want to define such quantities within the more general frameworkconsidered in this chapter. We begin with the definition

A covariant quantity is a local function that transforms under all local sym-metries with no derivatives of a transformation parameter.

Our considerations apply in a straightforward way to gauged internal symmetry, Lorentztransformations and SUSY, but local translations require special care and are mainly dis-cussed in Sec. 11.3.

One very important covariant quantity is the covariant derivative of a field φi (whetherelementary or composite) whose gauge transformation rule is of the form in (11.32). Theordinary derivative ∂µφ

i is certainly not a covariant quantity since

δ(ε)∂µφi = εA∂µ(TAφ

i) + (∂µεA)TAφ

i . (11.33)

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The second term spoils covariance but we can correct for this by adding a term involvingthe gauge fields and defining

Dµφi ≡ (∂µ − δ(Bµ))φi

=(∂µ −BA

µ TA)φi . (11.34)

The notation δ(Bµ) means that the covariant derivative is constructed by the specific pre-scription to subtract the gauge transform of the field with the gauge field itself as thesymmetry parameter. As is clear from the second line of (11.34) there is a sum over allgauge transformations of the theory. As an example of this construction we rewrite thespinor covariant derivative of (4.38) and transformation rule in the present notation23

δ(θ)Ψ = −θAtAΨ , DµΨ =(∂µ + AAµ tA

)Ψ . (11.35)

It is easy to check using (11.26) that the gauge transformation of the covariant derivativedoes not contain derivatives of the gauge parameters. Hence, the covariant derivative is acovariant quantity. We now prove the stronger result that gauge transformations commutewith covariant differentiation on fields φ for which the algebra is closed24. Thus we can apply(11.19). We first write an auxiliary relation which is (11.19) applied to φ as in (11.32), withε1 replaced by Bµ and ε2 by ε:

εAδ(Bµ)(TAφ)−BAµ δ(ε)(TAφ) = εBBA

µ fABC(TCφ) . (11.36)

We then write, using (11.26) in the second line and (11.36) in the third,

δ(ε)Dµφ = ∂µ(εA(TAφ)

)−BA

µ δ(ε)(TAφ)−(δ(ε)BA

µ

)(TAφ)

= εA∂µ(TAφ)−BAµ δ(ε)(TAφ)− εCBB

µ fBCA(TAφ)

= εADµ(TAφ) . (11.37)

Hence,

Gauge transformations and covariant derivatives commute on fields on which thealgebra is satisfied.

In summary we repeat the definition of the covariant derivative in slightly more general form.

The covariant derivative of any covariant quantity is given by the operator

Dµ = ∂µ − δ(Bµ) , (11.38)

acting on that quantity. The instruction δ(Bµ) means compute all gauge trans-formations of the quantity, with the potential BA

µ as the gauge parameter.

23The gauge coupling of Ch. 4 is now set to g = 1.24Remember that this is not a trivial statement, see the remarks above on open algebras. Thus, e.g. in

the chiral multiplet without auxiliary fields, (8.13) implies that the statements below do apply on Z, butnot for χ.

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Ex. 11.7 Symmetries of the nonlinear σ-model are generated by Killing vectors kiA(φ) whichin general determine a Lie algebra as defined in (11.22). The elementary fields φj are localcoordinates of the target space. Suppose that the symmetry is gauged. Show that the covariantderivative Dµφ

i = ∂µφi − AAµk

iA transforms as

δDµφi = θADµk

iA = θA

(∂jk

iA

)Dµφ

j . (11.39)

We can use the covariant derivative to define the next important set of quantities inany gauge theory of an algebra. For each generator of the algebra the curvature Rµν

A is asecond rank antisymmetric tensor which is also a covariant quantity. In Yang-Mills theorythe curvature was called a field strength and was defined in (4.39),(4.40) as a commutatorof covariant derivatives acting on a covariant field. We now proceed in the same way in thismore general framework. Using (11.37) and (11.36) (with now ε replaced by Bν), we obtain

[Dµ,Dν ] = −δ(Rµν) ,

RµνA = 2∂[µBν]

A +BνCBµ

BfBCA . (11.40)

Here again δ(Rµν), means that one takes a sum over all gauge symmetries and replaces theparameters εA with Rµν

A.Curvatures are covariant quantities, which transform as

δ(ε)RµνA = εCRµν

BfBCA . (11.41)

They satisfy Bianchi identities, which are, using the definition (11.38),

D[µRνρ]A = 0 . (11.42)

The curvatures and Bianchi identities discussed in Sec. 6.8 are a special case of (11.40)in which the Lie algebra is the algebra of the Lorentz group so(D − 1, 1). Here is a smallexercise to verify this.

Ex. 11.8 Obtain the curvature tensor (6.63) for Lorentz symmetry using the approach ofthis chapter. Use the definition (11.40) and the structure constants in table 6.

We mentioned earlier that gauged local translations, generated by Pa, are more subtle andrequire special treatment. The following exercise will illustrate this.

Ex. 11.9 Translations occur on the right-hand side of the commutators [Mab, Pc] and Qα, Qβ,see table 6. Use this information to show that the curvature associated to the translation gen-erator P a is

Rµνa = 2∂[µeν]

a + 2ω[µabeν]b − 1

2ψµγ

aψν . (11.43)

The last term is the torsion tensor of N = 1, D = 4, see (9.15), which, by (6.57), is theantisymmetric part of the connection. Hence, the latter equation is just the antisymmetricpart of (6.51) in [µν], which is equivalent to the first Cartan structure equation (6.43).Therefore, the curvature for spacetime translations vanishes:

Rµνa = 0 . (11.44)

The last exercise indicates already that something is special when local translations areincluded. In the next section we will discuss the special features of such theories.

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11.3 Gravity as a gauge theory

The main issue that we have to address here is the reconciliation of the concept of gaugetheories developed in chapter 11 with general coordinate transformations, which we studiedin chapter 6.

11.3.1 Gauge transformations for the Poincare group

The Lie algebra of the Poincare group includes both translations Pa and Lorentz transfor-mations Mab. In theories of gravity both symmetries are gauged by parameters ξa(x) andλab(x), which are arbitrary functions on the spacetime manifold.

Special issues arise for local translations. The first issue is quite simple. Under globaltransformations of the Poincare group, it was shown in Ch. 2 that a scalar field in Minkowskispacetime transforms as

δ(a, λ)φ(x) =[aµ∂µ − 1

2λµνLµν

]φ(x) = [aµ + λµνxν ] ∂µφ(x) . (11.45)

In curved spacetime we replace aµ → ξµ(x) which is an arbitrary spacetime dependentfunction. The effect of the second term, which is sometimes called the ‘orbital part’ of aLorentz transformation, can be included in ξµ(x). Rather than taking aµ and λab as thebasis for gauge transformations, we use ξµ(x) = aµ(x) + λµν(x)xν and λab(x) as a basis ofindependent transformations. Thus (11.45) becomes

δgct(ξ)φ(x) = ξµ(x)∂µφ(x) , (11.46)

in curved spacetime. In a similar fashion we extend this rule to spinors and frame vectors as

δgct(ξ)Ψ(x) = ξµ(x)∂µΨ(x) ,

δgct(ξ)Va(x) = ξµ(x)∂µV

a(x) . (11.47)

Fields referred to local frames effectively transform as scalars under diffeomorphisms. Theframe vectors do transform under Lorentz transformations separately.

For a vector field we have a choice of using either the coordinate basis or local frame.The components are related by Vµ = eaµVa. The components Vµ transform under generalcoordinate transformations by the Lie derivative, as explained in (6.3), namely

δgct(ξ)Vµ = LξVµ = ξν∂νVµ + Vν∂µξν . (11.48)

This already includes the global Lorentz transformations of the vectors. Thus we do nothave to include further Lorentz transformations.

Further, we introduce the frame vector parameter ξa(x) related to ξµ by ξa = ξµeaµ. Then

(11.46) can be rewritten as

δφ(x) = ξa(x)eµa(x)∂µφ(x) = ξa(x)∂aφ(x) , (11.49)

with ∂aφ = eµa∂µφ. This defines the local translations, and we have seen that they haveabsorbed the ‘orbital part’ of Lorentz transformations. Only the spin part remains as Lorentztransformations.

99

The rule for local Lorentz transformations is simple to state and is implicit in the discus-sion of Ch. 6.Rule – Only fields carrying local frame indices transform under local Lorentz transformations.The transformation rule involves the appropriate matrix generator. Thus for scalars φ,spinors Ψ, and frame vectors V a we have

δ(λ)φ = 0 ,

δ(λ)Ψ = −14λabγabΨ ,

δ(λ)V a = −λabV b , δ(λ)Va = Vbλba . (11.50)

Thus the main results of this section are to establish ξa and λab as our ‘basis’ for gaugeparameters of the Poincare group and to define the associated transformation rules of thefields we will encounter.

Note that (11.48), (11.47) and (11.50) are compatible if the transformation rule of theframe field is

δeaµ = ξν∂νeaµ + eaν∂µξ

ν − λabeµb . (11.51)

We have combined Lorentz and coordinate transformations, and we have simply applied thepreviously stated rules to a field with both coordinate and frame indices. To complete thelist of transformation rules we include the rule for the transformation of the spin connectionωabµ , namely

δωµab = ξceµcωµ

ab + ∂µλab − λacωµ

cb + ωµacλc

b . (11.52)

The Lorentz term is just the infinitesimal limit of (6.40).

11.3.2 Covariant derivatives and general coordinate transformations

It is easy that the definition (11.38) needs to be changed when the gauge group includeslocal translations. Otherwise we would write the covariant derivative of a scalar field (withno internal symmetry present) as

Dµφ = ∂µφ− eaµ(x)∂aφ(x) = 0 . (11.53)

So the previous definition is rather useless. It is not hard to repair the definition by removinggeneral coordinate transformations from the sum over gauge transformations. Thus, fromnow on, we define the covariant derivative of any field φ as

Dµφ ≡ ∂µφ− δ(Bµ)TAφ , (11.54)

with general coordinate transformations omitted in the sum over the gauge group indices A.The same restriction applies to the gauge transformation δ(ε)φ. We discuss below how thischanges the rules established in Sec. 11.2. We use the term ‘standard gauge transformations’to refer to the set of gauge transformations that remain in (11.54). These include localLorentz, local supersymmetry, and local internal symmetry transformations.

A second peculiar feature of translations was encountered in Ex. 11.9. Namely thecurvature component for local translations in gravity or supergravity vanishes, see (11.43).

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From now on we will always impose (11.44) as a constraint, which determines the connectionin the presence of torsion due to the gravitino field.25 We mentioned already that thisconstraint, which we found there as the first Cartan structure equation, can be solved toexpress the spin connection in terms of the frame field and the torsion tensor. Thus thespin connection becomes a ‘composite gauge field’ in the formalism we are now developing.Other examples of composite gauge fields appear in the gauging of the conformal groupand superconformal groups, which are important applications of this formalism in order tosimplify the construction of matter-coupled supergravities. In both cases composite fieldswill be determined by curvature constraints.

We now make a further change in the basis of gauge transformations. Namely we replacegeneral coordinate transformations by gauge covariant general coordinate transformations.To motivate this we consider a set of scalar fields φi transforming under an internal symmetryas δ(θ)φi(x) = −θA(x)tA

ijφ

j. Using the previous definition their transformation under acoordinate transformation would be

δ(ξ)φi = ξµ∂µφi . (11.55)

This is correct, but it has the undesirable property that it does not transform covariantlyunder internal symmetry. We fix this by adding a field dependent gauge transformation andthus define

δcgct(ξ)φi ≡ ξµ∂µφ

i + (ξµAµA) tA

ijφ

j . (11.56)

A good indication that this is a quite natural modification already came in Ex. 8.4, wherereaders showed that such field dependent gauge transformations occur in the commutatorof global SUSY transformations in a SUSY gauge theory. This led to the concept of softalgebras in (11.28). It also appeared in Sec. 9.5 when we first discussed the algebra of localsupersymmetry. Therefore for any field that transforms under one or more of the standardgauge transformations, we define its ‘covariant general coordinate transformation’ by

δcgct(ξ) = δgct(ξ)− δ(ξµBµ) . (11.57)

In every case the parameter of the ‘standard gauge transformation’ is the scalar product ofξµ with the standard gauge field Bµ

A.We now discuss the form of the covariant general coordinate transformations on the

several types of fields which we require.

Coordinate scalars. Note that the two terms in the example (11.56) can be grouped intothe standard covariant derivative, so we can write

δcgct(ξ)φ = ξµDµφ = ξaDaφ . (11.58)

We thus find that these scalars are also covariant with respect to covariant generalcoordinate transformations, i.e. the cgct does not lead to a derivative on the parameter,and cgct transform the scalars to

Daφ = eaµDµφ , Dµφ = ∂µφ−BA

µ (TAφ) . (11.59)

25For application to ordinary gravity, we just set ψµ = 0.

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The same situation occurs for the spinor fields of N = 1 multiplets, which are alsogeneral coordinate scalars when they are coupled to supergravity. Consider the fermionχ of a chiral multiplet and the gaugino λ of an abelian gauge multiplet. For themthe gauge covariant translation and subsequent covariant derivative requires a field-dependent SUSY transformation and thus involves their SUSY partners Z, F , Aµ(x)and D, as well as the gravitino ψµ.

δcgct(ξ)PLχ = ξµPL(∂µχ+1

4ωµ

abγabχ−1√2γν∂νZψµ − Fψµ) = ξµPLDµχ ,

δcgct(ξ)λ = ξµ(∂µλ+ 1

4ωµ

abγabλ+ 14γρσFρσψµ − 1

2iγ∗D

)= ξµDµλ . (11.60)

The frame field. The frame field eaµ is the gauge field of local translations. Its standardgauge transformations are defined in (11.26). As described until now, those standardgauge transformations are local Lorentz and local SUSY transformations. However, forother applications one includes conformal and superconformal transformations. There-fore we now use the general notation in which any gauge field is denoted by Bµ

A. Asdiscussed above general coordinate transformations are excluded in a sum with gaugeindex A, but they should otherwise be kept. Therefore we define the sum over index Bbelow to include only standard gauge transformations. Local translations (with gaugeparameter ξa) are indicated separately. We indicate these by lower case indices (para-meter ξa). The computation is in fact a formalization of the computation that we didin Sec. 9.5, and leads to a simple result for δcgct(ξ)e

aµ:

δcgct(ξ)eaµ = ξν∂νe

aµ + eaν∂µξ

ν − ξνBCν Bµ

BfBCa − ξνBC

ν eµbfbC

a

= ∂µξa + ξν

(∂νe

aµ − ∂µe

aν −BC

ν BµBfBC

a −BCν e

bµfbC

a − ecνBBµ fBc

a)

+ξcBBµ fBc

a

= ∂µξa + ξcBµ

BfBca − ξνRµν

a . (11.61)

As discussed above, the last term vanishes as a constraint on the spin connection. Theresult is that the cgct of the frame field is given by the first two terms. This agreesexactly with the general rule (11.26) in which the frame field is considered to be thegauge field of translations. This confirms the identification of the cgct as the localtranslations. Thus, its cgct act also as if they were translations following the rule(11.26).

Gauge fields. We now compute the cgct for the gauge field BµA of a standard gauge trans-

formation. For simplicity we assume that the structure constant faBA = 0, , where A

is the particular symmetry index of interest, the index a indicates local translations,and B is general. This assumption is satisfied in the N = 1 SUSY algebra26, as shownin Table 6 above. The cgct of BA

µ is obtained from combining (11.58) and (11.26)

δcgct(ξ)BAµ = ξν∂νB

Aµ +BA

ν ∂µξν − ∂µ

(ξνBA

ν

)− ξνBC

ν BµBfBC

A = ξνRνµA . (11.62)

26In some situations, this simplification is not generally valid.

102

Thus the cgct of a gauge field involves its curvature. An example already appeared inthe commutator of global SUSY transformations on the Yang-Mills potential in (8.24).

There are important cases in supergravity in which (11.26) (with indices restricted to stan-dard gauge transformations) is not complete, so that additional terms appear in (11.62). Toallow for this, we write the more general form

δ(ε)BAµ = ∂µε

A + εCBµBfBC

A + εBMµBA . (11.63)

One case is where gauge transformations do not satisfy faBA = 0, and the latter term thus

contains terms of the form εBeaµfaBA as we will encounter later. Another example is the

Yang-Mills gauge field AAµ , whose transformation law under supersymmetry in (8.22) is notrelated to its gauge properties. This is due to the presence of the non-gauge field λA in thegauge multiplet. In this case the modified gauge transformation law reads

δ(ε)AAµ = ∂µεA + εCAµ

BfBCA + 1

2(εγµλ

A) . (11.64)

In this case the new term involves only the local SUSY parameter εα. Formally, we thushave for the latter case (with spinor index α referring to supersymmetry)

MµαA = 1

2

(γµλ

A)α. (11.65)

Other modifications can also occur, and they lead to a change in the definition of the cur-vature which appears in (11.62) which we discuss in the next section.

In practical calculations it turns out that the MµBA are the important terms, while the

other parts of the transformations (11.63) automatically appear in the formalism of covariantderivatives which we discuss next.

11.3.3 Covariant derivatives and curvatures in a gravity theory.

In this section we will find out what are the modifications that we need to the treatmentof covariant derivatives and curvatures in section 11.2. We maintain the same definitionof a covariant quantity as we saw on page 96. In particular, as the general coordinatetransformation of a covariant quantity should not involve a derivative on the parameter ξµ,a covariant quantity has to be a world scalar. Hence, the covariant derivative that we wantto use is Daφ as defined in (11.59), rather than Dµφ.

The modifications that we saw for covariant derivatives have their counterpart for cur-vatures. First of all we have to distinguish again translations from standard gauge transfor-mations. Hence, we define now curvatures

rµνA = 2∂[µBν]

A +BνCBµ

BfBCA , (11.66)

where the sums over B and C involve only standard gauge transformations. Secondly, ifgauge fields transform with extra terms as in (11.63), one has to ‘covariantize also for these.Hence the modified curvature is

RAµν = rµν

A − 2B[µBMν]B

A . (11.67)

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Finally, as argued above for the covariant derivative, the covariant curvature should carryLorentz indices rather than world indices in order to be a world scalar. Thus

RabA = ea

µebνRµν

A . (11.68)

11.3.4 Calculating transformations of covariant quantities

There are useful tricks to obtain gauge transformations of covariant quantities. The tricksinvolve the knowledge of the following principles:

1. Da on a covariant quantity is a covariant quantity, and so is Rab.

2. The gauge transformation of a covariant quantity does not involve a derivative of aparameter.27

3. If the algebra closes on the fields, then the transformation of a covariant quantity is acovariant quantity, i.e. gauge fields only appear either included in covariant derivativesor in curvatures.

We will refrain from giving formal proofs of these statements here. The fact that we know inadvance that explicit gauge fields only occur inside covariant derivatives simplifies our live.

Consider scalar fields φi in the context of gravity and a gauge group. Their covariantderivatives are thus as in (11.59), where the BA

µ = AAµ are the usual Yang-Mills gauge fields.The scalar fields transforms e.g. under these gauge transformations as

δφi = −θA(tA)ijφj . (11.69)

As explained in Sec. 11.3.1 the Lorentz transformations of a scalar are absorbed in thegeneral coordinate transformations, hence φi can be considered as invariant under localLorentz transformations. Now we want to calculate the transformations of the covariantderivative Daφ. First principles say

δDaφ = (δeaµ)Dµφ+ ea

µ∂µδφ− eaµ(δAAµ )(TAφ)− ea

µAAµ δ(TAφ) . (11.70)

The rule (11.26) with the commutator [P,M ] as in table 6 implies the Lorentz transforma-tions of the frame fields that we already know:

δeµa = −λabeµb , i.e. δea

µ = −λabebµ . (11.71)

Later, we will see that a similar term has to be taken into account for the dilatation trans-formation of a covariant derivative when one considers the conformal group. This term willlead to the expected value of the Lorentz transformation of a covariant derivative of scalars.The Lorentz transformation acts as indicated by the appearance of the a index:

δLor(λ)Daφi = −λabDbφ

i . (11.72)

27This is in fact an empty statement if you remember the definition, but we remind this here to have allthe important facts together for further reference.

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When we insert (11.69) in the second term in (11.70), we can forget about the ∂µε termbecause the first principle says that Daφ is a covariant quantity and the second principlesays that then its transformations should not have derivatives on the parameter. Thus, weknow that these terms should cancel finally, and we can already omit them in the calculation.Hence, this term leads to the transformation under the gauge group

δgaugeDaφi = −θA(tA)ijDaφ

j . (11.73)

At the r.h.s. we write Daφj because the third principle says that the result should be a

covariant quantity, hence the derivative should combine with terms that have explicit gaugefields to get to this result. Then we can forget about the third and fourth terms. Thefourth clearly leaves explicit gauge fields and they are excluded by the third principle. Thetransformation of the Yang-Mills gauge fields in the third term involve either derivatives ofthe parameter or explicit gauge fields. So we know that these cancel against other termsthat we neglected. In this case, it is easy to check that this is indeed the case, but for morecomplicated calculations in supergravity we will be happy that we have these principles thatsave us a lot of calculational work. An illustration is provided in Appendix 11.D. Let usfinally remark that in the calculation, apart from the transformation of the scalar, we onlyused the Lorentz transformation of the vierbein (11.71). This is so because this was the onlyterm where gauge fields transform in terms without derivatives and without explicit gaugefields of standard gauge transformations.

Ex. 11.10 One can derive a general formula for the transformation of curvatures, correcting(11.41) for the effects that gauge fields transform with matter-like terms. To apply themethods explained earlier, the decomposition in (11.67) is most useful. Indeed, explicit gaugefields appear in rab

A only quadratically. Show that

δ(ε)RabA = εBRab

CfCBA + 2εBD[aMb]B

A − 2εCM[aCBMb]B

A . (11.74)

Similarly, show that the Bianchi identity becomes

D[aRbc]A − 2R[ab

BMc]BA = 0 . (11.75)

11.D Appendix: example with supergravity

We illustrate here the use of the tricks with covariant derivatives and curvatures in a settingthat is a bit advanced for the readers of this chapter, in the sense that it uses supergravity.

The first exercise clarifies the usefulness of the theorem.

Ex. 11.11 Consider the scalars Z of chiral multiplets that we will embed in supergravity,and we assume moreover that they transform under a Yang-Mills gauge group as δYMZ =−θAtAZ. This means that the full set of gauge transformations contains (covariant) generalcoordinate transformations and as standard gauge transformations: supersymmetry, Lorentztransformations and Yang-Mills transformations. The definition of the covariant derivativeon Z is therefore

DaZ = eaµ

(∂µZ −

1√2ψµPLχ+ AAµ tAZ

). (11.76)

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Prove that (assuming here that the gravitino has only the gauge transformations that followfrom the algebra) this leads to

δ(ε)DaZ =1√2εDaPLχ− θAtADaZ

+12εγaλ

AtAZ

−λabDbZ . (11.77)

The first line is the easy one that follows from commutating transformations and covariantderivatives. The second line is due to the extra term in the supersymmetry transformationof the gauge vector, and the third line is the one due to transformation of the vierbein aswritten above.

Ex. 11.12 To illustrate (11.68), see that in a supersymmetric theory the covariant fieldstrength of a vector in an Abelian vector multiplet is

FabA = ea

µebν(2∂[µA

Aν] − ψ[µγν]λ

A). (11.78)

We illustrate now the calculation of the transformation of a covariant field strength bycalculating the supersymmetry transformation of Fab. The first principle implies that this isa covariant quantity. Though we have not yet proven the embedding of the vector multipletin supergravity, we assume now that the full supersymmetry rule of the gauge field AAµ isthe one given in (8.22). Furthermore let us assume that the gravitino has, apart from thetransformation determined by the algebra an extra term that we shall denote as Υµ, whoseform is irrelevant for this exercise:

δsusy(ε)ψµ =(∂µ + 1

4γabωµab

)ε+ Υµε . (11.79)

When considering the transformation of AAν in the first term of (11.78), we do not have toconsider that the derivative ∂µ acts on the parameter. Indeed, principle 2 implies that suchterms will at the end anyway disappear. The derivative thus acts on the gaugino λ. As thesupersymmetry algebra is supposed to be closed on the vector multiplet, the third principleimplies that this derivative should become a covariant derivative. This leads to

δsusy(ε)Fab = εγ[bDa]λA + . . . . (11.80)

Now we still have to consider the second term in (11.78). An important simplification isthat we do not have to act with δsusy on λA. Indeed, such terms would leave an explicit ψµin the result, and the principle 3 says that such terms do not appear. Thus, we need onlyto consider (11.79). With the same principles, one sees that only the last term is relevant asthe other contain a derivative on the parameter or an explicit gauge field ωµab. Hence thefinal result is

δsusy(ε)Fab = εγ[bDa]λA + λAγ[bΥa]ε . (11.81)

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The principles determine also that the modification in a supergravity theory of the trans-formation law of the gaugino as given in (8.22) will involve the covariant curvature, i.e.

δλA =[−1

4γabFA

ab + 12iγ∗D

A]ε . (11.82)

As we have already calculated the supersymmetry transformation of Fab, it is easy to calculatethe commutator of two supersymmetries on the gaugino. In fact, it can be compared to thecalculation in the global case. If the student still has its/her calculation for exercise 8.4,

(s)he has only to replace each F by F and Dµ by Dµ, and nothing changes, except for thecontribution of the Υ-term to (11.81). This shows how these tricks are used in practice.

107

12 Survey of supergravities

We have meanwhile seen a few supergravity theories and have written down some generalaspects of gauge theories that may include local supersymmetry. We also studied already inCh. 3 the properties of spinors in various dimensions. All this information is sufficient to getalready an idea of which supergravity theories can exist and what are the general propertiesthat these possess. This will be the subject of the present chapter.

12.1 Minimal and extended superalgebras

The minimal supersymmetry algebra is the one that we saw in (11.17):

Qα, Qβ = −12(γa)αβPa . (12.1)

The supersymmetries commute with translations and are a spinor of Lorentz transformations.We have seen that in many cases the final algebra is more complicated due to the appearanceof structure functions, but let us first restrict ourselves to strict algebras, i.e. the ones thatare obtained for ‘vanishing fields’.

12.1.1 4 dimensional minimal algebras

Let us first restrict to 4 dimensions. The supercharges Qα are Majorana spinors and we canalso choose to represent them by the chiral projections PLQ, and PRQ, which is then thecharge conjugate. The non-vanishing part of (12.1) is then only the anticommutator betweentwo supercharges of opposite chirality, i.e.

(PLQ)α, (PRQ)β = −12(PLγ

a)αβPa . (12.2)

Extended superalgebras means that there are several supercharges, with a label i = 1, . . . ,N .It is in that case convenient to use the position of the index i up or down to indicate at oncethe chiral parts, thus

Qi = PLQi , Qi = PRQi . (12.3)

The Majorana spinors are thus Qi +Qi, and Qi is the charge conjugate of Qi. The minimalextended algebra is then

Qαi, Qβj

= −12(PLγ

a)αβδjiPa . (12.4)

12.1.2 Central charges in 4 dimensions

For extended supersymmetry the algebra can be modified by ‘central charges ’. Centralcharges appeared first in the classical work of Haag- Lopuszanski–Sohnius. The simplestexample is for N = 2, where (12.4) can be extended with a nonvanishing commutator fortwo supercharges of the same chirality:

Qαi, Qβj = εijPLαβZ ,Qα

i, Qβj

= εijPRαβZ . (12.5)

108

The generators Z and its complex conjugate Z commute with everything else and are thusreally ‘central’. The matrices PL are symmetric in the spinor indices, and thus one needs anantisymmetric εij to make the superalgebra consistent. Therefore, this is only possible forextended supersymmetry. These central charges are important for supersymmetric solutions.Indeed, applying the supersymmetry algebra to such a solution must give zero. Therefore,with the minimal algebra (12.4) this can only be for solution on which Pa vanish, and whichthus have zero mass. In the presence of central charges that do not vanish on these solutions,the right-hand side of (12.4) can be cancelled by that of (12.5).

12.1.3 Superalgebras in higher dimensions

When we consider (12.1), we see that this in view of (3.57) only consistent with symmetryif t1 = −1. This is according to (3.50) exactly the condition to allow (pseudo-)Majoranaspinors. Furthermore, according to (3.43) the left-right projection in (12.2) is only consistentfor 4 or 8 dimensions, but not e.g. for 10 dimensions. Indeed, in 4 or 8 dimensions we have,due to t0 = t4 = t8 that

(PRQ)α = (PR)αβQβ = −t0Qβ(PR)βα , (12.6)

while for 10 (or 6 dimensions) we have t10 = t6 = t2 = −t0 that

(PRQ)α = (PR)αβQβ = −t0Qβ(PL)βα . (12.7)

Hence, for 10 or 6 dimensions the non-zero commutator is between left and left chiral super-symmetries or between right and right supersymmetries. Hence in these cases one can havethe supersymmetry with only left-handed or only right-handed supersymmetry. This is alsoconsistent with the fact that in 10 dimensions Majorana spinors can be chiral (Majorana-Weyl spinors, see table 2). Thus in this case one can have a chiral supersymmetry algebra(called type I), or when one has two supersymmetries one can choose to have them of dif-ferent chirality (called IIA) or of the same chirality (called IIB). As according to table 2the minimal spinors are 16 dimensional, we cannot have more than 2 supersymmetries fora supergravity field theory in 10 dimensions. Indeed, remember the argument in Sec. 10.1that dimensional reduction to 4 dimensions should not give more that N = 8, which meansat most 4× 8 = 32 real components in the spinors.

In the cases that t1 = 1, i.e. when there are no Majorana fermions, then (12.1) is notconsistent with the symmetry of an anticommutator. This is for the case for the dimensions5, 6 and 7 when according to table 2 we have to use symplectic spinors. Hence in thesecases there is an antisymmetric tensor εij that appears in the reality condition, see (3.48).Therefore, in these cases we can write a consistent algebra as

Qiα, Q

jβ

= (γa)αβε

ijPa . (12.8)

In the case of 6 dimensions, we mentioned already that a chiral projection of the super-symmetries is consistent. Hence in this case one can have algebras of the type (p, q) withp chiral and q antichiral supersymmetries. In order to stay in the limit of 32 supercharges,one should have that p+ q ≤ 4.

109

12.1.4 ‘Central charges’ in higher dimensions

The name ‘central charges’ has been generalized to include other generators that can appearin the anticommutator of supersymmetries similar to (12.5), but that are possibly not Lorentzscalars. E.g. in D = 11 the properties of the spinors allow us to extend the anticommutatoras

Qα, Qβ = γµαβPµ + γµναβZµν + γµ1···µ5

αβ Zµ1···µ5 . (12.9)

The allowed structures on the right-hand side are determined by the last entry in table 2(remember that this is modulo 4, which thus allows the 5-index object). The ‘central charges’Z are no longer Lorentz scalars, and thus do not commute with the Lorentz generators.They are therefore not ‘central’ in the group-theoretical meaning of the word, but play inthe physical context the same role as the ones in (12.5). They also allow the construction ofsupersymmetric solutions by the mechanism mentioned in Sec. 12.1.2, and therefore got alsothe name ‘central charges’ despite their non-zero commutator with the Lorentz generators.The second and third terms in (12.9) are in this way important for the existence of M2 andM5 branes. In general, branes can be classified by considering the possibilities for centralcharges.

12.2 The R-symmetry group

R-symmetry is the name that is given for symmetries between the different supersymmetrygenerators and commute with the Lorentz generators. The Lorentz algebra itself already isa transformation between the different supersymmetry generators according to[

Mab, Qαi]

= −12(γab)α

βQβi . (12.10)

We now investigate the existence of other transformations TA that mix the supersymmetries.We investigate this first in 4 dimensions. We parametrize the action of the TA on thesupersymmetries by matrices (UA)i

j and (UA)ij:

[TA, Qαi] = (UA)ijQαj ,

[TA, Q

iα

]= (UA)ijQ

jα . (12.11)

The latter equation is related to the the former by charge conjugation, thus (UA)ij is thecomplex conjugate of (UA)i

j. The consistency should be determined by Jacobi identities.The Jacobi identity [TTQ], i.e.28

[[TA, TB], Q] = [TA, [TB, Q]]− [TB, [TA, Q]] → fABCUC = [UB, UA] , (12.12)

imply that the matrices UA form a representation of the algebra of the TA. The generatorsTA should commute with Pa as follows already from the Coleman–Mandula theorem, and as

28See that super-Jacobi identities are most easily written in this form: the first term on the right-handside has the same order of the generators as the left hand side, but other arrangement of brackets, then themissing term should satisfy the obvious symmetry property, in this case antisymmetry in TA versus TB , in(12.13) symmetry between the two Q-operators.

110

follows from the fact that they are external to the Poincare algebra. This can be used in the[TQiQ

j] Jacobi identity:

[TA, Qαi, Qβj] = [TA, Qαi], Qβ

j+ [TA, Qβj], Qαi . (12.13)

This leads to

0 = (UA)ij + (UA)j i , i.e. (UA)i

j = −(UA)j i ≡ −((UA)j

i)∗. (12.14)

Thus the matrices U are anti-Hermitian matrices. These are the generating matrices ofU(N ).

For N = 1 this leaves only the possibility that up to normalization U11 = −U1

1 = i.This gives for the Majorana Q = QL +QR = Q1 +Q2

[T,Q] = −iPLQ+ iPRQ = −iγ∗Q . (12.15)

Hence, this T corresponds to the R-symmetry that you encountered in the lectures of P.Fayet.

We can make a general statement about the automorphism groups in view of table 2.The 4-dimensional example is general for even dimensions that have Majorana spinors, butno Majorana–Weyl spinors, i.e. 4 and 8 dimensions. In fact, in this formulation we have notused the Majorana property, but rather the Weyl-spinor formulation. In all even dimensionswe can use Weyl spinors, and this leads to spinors of the same dimension as the Majoranaones (they are the same in another notation).

When we have the possibility of Majorana–Weyl spinors, then the reality condition saysthat the U for the left-chiral and those for the right-chiral spinors should each be real. Thenon-trivial Q,Q anticommutator is in this case also between supersymmetries of equalchirality, which leads analogous to (12.14) now to a condition of antisymmetry for each ofthem separately. Therefore, we obtain SO(NL)× SO(NR).

For odd dimensions, we can not use the chiral formulation, and if there are Majoranaspinors, the result is similarly SO(N ). If there are symplectic Majorana conditions, thenthe same argument of the Jacobi identity [TQQ] leads to the preservation of the symplecticmetric, and the automorphism algebra is thus reduced to USp(N ) (with even N ).29 If thereare SW spinors, then there are two such factors, for the left and for the right-handed sector.In summary, the automorphism groups are obtained from the entries of table 2:

D = 10 : MW : SO(NL)× SO(NR) ,D = 9 : M and D odd : SO(N ) ,D = 8 and D = 4 : M and D even : U(N ) ,D = 7 and D = 5 : S ; USp(N ) ,D = 6 : SW : USp(NL)× USp(NR) .

(12.16)

29There is often confusion about the notation of Sp groups. We use notation such that Sp(2) is the smallestsymplectic group.

111

12.3 Multiplets

12.3.1 On- and off-shell multiplets and degrees of freedom

It may already be clear to the reader from the construction of multiplets in Sec. 8.4 or fromthe explicit examples, that the multiplets have an equal number of bosonic and fermionicdegrees of freedom. More strictly stated, the following theorem holds.30

Theorem: There are an equal number of bosonic and fermionic degrees of free-dom in any realization of the supersymmetry algebra (12.1) when translations arean invertible operation.

To interpret this theorem we remind the reader to the terminology of on-shell and off-shelldegrees of freedom that we introduced in Sec. 5.1. But first, let us explain the origin of thistheorem.

Consider the commutator (12.1) on the space of all bosons. The first Q transforms thespace to a space of fermions, while the second one brings us back to bosons, translated byPµ.

bosonsQ−→ fermions

Q−→ bosons translated by Pµ

The latter is an invertible operator, which proves that the last space is as large as the firstone, and thus also the middle one, the space of the fermions has the same dimension. Thisproves the well-known fact that there are an equal number of bosonic and of fermionic states.It should hold when the algebra (12.1) holds. So this equality of bosonic and fermionic statesshould hold e.g. for on-shell states. I.e. for the chiral multiplet, without the auxiliary fieldF we can apply this only for on-shell states. Then we count 2 bosonic states for the complexZ, and also the fermions have 2 on-shell degrees of freedom. So we have a 2 + 2 on-shellmultiplet. When the auxiliary field F is included, the algebra is also satisfied off-shell. Thuswe have in this case the 4 off-shell degrees of freedom of χ, while the complex Z and F givetogether also 4 real off-shell degrees of freedom. In this case, we say that we have a 4 + 4off-shell multiplet. These two ways of counting are called on-shell counting and off-shellcounting.

On the other hand, the equality of boson and fermion states is not a general fact of asuperalgebra. One needs the invertibility of the square of fermionic generators. This doesnot hold for massless states in an Euclidean spacetime, or in 1 dimension. In these cases theP 2 = 0 condition for a massless state implies P = 0, and there is thus no invertibility.

For on-shell degrees of freedom it is clear that the gauge degrees of freedom are notincluded. But they should not be included in off-shell counting either. This is the way thatwe defined the number of off-shell degrees of freedom. Why we do so should be clear from thealgebra. The theorem holds only when the Q,Q = P algebra holds. When there are gaugesymmetries, we saw already in the simplest case (illustrated in (11.28)) that the algebra ismodified by gauge transformations. Thus, in order to have the minimal algebra we will haveto eliminate the extra gauge terms. Therefore we have to subtract gauge degrees of freedom

30The algebra (12.1) is part of the extended algebras, so that this theorem holds also for extended super-symmetry, and can be generalized also for the case of symplectic structures as in (12.8).

112

in all countings. E.g., a gauge vector in 4 dimensions would count off-shell for 3 degrees offreedom, balancing the 4 off-shell ones of the gaugino and the 1 real degree of freedom ofthe auxiliary field D.

12.3.2 Multiplets in 4 dimensions

We mentioned already in Sec. 8.4 that in 4 dimensions multiplets of global supersymmetryare restricted to N ≤ 4 and those of local supersymmetry to N ≤ 8 by the restrictions offield theory. Here are the most important multiplets of on-shell fields.

The supergravity multiplet is the minimal multiplet that contains the graviton. It con-tains the set of fields that represents the spacetime susy algebra and has gauge fieldsfor the supersymmetries. The number of fields was given in table 3. These are on-shellmultiplets. The fields of spin s > 0 have 2 degrees of freedom (helicity +s and −s).Therefore you can understand e.g. the 2 rather than 1 in the entry of spin 0 for N = 4to obtain an equal number of bosonic as fermionic degrees of freedom. Of course, youmight have understood this number when you solved Ex. 8.5 using the constructionsexplained in Sec. 8.4.

Vector multiplets are the multiplets with fields up to spin 1. They exist for N ≤ 4.The multiplet for N = 1 is the one that we encountered in Sec. 8.3, and was calledthe gauge multiplet. Their content in general is given in table 4. The gauge fieldsof these multiplets can gauge an extra Yang-Mills group that is not contained in thesupersymmetry algebra as we saw it in already in Sec. 8.3. In the final soft algebra,however, some mixing always occurs as we saw in (11.28).

Chiral and hypermultiplets are multiplets with only spins 0 and 1/2. For N = 1 theseare the chiral multiplets that we know from Sec. 8.2. For N = 2 a similar multipletis the ‘hypermultiplet’. Table 4 contains the field content of these multiplets. Theydo not exist for N > 2. These multiplets may also transform under the gauge groupdefined by the vector multiplet, as we have illustrated in Sec. 8.3. They form thus arepresentation of these gauge groups.

The vector and chiral multiplets are called matter multiplets. These exist for global supersym-metry, and they can be coupled to supergravity theories (which is then called ‘matter-coupledsupergravity’) as we will see later in the book.

Let us still remark that in global supersymmetry one can only have compact gauge groupsif one requires positive kinetic energies, but in supergravity some non-compact gauge groupsare possible without spoiling the positivity of the kinetic energies. However, the list ofpossible non-compact groups is restricted for any N .

12.3.3 Multiplets in more than 4 dimensions

We saw in Sec. 10.2 how the fields of the N = 8 supergravity multiplet are all contained inthe simpler multiplet in 11 dimensions. The relation was made by dimensional reduction on

113

the torus. Remember that the number of components of the supersymmetry did not changein such a torus dimensional reduction. We started with the 32-component supersymmetry in11 dimensions, and ended up with N = 8 in 4 dimensions, which contains again 8× 4 = 32real components. Hence the multiplets that we saw above in 4 dimensions can often beobtained in other ways from higher-dimensional multiplets.

With 16 supersymmetries, global supersymmetry is possible. One can have vector mul-tiplets, or tensor multiplets for (2, 0) supersymmetry in 6 dimensions. In D = 5, the tensormultiplets are dual to vector multiplets at the level of zero gauge coupling constant. Gaugingbreaks this duality. Supergravity theories with 16 supersymmetries may contain a numberof these multiplets. The model is fixed once one gives the number of matter multiplets andthe gauging that is performed by the vectors.

Theories with 8 or 4 supersymmetries are not fixed by the discrete choices of numberof multiplets and gauging. In these cases the model depends on some functions that canvary by infinitesimal variations. It is in these models that auxiliary fields are most useful.E.g. for the chiral multiplets that we mentioned before, a holomorphic function W (Z) canbe introduced as we saw in (8.5). Similarly, the kinetic terms can be generalized dependingon an arbitrary function K(Z,Z∗), which will play the role of a Kahler potential for thegeometry determined by the scalars.

114

Tab

le7:

Supe

rsym

met

ryan

dsu

perg

ravi

tyth

eori

esin

dim

ension

s4

to11

.A

nen

try

repre

sents

the

pos

sibilit

yto

hav

esu

per

grav

ity

theo

ries

ina

spec

ific

dim

ensi

onD

wit

hth

enum

ber

of(r

eal)

sup

ersy

mm

etri

esin

dic

ated

inth

eto

pro

w.

We

firs

tre

pea

tfo

rev

ery

dim

ensi

onth

ety

pe

ofsp

inor

sth

atca

nb

euse

d.

Eve

ryen

try

allo

ws

diff

eren

tp

ossi

bilit

ies.

Theo

ries

wit

hm

ore

than

16su

per

sym

met

ries

can

hav

ediff

eren

tga

ugi

ngs

.T

heo

ries

wit

hup

to16

(rea

l)su

per

sym

met

ryge

ner

ator

sal

low

‘mat

ter’

mult

iple

ts.

The

pos

sibilit

yof

vect

orm

ult

iple

tsis

indic

ated

wit

h♥

.T

enso

rm

ult

iple

tsinD

=6

are

indic

ated

by♦

.M

ult

iple

tsw

ith

only

scal

ars

and

spin

-1 2fiel

ds

are

indic

ated

wit

h♣

.A

tth

eb

otto

mis

indic

ated

whet

her

thes

eth

eori

esex

ist

only

insu

per

grav

ity,

oral

sow

ith

just

glob

alsu

per

sym

met

ry.

Dsu

sy32

2420

1612

84

11M

M

10M

WII

AII

BI ♥

9M

N=

2N

=1

♥

8M

N=

2N

=1

♥

7S

N=

4N

=2

♥

6SW

(2,2

)(3,1

)(4,0

)(2,1

)(3,0

)(1,1

) ♥(2,0

) ♦(1,0

)♥,♦,♣

5S

N=

8N

=6

N=

4♥

N=

2♥,♣

4M

N=

8N

=6

N=

5N

=4

♥N

=3

♥N

=2

♥,♣

N=

1♥,♣

SU

GR

ASU

GR

A/S

USY

SU

GR

ASU

GR

A/S

USY

115

12.4 Basic theories: dimensions and supersymmetries

Table 7 gives an overview on supersymmetric theories in Minkowski spacetimes and withpositive definite kinetic terms. The most relevant source in this respect is the paper ofStrathdee that analyses the representations of supersymmetries. Remember also Sec. 8.4 forthe case of 4 dimensions.

Supersymmetric field theories of this type in 4 dimensions are restricted to fields withspins ≤ 2. This is the restriction to N ≤ 8 or up to 32 supersymmetries as an elementary(real) spinor in 4 dimensions has 4 components, see table 2. The same table shows thatone can not have more than 11 dimensions if the supersymmetries are restricted to 32 (atleast in spacetimes of Minkowski signature). We are considering here the supersymmetriesthat square to general coordinate transformations. The 11-dimensional theory is the basisof ‘M-theory’, and is therefore indicated as M in the table.

Going down vertically in the table is obtained by Kaluza-Klein reduction. That meansthat one splits all the fields in representations of the lower dimensional Lorentz group. E.g.the spinors of the 11-dimensional M-theory split in one right-handed and one left-handedspinor. This is the theory of the massless sector of IIA string theory, and that is why wehave indicated it as IIA. The massless sector of IIB theory involves doublets of spinors of thesame chirality (thus also with 32 real supersymmetries). That theory is not the dimensionalreduction of an 11-dimensional theory, as indicated by its place in the table.

Elementary supersymmetric theories in 10 dimensions involve only 16 supersymmetries.They appear in superstring theories with open and closed strings. Apart from the supergrav-ity multiplet which involves the graviton, one can also have a vector multiplet. The existenceof the simplest matter multiplets (representations of supersymmetry that do not involve thegraviton) is indicated in the table. Vector multiplets involve a vector, an elementary spinorand possibly scalars. Tensor multiplets involve antisymmetric tensors Aµν and other fields.For dimensions 5 and 4 these are dual to vectors and scalars. Therefore, tensor multipletsare not explicitly indicated for these lower dimensions.

One can have theories with these matter multiplets also for global supersymmetry. Globalsupersymmetry is only possible with up to 16 supersymmetries. Again, this can be under-stood in 4 dimensions, because for N > 4 one needs fields of spin-3

2. The latter are in field

theory only possible when they are gauge fields of a local supersymmetry (gravitinos), whichthen need gravitons for the gauge fields of the translations that appear in the commutatorof two supersymmetries.

In the dimensions lower than 10 we indicate the theories by a numberN that indicates thesymmetry group rotating different supersymmetries. For 6 dimensions, as in 10 dimensions,there are spinors of different chirality, and one has to distinguish the number of left andright-handed spinors. The simplest case is with 8 supersymmetries. They have to be all ofthe same chirality. The theory is then called (1, 0) and would be N = 2 in the terminologyused in 5 dimensions. With 16 supersymmetries, one can have (1, 1) or (2, 0). These are theanalogues of IIA and IIB, respectively, in 10 dimensions. For more supersymmetries, thereis a subtlety. The (2, 1) and (2, 2) theories can be constructed using a metric tensor gµν(x).For the (4, 0), (3, 1) or (3, 0) theories, this field is not present, but is replaced by a more

116

complicated representation of the Lorentz group. Thus, these theories are different in thesense that they are not based on a dynamical metric tensor. They have not been constructedbeyond the linear level.

If one constructs in 4 dimensions a field theory with N = 7, then it automatically hasan eighth local supersymmetry. That is why it is not mentioned in the table. Similarly ifone constructs a global supersymmetric theory with N = 3, it automatically has a fourthsupersymmetry. However, in this case, there is the possibility of having only three of thefour local. Thus N = 3 is only meaningful in supergravity. This explains the lowest line ofthe table.

12.5 Deformations and gauged supergravities

Table 7 determines what sort of theories are possible, but the theories are far from uniquefor every entry. E.g. for all the entries in which there are matter multiplets indicated inthe table, one first of all can choose whether and how many multiplets appear in the theory.In some cases there are also various representations of the same physical (on-shell) theory.E.g. for the chiral multiplet that we saw before, there is an alternative where one of thescalars is replaced by an antisymmetric tensor. But we saw before that in 4 dimensions aphysical antisymmetric tensor has the same degrees of freedom as a scalar. Therefore, atleast when we talk about the kinetic terms, we do not describe new theories when we involveantisymmetric tensor multiplets. That is why these were not mentioned in the table.

If we still concentrate on the kinetic terms of the fields, then the theories with morethan 8 supersymmetries are unique once we have determined how many multiplets are in thetheory. But for 8 or less supersymmetries, the kinetic terms are still determined by functionsthat can be continuously deformed. E.g. for the chiral multiplet, the kinetic terms dependon a so-called Kahler potential.

Up to here, the theories that we described are determined by their kinetic terms. Theyall exist in a version without a potential for the scalar fields, or a cosmological constant. Thiswe call the undeformed theories. These are all classified and well-known. However, thereare many deformations. Most of these are known as gauged supergravities, but there arealso other deformations. E.g. for N = 1, the addition of a superpotential is a deformationunrelated to gauging. Similarly there is a massive theory of Romans of D = 10 supergravity,and several supergravities have a version with a cosmological constant.

The most common deformations are the so-called gauged supergravities. In string theorythey appear due to fluxes. These are the theories in which some of the vectors gauge a non-Abelian group. The coupling constant gives then the deformation. The number of generatorsof the gauge group is equal to the number of vector fields.31 This counting includes as wellvectors in the supergravity multiplet and those in vector multiplets. In fact, in general theirkinetic terms are mixed and thus they do not have to be distinguished. The gauge groupis in principle arbitrary, but to have positive kinetic terms gives restrictions on possiblenon-compact gauge groups for any supergravity. The operations of this group can act on

31One could say less or equal, but considering that any vector transforms at least as δWµ = ∂µε, even ifthe ε transformation does not act on anything else, we can say that there is a U(1) factor.

117

various fields. The easiest example is when in N = 1 we consider the non-Abelian vectormultiplet. As we have seen, the transformations can act on the chiral multiplet. In a globalsupersymmetric theory these gauge groups commute with the supersymmetries. However,in supergravity they always mix with the R-symmetry group.

Such gaugings have various consequences:

1. There are extra transformations of the fermions. In global supersymmetry we can seethis because the auxiliary field D of the vector multiplet gets a non-zero value. Theseare sometimes called ‘fermion shifts’.

2. A scalar potential is generated. As we will see later, this is related to the previousitem as the potential is built out of the squares of the fermion shifts.

3. Similarly, there are other new terms in the action like fermion mass terms.

The gauged groups originate always from promoting global symmetries of the basic the-ories to gauged symmetries. Thus they form a subset of these global symmetries. Whenthere are scalars, the action of global symmetries on the scalars is generated by the Killingvectors, as we saw at the end of section 11.1.1. For spin-1 fields in 4 dimensions, there is thesymplectic group of dualities, and the gauge symmetries should be embedded in this group.There are generalizations of this for other dimensions.

For the basic theories we could restrict ourselves to one representation of physical fields ofa certain spin. E.g. in 5 dimensions, an antisymmetric tensor is dual to a vector, and we canomit it. But this does not hold anymore for the deformed theories, e.g. the gauged super-gravities. This can be understood by realizing that spin-1 fields AAµ transform always in theadjoint representation of the gauge group. However, the antisymmetric tensors allow otherrepresentations. Hence, by considering also these fields, one can get more general gaugedtheories. One can also understand this by realizing that the duality between antisymmetrictensors and vectors holds only for the massless case. The gauging allows us to generate massterms.

A further remark is that the distinction between deformed theories and gauged theoriescan depend on the way that the theory is looked upon. Massive theories can be obtainedfrom gauge-fixed gauge theories. An example are the Fayet–Iliopoulos (FI) terms. They aredeformations in the sense that they can be added without changing the gauge transformationsof the physical fields. On the other hand, in the superconformal framework they appear bythe fact that gauge transformations act on the compensating multiplets, to which we willreturn later. Thus the deformed theories are often also included in the terminology ‘gaugedsupergravities’.

The last years, various new results have been obtained in this direction. A completecatalogue of theories is not yet known. However, we believe that all supersymmetric fieldtheories (with a finite number of fields and field equations that are at most quadratic inderivatives) belong to one of the entries in table 7.

118

12.6 General characteristics of the actions

The full action of a supergravity theory is very complicated. It contains 4-fermion couplings,couplings between fermions and vectors (as dipole moments), . . . . We show here some generalstructure of the bosonic terms in 4 dimensions. The basic theories contain (we thus do notconsider antisymmetric tensors) the graviton, represented by the vierbein eaµ, a number ofvectors AAµ with field strengths FA

µν , a number of scalars ϕu, N gravitinos ψiµ, and a numberof spin 1/2 fermions. The pure bosonic terms of such an action are32

e−1Lbos = 12R + 1

4(ImNAB)FA

µνFµνB − 1

8(ReNAB)e−1εµνρσFA

µνFBρσ

−12guv(ϕ)Dµϕ

uDµϕv − V (ϕ) . (12.17)

We factorized the determinant of the vierbein to the left hand side. The first term gives thepure gravity action. Then there are the kinetic terms for the spin-1 fields, which we discussedalready in section 4.1.3. Possibly there is an extra Chern-Simons term. The scalars havekinetic terms determined by a symmetric tensor guv(ϕ). This is the main ingredient forthe identification of geometries, which we will discuss in section 12.7. The scalars couple‘minimally’ to the vectors with a covariant derivative for the gauge symmetries as in (8.20).Finally, there is the potential V (ϕ) for the scalars with properties that are determined bythe supersymmetry.

12.7 Scalars and geometry

One defines a manifold, which we indicate as M, with dimension equal to the number ofscalars in the model. The scalars ϕu give a chart in this manifold. The values of a scalar at aspacetime point x thus determine a submanifold of M, parametrized by ϕu(x). The metricof spacetime is gµν(x) = eaµ(x)ηabe

bν(x). The metric on M is guv(ϕ). This metric is part of the

definition of the theory. This is to be contrasted with gµν(x), which is, together with ϕu(x), adynamical field. On the other hand, the induced metric on spacetime is guv(ϕ) (∂µϕ

u)(∂νϕv)

at ϕ = ϕ(x), and its contraction with the (inverse) spacetime metric and its determinant√ggµν appears in the action.

The scalar manifold can have isometries, i.e. symmetries of the induced metric ds2 =guv(ϕ) dϕu dϕv. Usually these symmetries are extended to a symmetry of the full action(there are counterexamples, but they are rare). This group is then called the U-dualitygroup. The scalars and the vectors are connected via the tensor NAB(ϕ). Therefore theisometries act as duality transformations in the vector sector, and as such must belong tothe Sp(2m,R) group (in 4 dimensions). This gives a restriction of possible U-duality groups,which are for D = 4 restricted to be a subgroup of Sp(2m,R) for scalars that belong tomultiplets including vectors. Actually, to count the m vectors, we have to include those ofthe gravity multiplet. A subgroup of the isometry group, at most of dimension m, can thenbe gauged. This means that the vectors couple to the scalars in the covariant derivative

32We use here the notation NAB for the metric of the vectors, as it is mostly used in extended supersym-metry. This matrix thus replaces fAB = iNAB that we used in section 4.1.3.

119

using in (8.20) the transformations of the scalars under these isometries. There are morecorrections due to this gauging in the fermionic sector.

The type of geometries that occur, depend on the number of supercharges. For allthe theories with more than 8 supersymmetries, all the scalar manifolds are (non-compact)symmetric spaces. This means that they are of the form G/H with G a non-compactgroup and H its maximal compact subgroup. These are shown in table 8. Notice that Hcontains always the R-symmetry group, and has another factor in case that there are mattermultiplets.

The theories with 4 supersymmetries (N = 1 in 4 dimensions, but one might also considerlower-dimensional theories) the manifold can be an arbitrary Kahler geometry, a geometrywith a (closed) complex structure. Arbitrary Kahler spaces are defined by a Kahler potentialK(z, z∗), see the lectures of A. Bilal. For any Kahler manifold there is such an N = 1 theory.

Beautiful structures emerge for theories with 8 supercharges (N = 2 if in D = 4). Thesetheories all belong to a class that was baptized special geometries, including some real,some Kahler geometries and all the quaternionic geometries. Especially, the scalars that bysupersymmetry are directly related to vectors have a geometrically distinct structure, specialKahler geometry. This is a subclass of the Kahler geometries discussed above, with an extrasymplectic symmetry structure related to the duality transformations of the vectors shown insection 4.1.3. Scalars in hypermultiplets exhibit quaternionic structures, with many relationswith special Kahler manifolds.

Specifically, the manifolds that occur in supergravity actions are

D = 6 :O(1, n)

O(n)× quaternionic-Kahler manifold

D = 5 : very special real manifold× quaternionic-Kahler manifold

D = 4 : special Kahler manifold× quaternionic-Kahler manifold. (12.18)

These geometries determine the general couplings of supergravity to matter multipletsin D = 6, D = 5 and D = 4. There exist also versions of these geometries for globalsupersymmetry, leading to rigid (the terminology ‘rigid’ is here more common than theterminology global that we use in this book) Kahler manifolds and hyper-Kahler manifolds.

120

Tab

le8:

Sca

lar

geom

etri

esin

theo

ries

with

mor

eth

an8

supe

rsym

met

ries

(and

dim

ension≥

4).

The

theo

ries

are

order

edas

inta

ble

7.N

ote

that

theR

-sym

met

rygr

oup,

men

tion

edin

(12.

16),

isal

way

sa

fact

orin

the

isot

ropy

grou

p.

For

mor

eth

an16

sup

ersy

mm

etri

es,

ther

eis

only

auniq

ue

sup

ergr

avit

y(u

pto

gaugi

ngs

irre

leva

nt

toth

ege

omet

ry),

while

for

16an

d12

sup

ersy

mm

etri

esth

ere

isa

num

bern

indic

atin

gth

enum

ber

ofve

ctor

mult

iple

tsth

atar

ein

cluded

.

D32

2420

1612

10O

(1,1

)SU

(1,1

)U

(1)

9S`(

2)

SO

(2)⊗

O(1,1

)O

(1,n

)O

(n)⊗

O(1,1

)

8S`(

3)

SU

(2)⊗

S`(

2)

U(1

)O

(2,n

)U

(1)×

O(n

)⊗

O(1,1

)

7S`(

5)

USp(4

)O

(3,n

)U

Sp(2

)×O

(n)⊗

O(1,1

)

6O

(5,5

)U

Sp(4

)×U

Sp(4

)F

4

USp(6

)×U

Sp(2

)E

6

USp(8

)SU∗(4

)U

Sp(4

)SU∗(6

)U

Sp(6

)O

(4,n

)O

(n)×SO

(4)⊗

O(1,1

)O

(5,n

)O

(n)×

USp(4

)

5E

6

USp(8

)SU∗(6

)U

Sp(6

)O

(5,n

)U

Sp(4

)×O

(n)⊗

O(1,1

)

4E

7

SU

(8)

SO∗(1

2)

U(6

)SU

(1,5

)U

(5)

SU

(1,1

)U

(1)×

SO

(6,n

)SU

(4)×

SO

(n)

SU

(3,n

)U

(3)×

SU

(n)

121