Santiago Lectures onSupergravity
Joaquim GomisBased on the SUGRA book of Dan Freedman
and Antoine Van Proeyen to appear in Cambridge University Press
Public Material
Lectures on supergravity, Amsterdam-Brussels-Paris doctoral school, Paris 2009, October-November 2009: PDF-file.
http://itf.fys.kuleuven.be/~toine/SUGRA_DoctSchool.pdf
A. Van Proeyen, Tools for supersymmetry, hep-th 9910030
History and overview ofSupergravity
60’ and 70’s. Yang Mills theories, Spontaneous symmetry breaking. Standard model
Supersymmetry
Yu. Gol’fand , E. Lichtman (1971)J.L. Gervais and B. Sakita (1971)A,Neveu, J. Schwarz, P.Ramond (1971)D. Volkov, V. Akulov (1972)J. Wess, B. Zumino (1974)
History and overview ofSupergravity
• Yu. Gol’fand , E. Lichtman- Parity violation in QFT, 4d• J.L. Gervais and B. Sakita & A,Neveu, J. Schwarz String theory-Dual
models. Worls sheet supersymmetry 2d• D. Volkov, V. Akulov- Goldstone particles of spin ½? 4d• J. Wess, B. Zumino Supersymmetric field theory in 4d
Supergroup, superalgebra
History and overview ofSupergravity
• Super Poincare
Translations
Spinor supercharge
Lorentz transformations
Massless multiplets contains spins (s, s-1/2), for s=1/2, 1, 2,
R symmetry
History and overview ofSupergravity
Supergravity
Gauged supersymmetry was expected to be an extension of generalRelativity with a superpartner of the gravito call gravition
Multiplet (2,3/2)
S. Ferrara, D. Freedman, P. Van Nieuwenhuizen (1976)S. Deser, B. Zumino (1976)D. Volkov, V. Soroka (1973), massive gravitinos,..
Extensions with more supersymmetries and extension has beenconsidered, N=2 supergravity, special geometry. N=1 Supergravity in 11d
Index
• Scalar field and its symmetries• The Dirac Field• Clifford algebras ans spinors• The Maxwell and Yang-Mills Gauge fields• Free Rarita-Schwinger field• Differential geometry• First and second order formulation of gravity• N=1 Global Supersymmetry in D=4
Index
• N=1 pure supergravity in 4 dimensions• D=11 supergravity• Bogomol’ny bound• Killing Spinors and BPS Solutions
Scalar field
Noether symmetry leaves the action invariant
Symmetry transformations
Metric (-,+,+,+…+)
map solutions into solutions
General internal symmetry
Infinitesimal transformations
General internal symmetryCommutator of infinitesimal transformations
Spacetime symmetries
Vector representation
Relations among Lorentz transformations
Lorentz condition
Spacetime symmetries
Orbital part
Lorentz algebra
Noether chargesInfinitesimal Noether symmetry
Noether current
Noether trick. Consider
Noether charges
Hamiltonian formalism
For internal symmetries
Noether charges
At quantum level
The fundamental spinor representations
The transformation induces a Lorentz transformation
Properties
Hermitean matrix
The Dirac Field
Applying the Dirac operator
Clifford algebra
The Dirac FieldExplicit representation for D=4 in terms of
Finite Lorentz transformations
The Dirac Field
Dirac action
Equation of motion for adjoint spinor
Weyl spinors
Undotted components
Dotted components
Weyl spinors
Energy momentum tensor
where
Clifford algebras and spinors
• Clifford algebras in general dimensions
Euclidean Clifford algebras
Clifford algebras and spinors
Clifford algebras and spinors
,
Clifford algebras and spinors
The antysymmetrization indicated with […] is always with total weight 1
distinc indexes choices
properties
Clifford algebras and spinors
Levi-Civita tensor
Schouten identity
Practical gamma matrix manipulation
More generally
Practical gamma matrix manipulation
No index contractions
Useful to prove the susy invariance of the supergravity action
Reverse ordering
Practical gamma matrix manipulation
• Other useful relations
In general
Basis of the algebra for even dimensions
Other possible basis
The highest rank Clifford algebra element
Provides the link bewteen even and odd dimensions
Properties
Explicit representationsAssume
implies
Explicit representationsimplies
Weyl spinors
No explicity Weyl representation will be used in these lectures
Odd space dimension D=2m+1The Clifford algebra for dimension D=2m+1 can be obtained by reorganazingthe matrices in the Clifford algebra for dimension D= 2m
The rank r and rank D-r sectors are related by duality relations
Not all the matrices are independent
Odd space dimension D=2m+1
Symmetries of gamma matrices
implies
Explicit forms conjugation matrix
The possible sign factors depend on the spacetime dimension D modulo 8And on r modulo 4
For odd dimension C is unique (up to phase factor)
Symmetries of gamma matrices
Symmetries of gamma matrices• Since we use hermitian representations, the symmetry
properties of gamma matrices determines also itscomplex conjugation
Adjoint spinor• We have defined the Dirac adjoint, which involves the complex
conjugate. Here we define the conjugate of “any” spinor using thetranspose and the charge conjugation matrix
Symmetry properties for bilinears
More in general
Majorana flip
Adjoint spinor
We have the rule
In even dimensions for chiral spinors
Questions-Comments I, IIIn even dimensions there are two charge conjugation
conjugation matricesSupersymmetry selects Because the supersymmetry is in
D=4
the left hand side is symmetric in alpha, beta therefore the right should alsobe symmetric, since
Questions-Comments I, II• Unique irreducible representation of the Clifford algebra• Traces and the basis of the Clifford algebra
Friendly representationsRecursive construction of generating Clifford algebra for
D=2m
Which is really real, hermitian, and friendly representation
is also real. Adding it as gamma2 gives a real representation in D=3.
which can be used as gamma 2m in D=2m+1
• This construction gives a real representation in 4 dimensions
Friendly representations
This one has an imaginary This construction will not give real Representations in higher dimensions
Friendly representations
Real representation for Euclidean gamma matrices in D=8
Friendly representations
Spinor indexes
Note
Spinor indexesThe gamma matrices have components
Fierz rearrangement• In supergravity we will need changing the pairing of
spinors in products of bilinears, which is called Fierzrearrangement
Basic Fierz identity from
Expanding any A as
Fierz rearrangement
Using
We get
Where
Completeness relation
Is the rank of
Fierz rearrangement
Cyclic identities
Which implies the cyclic identity
Analogously one can prove
Cyclic identity useful to study the kappa invariance of M2 brane
Multiplying by four commuting spinors
Cyclic identities• Notice the vector Is light-like
Charge conjugate spinorComplex conjugation is necessary to verify that the lagrangian involvingspinor bilinears is hermitian.
In practice complex conjugation is replaced by charge conjugation
Charge conjugate of any spinor
It coincides withe Dirac conjugate except for the numerical factor
Barred charge conjugate spinor
Reality properties
For a matrix M charge conjugate is
Majorana spinors• Majorana fields are Dirac fields that satisfy and addtional
“reality” condition, whic reduces the number degrees offreedom by two. More fundamental like Weyl fields
Particles described by a Majorana field are such that particles andantiparticles are identical
Majorana field
We have which implies
Recall
which implies
Majorana spinors
In this case we have Majorana spinors. We have that the barred conjugatedspinor and Dirac adjoint spinor coincide
In the Majorana case we can have real representations for the gammaMatrices . For D=4
Two cases
Majorana spinors
We have B=1, then Implies
Properties
also
• In case
Pseudo-Majorana spinors
We have pseudo-Majorana spinors, no real reprsentations of gammamatrices
Mostly relevant for D=8 or 9
Weyl-Majorana spinors
The two constraints
are compatible since
We have Majorana-Weyl spinor
D=2 mod 8. Supergravity and string theory in D=10 are based in Majorana-Weylspinors
Consider (pseudo) Majorana spinors for D=0,2,4 mod 8
Incompatibilty of Majorana and Weyl condition
which implies
The “left” and “right” components of a Majorana spinor are related by chargeby charge conjugation
Symplectic-Majorana spinors
We can define sympletic Majorana spinors
For dimensions D=6 mod 8 we can show that the sympleticMajorana constraint is compatible with chirality
which implies
Majorana spinors in physical theories
for D=2,3, 4 mod 8 . Majorana and Dirac fields transform in the same way underLorentz transformations, but half degrees of freedom
For commuting spinors vanishes
Is a total derivative, we need anticommuting Majoranaspinors
The Majorana field satisfies the conventional Dirac equation
Majorana spinors in physical theories
Majorna action in terms of “Weyl” fields, D=4
equations of motion
D=4 Majorana spinors in terms of Weyl spinors
Weyl representation
implies
preserves the Majorana condition
U(n) symmetry of Majorana fields
there is a larger U(n) chiral symmetry
The symmetry is manifest if we use the chiral projections
n,1 ,-1
U(n) symmetry of Majorana fields
This is manifestly U(n) invariant
The Maxwell and Yang-Mills Gauge Fields
Gauge invariancesa) Relativistic covariance is maintainedb) The field equations do not determine certain longitudinal componentsc) We have constraints, that restrict the initial data
The classical degrees of freedom are the independent functions required as Initial data for the Cauchy problem of hyperbolic equations.
An elliptic equation Does not contain degrees of freedom
Abelian Gauge Field
• Principle of minimal couplingFor a complex Dirac spinor of charge q
Gauge field
Covariant derivative
Abelian Gauge Field
• Free gauge field
Free equation of motion
Noether identity signal of the gauge symmetry
Bianchi identity
D-1 off-shell degrees of freedom
Maxwell algebra• For constant electromagnetic field there is a
generalization of the Poincare group
together with the generators of Lorentz transformations
Abelian Gauge Field
• Degrees of freedomGauge fixing, eg Coulomb gauge
This condition does eliminate the gauge freedom
If
which implies
Maxwell equation
Abelian Gauge Field
• Degrees of freedomIn the Coulomb gauge
implies no degrees of freedom
Initial data
Off-shell degrees of freedom
helicity states
massless particle
Abelian Gauge Field
• The field strength verifies
Gauge invriant description that electromagnetic field describesmassless particles
Abelian Gauge Field• Hamiltonian counting of degrees of freedom
Primary constraints
Secondary constraint
First class constraints
Gauge fixing constraints
Dirac bracket and number of degrees of freedom , 8-4=4=2+2
Noether identities• Combination of equation of motion that vanish identically
Action
Gauge transformations
Variation of the action
Noether identities
equations of motion =0
Abelian Gauge Field
• QED
Abelian Gauge Field
• Dual tensorsIn D=4
Selfdual-antiselfdual
properties
Duality for the electromagnetic field
• Free case The Maxwell and Bianchi equations
are invariant under the transformation
property
Chern-Simmons action L =A \wedge dAtopological action
Equations of motion F=dA=0 flatconnections
Duality for the electromagnetic field
Interacting theory with one ableian gauge field and a complex scalar
Bianchi identity and equation of motion
Define the tensor
Duality for the electromagnetic field
These equations are invariant under the transformation
Where
is an SL(2,R) transformation
If Which is the transformation of the scalar
Duality for the electromagnetic field
Magnetic and electric charges appear as sources for the Bianchi identityand generalized Maxwell equation
Transforms like
Schwinger-Zwanziger quantization condition for dyons
We have
S-duality
group
Duality for the electromagnetic field
Duality for the electromagnetic field
Duality for the electromagnetic field
Dimension of the symplectic group m(2m+1)
Duality for the electromagnetic field
• Duality transformations-symmetries of one theory (S-duality)-transformations from theory to another theory (M-
theory applications)
Non-abelian Gauge FieldAn element of the gauge group in the fundamental representation
Gauge potential
Non-abelian Gauge Field
Covarint derivatives
Non-abelian Gauge Field
Bianchi identity
Where
Action
Non-abelian Chern Simons
Equation of motion
Internal Symmetry for Majorana Spinors
real and therefore compatible with the Majorana condition
Internal Symmetry for Majorana Spinors
• D=4 Complex representation, we have the highest rank element
The chiral projectiosn transform
Variation of the mass term
If G=SU(n), the mass term is preserved by SO(n)
The free Rarita-Schwinger field
Consider now a free spinor abelian gauge fieldwe omit the spinor indexes
Gauge transformation
This is fine for a free theory, but interacting supergravity theories are more restrictive .We will need to use Majorana and/or Weyl spinors
Field strenght gauge invariant
The free Rarita-Schwinger field
• ActionProperties: a) Lorentz invariant, b) first order in space-time derivativesc) gauge invariant, d) hermitean
The lagrangian is invariant up to a total derivative
The free Rarita-Schwinger field
• Equation of motion
Noether identities
Using
We can write the equations of motion as
The free Rarita-Schwinger field
• Massless particles
Noether identities Identically vanishes
In D=3 the equations of motion are No local invariant degrees of freedom and therefore no propgating particlesmodels
Equivalent gauge transformation
Off shell degrees of freedom
The free Rarita-Schwinger field
• Initial value problemThe gauge fixes completely the gauge
the equations of motion in components
The free Rarita-Schwinger field
• Initial value problem
We have also
The restrictions on the initial conditions are
As we can see from
The free Rarita-Schwinger field
• Initial value problemThe on-sheel degrees freedom are half of
In D=4, with Majorana conditions, we frind two states expected for aa masslees particle for any s>0. The helicities are +3/2 and -3/2
Degrees of freedom
• Scalar fieldsWe consider a massive complex scalar field
We have
• Spinor fields
Dimensional reduction
Consider D=2m, the spinors in D+1 have the same number ofcomponents
The sign of of m has no physical significance, since it can be changedBy a field redefinition with
Dimensional reduction
• Periodic and antiperiodic boundary conditions
Fourier expansion
The D+1 Dirac equation
implies
where
Dimensional reduction
• Periodic and antiperiodic boundaryconditions
With a
We see
Dimensional reduction
• Maxwell Field
Gauge fixing condition implies
Dimensional reduction
Dimensional reduction
• Maxwell Field
Degrees of freedom ( initial conditions) 2(D-1), the on-shell degreesof freedom are D-1
k=0, 2( D-2) corresponding to the vector and 2 associated to an scalar
It coincides with the counting of a massless gauge vector in D+1
Dimensional reduction
• Action of the massive gauge field
Dimensional reduction
• Rarita Schwinger FieldConsider a massless Rarita-Schwinger field in D+1 with D=2m.We assume is antiperiodic, so the Fourier series
modes involve only half-integer k
We choose the gauge
The reduced equations are
Dimensional reduction
• Rarita Schwinger FieldGives the equation of motion of a massive RS field
There are two constraints
Th equation of motion becomes
Dimensional reduction
Differential geometry
• The metric and the frame fieldLine element Non-degenerate metric
Frame field
Inverse frame field
Differential geometry
• Frame field
Vector under Lorentz transformations
Vector field
Dual form
Volume forms and integrationcan be integrated
Canonical volume form depends of the metric or frame field
Volume forms and integrationdV
Action for fields
Hodge duality of forms
Lorentzian signature
Euclidean signature
For D=2m it is possible the constraint of self-duality or antiself duality
Hodge duality of forms
Euclidean signature
Lorentzian siganture
Is a top form and can be integrated
p-forms gauge fields
Bianchi identity
p-forms gauge fieldsequations of motion, useful relation
Bianchi identity
A p-form and D-p-2 form are dual
Algebraic equation of motion
p-forms gauge fields
Off-shell degrees of freedom, number of compoents of a p-form in D-1Dimensions.
On-shell degrees of freedom
First structure equation
• Spin connection
same transformation properties that YM potential for the group O(D-1,1)
it is not a Lorentz vector. Introduce thespin connection connection one form
The quantity
transforms as a vector
Let us consider the differential of the vielbvein
First structure equation
• Lorentz Covariant derivatives
The metric has vanishing covarint derivative.
First structure equation
The geometrical effect of torsion is seen in the properties of an infinitesimalparallelogram constructed by the parallel transport of two vector fields.
For the Levi-Civita connection the torsion vanishes
Non-vanishing torsion appears in supergravity
First structure equation
• Covariant derivatives
The structure equationimplies
is called contorsion
First structure equation
• Covariant derivatives
Matrix representation of Lorentz transformation
Spinor representation
First structure equation
• The affine connectionOur next task is to transform Lorentz covariant derivatives to covariantderivatives with respect to general conformal transformations
,Affine connection
vielbein postulate
relates affine connection with spin connection
First structure equation
Covariant differentation commutes with index raising
For tensors in general
First structure equation
• The affine connection
For mixed quantities with both coordinate ans frame indexes, it isuseful to distinguish among local Lorentz and coordinate covariantderivatives
Vielbein postulate euivalent to
First structure equation
• Partial integration
The second term shows the violation of the manipulations of the integration byParts in the case of torsion
We have
from which
Second structure equation
• Curvature tensorYM gauge potential for the
Group O(D-1,1)
YM field strength. We define the curvature two form
Second structure equation
Bianchi identities
we have
First Bianchi identity, it has no analogue in YM
usual Bianchi identity for YM
useful relation
Ricci identities and curvature tensorCommutator of covariant derivatives
Curvature tensor
Second Bianchi identity
Ricci tensorRicci tensor
Scalar curvature R=
If there is no torsion
Useful relation
Hilbert action
Dimensional analysis and Planck units
The first and second orderformulations of general relativity
• Second order formalismField content,
Action
The first and second orderformulations of general relativity
• Variation of the action
Last term total derivative due
Einstein equations
The Einstein equations are consistent only if the matter tensor
plus no torsion
The first and second orderformulations of general relativity
• Conservation energy-momentum tensorThe invariance under diff of the matter action
equations of motion
which implies
The first and second orderformulations of general relativity
• Scalar and gauge field equations
Ricci form of Einstein field equation
The first and second orderformulations of general relativity
Useful relations
Particular case of
The first and second orderformulations of general relativity
• Matter scalarsL=
The first and second orderformulations of general relativity
In absence of matter Solution
fluctuations
The gauge transformations are obtained linearizing the diff transformations
The first and second orderformulations of general relativity
Degrees of freedom. Choose the gauge
Fixes completely the gauge from
We have
The equations of motion become
• Degree of freedom
The first and second orderformulations of general relativity
The non-trivial equations are
Constraints
Since
The number of on-shell degrees of freedom, helicities is
oi terrm
Symmetric traceless representation
The first and second orderformulations of general relativity
Field content,
Spin connection dependent quantity
The first and second orderformulations of general relativity
The total covariant derivative and the Lorentz covariant derivativecoincide for spinor field but not for the gravitino
The first and second orderformulations of general relativity
Constant gamma matrices verify
The curved gamma matrices transforms a vector under coordinate transformationsBut they have also spinor indexes
holds for any affine connection with or without torsion
• Curved space gamma matrices
The first and second orderformulations of general relativity
• Fermion equation of motion
Steps in the derivation of Einstein equation
We drop a term proportional to the fermion lagrangian because we use theequations of motion for the fermion
The first and second orderformulations of general relativity
From which we deduce the Einstein equation
The stress tensor is the covariant version of flat Dirac symmetric stress tensor
Follows from the matter being invariant coordinateand local Lorentz transformations
The first and second orderformulations of general relativity
• The first order formalism for gravity and fermionsField content
Fermion field
Variation of the gravitational action
We have used
Same form of the action asin the second orderformalism but now vielbein and spin conn independent
The first and second orderformulations of general relativity
• The first order formalism for gravity and fermionsIntegration by parts
Form the fermion action
The first and second orderformulations of general relativity
• The first order formalism for gravity and fermionsThe equations of motion of the spin connection gives
If we substitute
the right hand side is traceless therefore also the torsion is traceless
The first and second orderformulations of general relativity
• The first order formalism for gravity and fermions
The physical equivalent second order action is
Physical effects in the fermion theories with torsion and without torsionDiffer only in the presence of quartic fermion term.This term generates 4-point contact diagrams .
N=1 Global Supersymmetry in D=4
• Susy algebra
equivalently
at quantum level
N=1 Global Supersymmetry in D=4
• Susy algebraIn Weyl basis
In this form it is obvious the U(1) R symmetry
N=1 Global Supersymmetry in D=4
• Susy algebraWe choose a Majorana representation for which all spinors are real. In a quantum theory the real spinor charge Q becomes a hermitean operator.
If we take the trace
N=1 Global Supersymmetry in D=4
• BPS statesApart from the vacuum states, which preserve all supersymmetries, theOnly states preserving some supersymmetry are states with null momentum
Since and
We have a BPS state with n=2
N=1 Global Supersymmetry in D=4
• General properties about representations
4- momentum
One particle states preserving n-supersymmetries are in some representation of theClifford algebra generated by (4-n) Qs
Massive particles. In the rest frame
Thre is a unique 4 d irreducible representation. Therefore supermultiplets will be multiple of 4 states, In massless case n=2, supermultiplets multiple of two states
N=1 Global Supersymmetry in D=4
• In any supermultiplet of one-particle states, the numberof bosons equal to number of fermions
Creation and annihilation fermionic opearors
In the massless case we have only one set of fermion creation and annihilationOperator, so we have one boson and one fermion.
N=1 Global Supersymmetry in D=4
• Basic multiplets
or
• Basic multiplets
N=1 Global Supersymmetry in D=4
Gauge gravity multiplet
N=1 Global Supersymmetry in D=4
• Conserved super-currents
Equations of motion
If we use
vanishes due to Maxwell equation and Bianchi identity
N=1 Global Supersymmetry in D=4
• Susy Yang-Mills Theory
Equations of motionplus Bianchi identity
The current is conserved
Basic fields: gauge boson
N=1 Global Supersymmetry in D=4
Now we need a Fierz rearragement
Is the tensor rank of the Clifford basis element
For anticommuting Majorana spinors , each bilinear has a definiteSymmetry under the interchange of
Super Yang Mills
the choices are
Therefore the supercurrent is conserved. It also conserved in other situations
N=1 Global Supersymmetry in D=4
• Susy field theories of the chiral multiplet
N=1 Global Supersymmetry in D=4
• Transformations rules of the antichiral multiplet
N=1 Global Supersymmetry in D=4
• Action
W(Z) superpotential, arbitrary holomorphic function of Z
Complete action
Are not a dynamical field, their equations of motion are algebraicwe can eliminate them
N=1 Global Supersymmetry in D=4
• Wess-Zumino model
Eliminating the auxiliary field F
N=1 Global Supersymmetry in D=4
• The action is invariant under susy transformations
The conserved supercurrent is given by
N=1 Global Supersymmetry in D=4
• Susy algebraNote that the anticommutator is realized as the commutator of two
variations with parameters
for Majorana spinorsIf we compute the left hand side, this dones not the anticommutator of thefermionic charges because any bosonic charge that commutes with fieldwill not contribute
N=1 Global Supersymmetry in D=4
• Susy algebra
has beenused
N=1 Global Supersymmetry in D=4
• Susy algebra
Fierz rearrangement is required
We have recovered the susy algebra via the transformations of fields
N=1 Global Supersymmetry in D=4
Now the symmetry algebra only closes on-shell
the extra factor apart from translation is a symmetric combination of the equationof the fermion field
N=1 Global Supersymmetry in D=4
the different weights are implied by the relation
N=1 Global Supersymmetry in D=4
One can show that
In the WZ model
to theelemntary field with a superpotential
Super Yang Mills
• Susy transformationsThe variation of
Consider the transformations
in units of mass
Super Yang Mills• The last term of the variation vanishes by the Fierz rearrangement
The supercurrent coincides with the one obtained before
Super Yang Mills
In 10d there is a topological term in the right hand side. Tensor cahrges notCarried by any particle could, there is no direct contradiction with theColeman-Mandula theorem
N=1 Global Supersymmetry in D=4
• More SYM action
Susy transformations
real pseudoscalar field in the adjoint representation
N=1 Global Supersymmetry in D=4
• Internal symmetries
Commutator of susy transformations
the gauge field dependent transformation is
• Representations
N=1 Global Supersymmetry in D=4
New Casimir
In the rest frame
N=1 Global Supersymmetry in D=4
• Representations
Values of the Casimir
Y superspin
Clifford vacuum
supermultiplet
Number of staes is a mutiple of four
• Basics
Therefore we have diffeomorphism. Thus local susy requires gravity
fields
1) If there is some sort of broken global symmetry. N=1 D=4 supergravity coupledto chiral and gauge multiplets of global Susy could describe the physics ofelementary particles
2) D=10 supergravity is the low energy limit of superstring theory. Solutions ofSUGRA exhibit spacetime compactification
3) Role of D=11 supergravity for M-theory
4) AdS/CFT in the limit in which string theory is approximated by supergravity.correlations of the boundary gauge theory at strong coupling are available fromweak coupling classical calculations in five and ten dimensional supergravity
IIA SUGRA bosonic fields
Fermionic fields, non-chiral gravitino, non-chiral dilatino
5,6
Gauge coupling unification
• The universal part of supergravity. Second orderformalism
Is the torsion-free spin connection
We not need to include the connectiondue to symmetry properties
GR can be viewed as a “gauge” theory of the Poincare group
The action
Einstein ‘s vacuum equation
Not invariant under local Poncaire. Torsion =0 by hand
• Transformation rules
Variation of the gravitational action
The variation of the action consists of terms linear in From the frame fieldvariation and the gravitino variation and cubic terms from the fiedd variation
of the gravitino action
• Transformation rules for gauge theory point of view
Gauge prameters
gauge transformations
In the second order formalism, partial integrationis valid, so we compute by two
Finaly we have
Therefore the linear terms cancel
First Bianchi identity without torsin
,
• Supersymmetry symmetry properties at the level of theequations of motion
Free super Maxwell eq of motion
Susy transformations
For local SUSY transformationsin in the linear approximation
the right side vanishes if the Einstein equation is satisfied
Buchdal problem
The supersymmetry transform of the Einstein equation vanishes if thegravitino satisfies its equation of motion. For linear fluctuations aboutMinkowski this is true if the SUSY transformation of the metric
• First order formalismWe regard the spin connection as an independent variable. We want to get theEquations of motion for the spin connection
The spin connection equation ofmotion is
valid for D=2,3,4,10, 11 where Majorana spinors exist
,therefore we have
• First order formalism
The fifth rank tensor vanishes for D=4. For dimensions D>4 this term is notVanishing and is one the complications of supergravity
The equivalent second order action of gravity is
With
• Local supersymmetry transformationsThe second order action for N=1 D=4 is supergravity is complete and it islocal supersymmetry
which includes the gravitino torsionThe variation of the action contains terms which are first, third and fifth orderin the gravitino field. The terms are independent and must cancel separately
Consider an action which is a functional of three variables
In the first order formalism the fifth variation is avoided, but we need to specifythe transformation of the spin connection. This procedure is complicated whenmatter multiplets are coupled to supergravity
but let us use the equation of motion for the spin connection and chain rule
• At the end substitute
• Let us work in the 1.5 formalism and rewrite the action of the Rarita Schwinger part
Recall
valid only in 4 dimensions
the presence of torsion. Also the Ricci tensor is not symmetric
The left acting derivativecan be partially integrated and acts distributively
recall
Last term using
plus ,the result cancels the
The last term with the first Bianchi identity
first term
Fierz rearrangement
the left hand side is antisymmetric in only the termscontribute
Infinitesimal transformation of the frame field
covariant form
the susy parameter
the dots means a symmetric combination of the equations of motion
• Generalizations
Supergravity in dimensions different from four
D=10 supergravities Type IIA and IIB are the low energy limits of superstringtheories of the same name
Type II A and gauged supergravities appear in ADS/CFT correspondence
D=11 low energy limit of M theory that it is not perturbative
Generalizations
The only non-vanishing part of
is
Extendes superalgebras there are several supercharges
We introduce the notation
Generalizations
N=2 SUSY
are central
‘Central’ charges in higher dimensions
GeneralizationsMore supercharges
Central charges
Gauged SUGRA
Recall the Kaluza-Klein compactification on
The Fourier modes of the symmetric tensor gives
More generally we study compactifications of a D’
A compact d-dimensional space
KK compactification keeps the massless and massive modes
Dimensional reduction keeps only the massless modes. The truncation is consistentif the field equation of the heavy modes are not sourced by the light modes,
We consider a toroidal compactification
Assume D’=11, we have a 32 Majorana spinor
generate the Clifford algebra in 7d euclidean space. In this basis the gravitino
8 gravitinos 7x8=56 spin 1/2
Note the total number of fermions 64 is the particle representation of N=8 susyAlgebra. If D’>11 we will have in 4d spins >2 for which a consistent theory is notKnow for finite number of fields. Vasiliev
Field content
4d metric 7 spin 1 particles 28 scalars
35 scalars 7 scalars (dual)
21 vectors
Which is the field content of N=8 SUGRA in d=4
Gauge transformation of 3-form
Ansatz action
Initially we use second order formalism with torsion-free spin connection
Bianchi dentity
• Ansatz transformations
useful relationsUseful relations
To determine the constants we consider the free action (global susy)
• transformations
using
• transformations
and Bianchi identity we get
To determine c we compute the commutator of two susy transformations
If With gauge transformation given theparameter
• transformationsthe conserved Noether current is (coefficient of
Ansatz for the action and transformations in the interacting case. We introducethe frame field and a gauge susy parameter
• Action
We need to find the dots
• Action
We need to cancel this term of rank 9. Recall
Suggest to introduce a term in the action
• Action
Full action
The spinor bilinears have a special role. They are non-vanishing fot the classicalBPS M2 and M5 solutions .
Coordinates of superspace
generated by the vector fields
Note the sign difference with respect toSUSY algebra
• Transformation in components
Covariant derivative
the algebra of covariant derivatives is the same that the original susyalgebra
Consider
Superaction
F term
D term
Integration of Grassman variables
Bogomol’ny bound• Consider an scalar field theory in 4d flat space time
There are two vacua at
We expect a domain wall separating the region of two vacua
We look for an static configuration connecting the two vacua
Bogomol’ny bound
• BPS procedureThe potential V can be wriiten in terms of superpotential W
Energy density in terms energy momentum tensor
Total energy
Bogomol’ny bound
where
We have an energy bound
which is saturated if the first order equation, BPS equation is verified
In this case the energy is
Domain wall as a BPS solution
One can prove that this BPS solution is also a solution of the second orderequations of motion
Notice that the domain wall is non-perturbative solution of the equations of motion
If the theory can be embbed in a supersymmetric theory, the solutions of theBPS equations will preserve some supersymmetry
Effective Dynamics of the domain wallThe width of the domanin wall is If we consider fluctuations of
the scalar filed with wave length >>L the dynamics of the will be Independent of of the details of the wall.
The lagrangian up to quadratic fluctuations is
Let us do the separation of variables
Effective Dynamics of the domain wallTo study the small perturbations we we should study the eigenvalueproblem
Exits a zero mode
This zero mode corresponds to a massless excitation and it is associatedwith the broken translation invariance
The action for these fluctuations given by
It describe the accion of a membrane, 2-brane, at low energies
Effective Dynamics of the domain wallThe membrane action to all orders is given by
where is teh determinat of the induced metric
Supersymmetric domain wall
• WZ Action
W(Z) superpotential, arbitrary holomorphic function of Z
Complete action
Are not a dynamical field, their equations of motion are algebraicwe can eliminate them
Domain wall ½ BPS The susy transformations for the WZ model are
For the domain wall ansatz the transformation of the should be
Domain wall ½ BPS
This condition implies
and
Note that this supersymmetric calculation recovers the result of the bosonic BPSCalculation. Therefore the domain wall is ½ BPS
This result can be deduced from the anticommutator of spinorial charges
Classical Solutions of Supergravity
• The solutions of supergravity give the metric, vector fields and scalar fields.
• The preserved supersymmetry means some rigidsupersymmetry
Killing Spinors and BPS Solutions
• N=1 D=4 supergravityFlat metric with fermions equal to zero is a solution of supergravity with
The residual global transformations are determined by the conditions
The Killing spinors of the Minkowski background are the set of 4 independentconstant Majorana spinors. We have D=4 Poincare Susy algebra
Vacuum solution
Killing vectors and Killing spinors
Killing Spinors and BPS Solutions
• The integrability condition for Killing spinors
A spacetime with Killing spinors satisfies
Integrability condition
only if
Killing spinors for pp-wavesAnsatz for the metric
For H=0 reduces to Minkowski spacetime in light-cone coordinates
Flat metric in these coordinates
Note that is a covariant constant null vector
Killing spinors for pp-wavesThe frame 1-forms are
From the first Cartan structure equation we get the torsion free spinconnection one forms
and from the second one
Killing spinors for pp-wavesThe Killing spinor conditions are
explicitely
All conditions are verified if we take constant spinors with constraint
Since there are two Killing spinors.
Killing spinors for pp-wavesNotice
To complete the analysis we need the Ricci tensor. The non-trvial component is
Therefore the pp-wave is Ricic flat if and only if H is harmonic in the variables x,y
pp-waves in D=11 supergravityEleven dimensional supergravity with bosonic fileds the metric and thefour-form field strength has pp-wave solutions
pp-waves in D=11 supergravity
If we choose
They have at least 16 Killing spinors. If one choose
The number of Killing spinors is 32!, like
Spheres
The metric of the sphere is obtained as induced metric of the flat
SpheresFrame one forms
Spin connection. First structure equation
Curvature. Second curvature equation
Constant positive curvature
SpheresRecursive proocedure for higher dimensional sphres
Spheres
• Coset structure
Anti-de Sitter spacesimple solutions of supergravity
with negative constant solution
Anti-de Sitter space
Ads as a coset space
Anti-de Sitter space
• MC 1-form
Ads metric
Ads can be embedded in pseuo-Euclidean space
Anti-de Sitter space
metric
Anti-de Sitter space
Anti-de Sitter spaceNote that varies in (-R.R)
Local parametrization
Global parametrization
Anti-de Sitter space
Different embeddings
Anti-de Sitter space, covers the whole hyperbolid for
the algunlar variables the whole
New radial coordinate
Another possibility
It is conformalto the direct product of the real line, time coordinate, times theSphere in D-1 dimensions
Anti-de Sitter space
Anti-de Sitter space
The metric is conformal to the positive region of D dimensional Minlowski space with coordinates
Killing spinor are solutions of
Integrability condition
If we insert vanishes identically
IIt is a hint that AdS is a maximally supersymmetric space
Frame fields
Spin connection
The last term includes transverse indexes.