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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 70s. Yang Mills theories, Spontaneous symmetry breaking. Standard model

Supersymmetry

Yu. Golfand , 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. Golfand , 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 ? 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 Bogomolny 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 spi