Santiago Lectures on Supergravity - .Santiago Lectures on Supergravity ... Public Material Lectures

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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.

    A. Van Proeyen, Tools for supersymmetry, hep-th 9910030

  • History and overview ofSupergravity

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


    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


    Spinor supercharge

    Lorentz transformations

    Massless multiplets contains spins (s, s-1/2), for s=1/2, 1, 2,

    R symmetry

  • History and overview ofSupergravity


    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


    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


  • 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


  • Explicit representationsAssume


  • 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


    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


    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


    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


  • 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


    We get


    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


    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



  • 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