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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 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)

• 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

• Super Poincare

Translations

Spinor supercharge

Lorentz transformations

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

R symmetry

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

• 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

• 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

Clifford algebras and spinors

• Clifford algebras in general dimensions

Euclidean Clifford algebras

The antysymmetrization indicated with […] is always with total weight 1

distinc indexes choices

properties

Levi-Civita tensor

Schouten identity

Practical gamma matrix manipulation

More generally

No index contractions

Useful to prove the susy invariance of the supergravity action

Reverse ordering

• 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

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

Real representation for Euclidean gamma matrices in D=8

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

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

• 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

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

Interacting theory with one ableian gauge field and a complex scalar

Bianchi identity and equation of motion

Define the tensor

These equations are invariant under the transformation

is an SL(2,R) transformation

If Which is the transformation of the scalar

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

Dimension of the symplectic group m(2m+1)

• 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

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

• ActionProperties: a) Lorentz invariant, b) first order in space-time derivativesc) gauge invariant, d) hermitean

The lagrangian is invariant up to a total derivative

• Equation of motion

Noether identities

We can write the equations of motion as

• 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

• Initial value problemThe gauge fixes completely the gauge

the equations of motion in components

• Initial value problem

We have also

The restrictions on the initial conditions are

As we can see from

• 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

• Periodic and antiperiodic boundary conditions

Fourier expansion

The D+1 Dirac equation

implies

• Periodic and antiperiodic boundaryconditions

With a

We see

• Maxwell Field

Gauge fixing condition implies

• 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

• Action of the massive gauge field

• 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

• Rarita Schwinger FieldGives the equation of motion of a massive RS field

There are two constraints

Th equation of motion becomes

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

• Lorentz Covariant derivatives

The metric has vanishing covarint derivative.

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

• Covariant derivatives

The structure equationimplies

is called contorsion

• Covariant derivatives

Matrix representation of Lorentz transformation

Spinor representation

• 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

Covariant differentation commutes with index raising

For tensors in general

• 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

• 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

• Variation of the action

Last term total derivative due

Einstein equations

The Einstein equations are consistent only if the matter tensor

plus no torsion

• Conservation energy-momentum tensorThe invariance under diff of the matter action

equations of motion

which implies

• Scalar and gauge field equations

Ricci form of Einstein field equation

Useful relations

Particular case of

• Matter scalarsL=

In absence of matter Solution

fluctuations

The gauge transformations are obtained linearizing the diff transformations

Degrees of freedom. Choose the gauge

Fixes completely the gauge from

We have

The equations of motion become

• Degree of freedom

The non-trivial equations are

Constraints

The number of on-shell degrees of freedom, helicities is

oi terrm

Symmetric traceless representation

Field content,

Spin connection dependent quantity

The total covariant derivative and the Lorentz covariant derivativecoincide for spinor field but not for the gravitino

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

• 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

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 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 order formalism for gravity and fermionsIntegration by parts

Form the fermion action

• 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 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

• Susy algebraIn Weyl basis

In this form it is obvious the U(1) R symmetry

• 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

• 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

• 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

• 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.

• Basic multiplets

Gauge gravity multiplet

• Conserved super-currents

Equations of motion

If we use

vanishes due to Maxwell equation and Bianchi identity

• Susy Yang-Mills Theory

Equations of motionplus Bianchi identity

The current is conserved

Basic fields: gauge boson

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

• Susy field theories of the chiral multiplet

• Transformations rules of the antichiral multiplet

• 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

• Wess-Zumino model

Eliminating the auxiliary field F

• The action is invariant under susy transformations

The conserved supercurrent is given by

• 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

• Susy algebra

has beenused

• Susy algebra

Fierz rearrangement is required

We have recovered the susy algebra via the transformations of fields

Now the symmetry algebra only closes on-shell

the extra factor apart from translation is a symmetric combination of the equationof the fermion field

the different weights are implied by the relation

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

• More SYM action

Susy transformations

real pseudoscalar field in the adjoint representation

• Internal symmetries

Commutator of susy transformations

the gauge field dependent transformation is

• Representations

New Casimir

In the rest frame

• 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

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

• 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

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

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

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

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

• MC 1-form

Ads metric

Ads can be embedded in pseuo-Euclidean space

metric

Anti-de Sitter spaceNote that varies in (-R.R)

Local parametrization

Global parametrization

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

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.

Santiago Lectures on Supergravity - ICCUBicc.ub.edu/.../Santiago_Lectures_on_Supergravity.pdf ·...

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