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Supergravity Notes

18th APCTP Winter Schol on Fundamental Physics, Pohang, Korea

Henning Samtleben

Universite de Lyon, Laboratoire de Physique, ENS Lyon,

46 allee dItalie, F-69364 Lyon CEDEX 07, France

henning.samtleben@ens-lyon.fr

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Contents

1 Introduction 3

2 Remarks on symmetries and supersymmetries 4

2.1 Bosonic symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Global supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Local supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Pure (N = 1) supergravity in D= 4 dimensions 93.1 The vielbein formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 The Palatini action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 The supersymmetric action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Matter couplings inD= 4,N = 1 supergravity 20

4.1 The structure of the bosonic sector . . . . . . . . . . . . . . . . . . . . . . . . 204.2 N= 1 and Kahler geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.3 The scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.4 AdS supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Extended supergravity inD= 4 dimensions 25

5.1 Multiplet structure forN >1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 D= 4,N = 2 supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Supergravity in higher dimensions 30

6.1 Spinors in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2 Eleven-dimensional supergravity . . . . . . . . . . . . . . . . . . . . . . . . . 33

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1 Introduction

There are several reasons to consider the combination of supersymmetry and gravitation.

The foremost is that if supersymmetry turns out to be realized at all in nature, then it

must eventually appear in the context of gravity. As is characteristic for supersymmetry,

its presence is likely to improve the quantum behavior of the theory, particularly interesting

in the context of gravity, a notoriously non-renormalizable theory. Indeed, in supergravity

divergences are typically delayed to higher loop orders, and to date it is still not ruled out

that the maximally supersymmetric extension of four-dimensional Einstein gravity might

eventually be a finite theory of quantum gravity only recently very tempting indications

in this direction have been unvealed. On the other hand, an underlying supersymmetric

extension of general relativity provides already many interesting aspects for the theory itself,

given by the intimate interplay of the two underlying symmetries. For example, the possibility

of defining and satisfying BPS bounds opens a link to the treatment of solitons in general

backgrounds. Last but not least of course, supergravity theories arise as the low-energy limit

of string theory compactifications.

These lecture notes can only cover some very selected aspects of the vast field of supergrav-

ity theories. We shall begin with pure or minimal supergravity, i.e.N= 1 supergravityin four space-time dimensions, which is minimal in the sense that it is the smallest possible

supersymmetric extension of Einsteins theory of general relativity. Next, we will consider

its matter couplings, and try to understand the restrictions which supersymmetry imposes

on the possible construction of consistent supergravity theories. Subsequently we will study

supergravity with extended supersymmetry (N > 1) and in higher dimensions (D > 4). Inparticular, we will discuss D = 5 supergravity which plays a distinguished role in the holo-

graphic context, providing a holographic description of aspects of particular four-dimensional

gauge theories. Finally we will discuss BPS solutions of supergravity, thereby illustrating

the use of the supersymmetric structure underlying matter coupled Einstein gravity for the

explicit construction of solutions to the highly non-linear field equations.

There exist many introductions and reviews on the subject, of which we mention the

following articles:

P. van Nieuwenhuizen [1] for a general introduction, and in particular for the N = 1case,

Y. Tanii [2], especially for theories in higher dimensions and with extended supersym-metry,

B. de Wit [3], in particular for the maximal supergravities, H. Samtleben [4], in particular for the gauged supergravities,

and the recent book

D. Freedman and A. Van Proeyen [5].

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2 Remarks on symmetries and supersymmetries

2.1 Bosonic symmetries

Symmetries play a central role in our understanding of the fundamental theories. From field

theory, we are familiar with different types of such symmetries:

Space-time symmetries are related to the geometry of space-time. E.g. relativistic field

theories are based on the Poincare group

GP = SO(1, 3) T4, (2.1)

generated by Lorentz transformations and translations, the latter acting as

(x) = (x), (2.2)

on the fields. Elementary particles can be classified according to the irreducible represen-

tations of (2.1) which turn out to be labeled by two numbers which are the eigenvalues of

the associated Casimir operators and physically correspond to the mass and the spin of the

particles.

Internal symmetries are related to the fact that fields may take values in some internal

auxiliary space. E.g. consider a Lagrangian

L = 12

ii , (2.3)

describing a set of scalar fields i, labelled by an index i = 1, . . . , n, such that scalars may be

thought of as taking values in the vector space Rn . Obviously, this Lagrangian is invariant

under orthogonal rotations

i

= ij

j

, so(n). (2.4)Particular interactions, such asL =V(|ii|), preserve this SO(n) symmetry. With con-stant symmetry parameter , these are globalsymmetries.

More fundamental are localsymmetries, in which the symmetry parameters themselves

are functions of the space-time coordinates. Obviously, the Lagrangian (2.3) is no longer

invariant under (2.4) if depends on the coordinates, but there is a well-known construction

to render it invariant by introducing a gauge field with minimal couplings. More precisely,

with covariant derivatives defined by

Di i Aijj , (2.5)

the covariantized Lagrangian

L = 12

DiDi , (2.6)

is invariant under the combined gauge transformation

i = ijj , A

ij =

ij Aikkj+ ikAkj = Dij , (2.7)

of scalar and vector fields. Adding a standard Yang-Mills term FF to capture the dynam-

ics of the vector fields gives rise to a consistent SO(n)-invariant theory. This is a standard

construction of electrodynamics (for n= 2) or non-abelian Yang-Mills theory, based on the

gaugingof an underlying global symmetry group Ggauge.

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2.2 Global supersymmetry

A natural fundamental question in view of the different symmetry groups above is the fol-

lowing: could the space-time symmetries and the internal gauge symmetries of a given model

be embedded in some bigger unifying group

GP Ggauge Gunifying, (2.8)The answer is negative as stated by the famous theorem of Coleman and Mandula [6]. The

notable exception to this no-go theorem has been analysed in Haag, Lopuszanski and Sohnius

[7] upon inclusion of fermionic generators in the underlying algebra. The associated trans-

formations aresupersymmetries. Let us study an example.

Consider the free Lagrangian of two scalar fields S, P, and a Majorana spinor

L0 = 12

SS+

1

2P

P i / . (2.9)

Before getting into any details, this is probably a good place to pause and review our spinor

conventions.

We use mostly minus signature, with = diag(+, , , ) with gamma matrices

{, } = 2 . (2.10)

The combination

5 i0123 , (2.11)

squares to the identity matrix and can be used to define chiral (Weyl) spinors as the eigenvec-tors of the projectors P 12(I 5). Rather, we will use four-component Majorana spinors,defined by imposing the reality condition

!= TC , (2.12)

with the Dirac conjugated spinor defined as 0, and the charge conjugation matrixC satisfying

CCT = I = CC , (2.13)

and relating the gamma matrices to their transposed

C1C = . (2.14)

Spinor bilinears transform, such as , , etc., transform covariantly under the Lorentz

group, while the reality condition (2.12) implies the relations

= , = , = ,5= 5 , 5=

5 , etc.. (2.15)

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Coming back to the Lagrangian (2.3), we can now realize that it is invariant under a

particular symmetry relating bosonic and fermionic fields, acting as

S = i ,

P = 5 ,

= 1

2 /(S i5

P) , (2.16)

where the symmetry parameter is a Majorana spinor itself. Invariance of (2.3) follows by a

simple calculation: variation of the first two terms after partial integration gives rise to

S(i) P

5

, (2.17)

whereas variation of the third term yields

i //

S i5P = i S i5P , (2.18)and precisely cancels (2.17). This is the first and simplest example of a supersymmetrythat

we meet. From it, we can already infer some characteristic features of supersymmetry.

Supersymmetry transforms fields of different spins and different statistics into eachother. In this example, the supermultiplet is made from a complex scalar S+iPanda Majorana spinor, satisfying the Klein-Gordon and the Dirac equation, respectively.

The number of bosonic degrees of freedom equals the number of fermionic degrees of

freedom. This is a general property of supersymmetric theories.

Mass dimensions of the scalar and spinor fields in D = 4 are [S] = 1 and [] = 32 ,respectively. From (2.16) we thus read off that the supersymmetry parameter comes

with mass dimension [] = 12.

The commutator of two supersymmetry transformations can easily be evaluated e.g. onthe scalar field S

[1 , 2 ] S = i2/S 1 S , (2.19)

and yields a translation with parameter =i21, which indeed is among thebosonic symmetries of the Lagrangian (2.3). We can conclude in general, that super-

symmetric theories are necessarily translation invariant. In terms of the associated

supergenerators Q, the algebra (2.19) reads

[Q, Q] = ()P , (2.20)

with the generator P of translations.

In general, interactions break the supersymmetry, there are however non-trivial inter-actions that can be added to (2.3) preserving invariance under supersymmetry upon

simultaneous modification of the supersymmetry transformation rules (2.16). Consider

Yukawa and scalar interactions

Lint = ig2

S+ i5P

+

8 S2 + P2

2

, (2.21)

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it can be shown that the LagrangianL0 + Lint is still supersymmetric upon propermodification of the supersymmetry transformation rules (2.16), if the two coupling

constantsg and are related as

= g2 . (2.22)

Interestingly, at this value also the quantum behaviour of the theory changes drastically:the leading divergence in the one-loop contribution to the scalar propagator comes with

a factor (g2) and thus drops out in the supersymmetric case (2.22). This is a genericfeature of supersymmetric theories: divergencies are milder (or sometimes even absent!)

due to cancellations between contributions from bosonic and fermionic origins.

2.3 Local supersymmetry

Let us now allow the supersymmetry parameter to depend on space-time coordinates, i.e.

consider localsupersymmetry instead of the global one introduced above. Upon generalising

(x), (2.23)the commutator of two supersymmetry transformations again leads to a transformation of

the form (2.19),

[1 , 2 ] S = i2/S 1 S , (2.24)with the difference that now the resulting translation parameter via (2.23) also depends on

the space-time coordinates. The r.h.s. of (2.24) thus represents the action of an infinitesimal

local diffeomorphisms on space-time. We conclude that a locally supersymmetric theory will

necessarily be diffeomorphism invariant and thus requires to treat the space-time metric as

a dynamical object. The simplest theory which achieves this is Einsteins general relativitywhich we shall consider in the following although higher order curvature corrections may

equally give rise to supersymmetric extensions. The reverse is also true: coupling of a theory

with rigid supersymmetry to gravity inevitably leads to a theory with local supersymmetry.

The easiest way to observe this is by considering the commutators of local Lorentz transfor-

mations and supersymmetry. Since the parameter transforms as a spinor under the Lorentz

group, this commutator leads to local supersymmetry transformations.

The dynamical space-time metric is described by the spin-2 state appearing in the su-

pergravity multiplets. Consider e.g. the supergravity multiplet inN = 2 supersymmetry,which contains a spin-2 state g giving rise to the graviton, together with a vector-spinor

of spin 3

2 denoted

, and a spin-1 gauge field A. If supersymmetry is unbroken, thesestates will be degenerate in mass. The fact that under extended supersymmetry bosonic

fields of different spin join in a single multiplet is quite remarkable. It shows that from a

supercovariant point of view their respective interactions are truly unified: we may thus

be able to recover the way gravity works from our knowledge of gauge theory by exploiting

the underlying supersymmetric structure.

We can now try to construct the extension of (2.3) that is invariant under a local exten-

sion of (2.16) in precise analogy to the construction of gauge theory (2.6) by introduction of

covariant derivatives. Even though for supersymmetry, this construction is far more compli-

cated and eventually leads to the introduction of an infinite number of additional couplings,

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it is quite instructive to work out in detail the first steps. Let us first reconsider the above

supersymmetry calculation and see what goes wrong with the invariance of (2.3) under (2.16)

when is coordinate-dependent. Whereas the contributions from (2.17) can still be brought

into this form, the terms from (2.18) now give an extra contribution due to the fact that the

derivative also hits the supersymmetry parameter :

L0 = i /S i5P () J () . (2.25)Indeed, we can identify J with the (spinor-valued) Noether current associated with the

global symmetry (2.16) of the Lagrangian. Just as in (2.6), this contribution was cancelled

by introduction of a gauge field A and extension of the Lagrangian by JA, the form of

(2.25) suggest to introduce a spinorial gauge field transforming as

= 1

, (2.26)

with a Noether coupling to the Lagrangian according to

LNoether = J

= i

/S i5P. (2.27)The dimensionalful coupling (with [] =1) has been introduced in (2.26) in order tomake up for the difference of mass dimension in a fermion and the supersymmetry parameter.

The field is a space-time vector whose entries are 4-component Majorana spinors, i.e. it

represents a 44 matrix. Counting of representation shows that it is a spin- 32 representation ofthe Poincare group. This field is called the gravitino and naturally arises in the supergravity

multiplets.

However, unlike the case of gauge theory (2.6), the construction does not stop here. The

-variation of (2.27) correctly cancels the unwanted contribution from (2.26), however the

-variation of (2.27) leads to new unwanted terms given by

LNoether i2

/

S+ i5P

/

S i5P . (2.28)Collecting the terms quadratic in Sand in P, we identify

i2

(SS+ P P) = i (T(S) + T(T)) , (2.29)

where we have used the gamma-matrix identity1

() = 2(

) g , (2.30)

and expressed the result in terms of the energy-momentum tensor of a free scalar field

T(S) SS 12

gSS . (2.31)

In order to cancel the new variation (2.29), one is thus obliged to introduce another new field

of index structure h=h() with supersymmetry variation

h = i i , (2.32)1Here, and in the following, symmetrizations and antisymmetrizations are indicated by round and square

brackets, respectively, and are always normalised to a total weight of 1, i.e. XAYB = X[AYB] + X(AYB),

etc..

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and coupling to the Lagrangian as

LNoether,2 = 2

h (T(S) + T(T)) . (2.33)

Putting together (2.3) and (2.33), we recognise the first terms in the expansion of the La-

grangian for scalar fields on a curved background

L = 12

|g| g (SS+ P P) , (2.34)upon expansion of the metric into

g = + h+ O(2),=

|g| g = h +1

2 h+ O(2) (2.35)

Continuing this procedure to all orders would eventually reproduce the infinite series corre-

sponding to (2.34) with taking the role of the gravitational constant. In accordance with

the general arguments from the supersymmetry algebra, we thus observe explicitly that local

supersymmetry implies the coupling of the fields to a dynamical metric, whose dynamics inturn is described by the Einstein field equations.

Rather than going through this lengthy procedure we will in the following make use of our

knowledge about Einstein gravity and its covariant matter couplings in order to construct

locally supersymmetric theories. We will first construct the minimal supergravity theory

in four dimensions, i.e. the theory describing only the graviton and the associated gravitino

which together form the smallestN= 1 supergravity multiplet. Our task shall be to constructa locally supersymmetric theory with this field content, such that by setting 0 we recoverEinsteins general relativity. I.e. we will work out the Lagrangian

L=

LEH+

L[] , (2.36)

which consists of a part describing the graviton by the Einstein-Hilbert action, and another

part describing its coupling to the gravitino.

3 Pure (N = 1) supergravity in D = 4 dimensionsIn this section, we will construct the minimal supersymmetric extension of standard Einstein

gravity in four space-time dimensions. In order to describe the coupling of spinor fields to

gravity, we first need to review the vielbein formalism of general relativity.

3.1 The vielbein formalism

We shall work in units where = = c = 1. The standard formulation of general relativity

includes the metric tensorgtogether with the Levi-Civita connection , defining covariantderivatives of tensors, as e.g. in the case of a vector field

X=X X , (3.1)implying thatX transforms as a tensor under diffeomorphisms. Further requiring thatthe metric is covariantly constant

g= 0 , (3.2)

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defines the Christoffel symbols as

= 12g

(g + g g) + K , (3.3)

whereKis the so-called contorsion, defined in terms of a torsion tensorT

=T[] as

K= 12(T

+T

+T

). The torsion tensor is not fixed by the metricity condition (3.2)

and usually set to zero, which amounts to having symmetric Christoffel symbols = ().Supergravity in its simplest formulation however exhibits a nontrivial torsion bilinear in the

fermionic fields, as we shall see.

The Riemann tensor can be expressed in terms of the Christoffel symbols; it is given by

R = 2[]+ 2

[|

|], (3.4)

where A[|B|] := 12(AB AB). From this one computes the Ricci tensor

R=R

,

and the Ricci scalarR= gR.

The Einstein-Hilbert action is given by the Lagrangian density

LEH= 14

|g|R , (3.5)

where g denotes the determinant of the metric, and the field equations are the vacuum

Einstein equations

R 12Rg= 0 . (3.6)In order to deal with spinors, whose transformation rules are difficult to generalise to curved

backgrounds, it is helpful to reformulate this theory in the so-called vielbein formalism.

The idea behind this is to consider a set of coordinates that is locally inertial, so that one

can apply the usual Lorentz behaviour of spinors, and to find a way to translate back to

the original coordinate frame. To be a bit more precise, let y a(x0; x), a= 0, . . . , 3, denote a

coordinate frame that is inertial at the space-time pointx0. We shall call these the Lorentz

coordinates. Then

e a (x0) :=y a(x0; x)

x

x=x0

,

compose the so-called vielbein, or tetrad in our case where atakes four different values. Under

general coordinate transformations x

x, the vielbein transforms covariantly, i.e.

e a (x) =

x

xe a (x) , (3.7)

while a Lorentz transformationy a ya =yb ab leads to

e a (x) =e b

(x) ab . (3.8)

In particular, the space-time metric can be expressed as

g(x) =e a

(x)e b (x) ab ,

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in terms of the Minkowski metricab= diag(1, 1, 1, 1). The vielbeine a thus takes lowerLorentz (or flat) indices a, b to lower indices in the coordinate basis (or curved) indices

, . Its inverse e a performs the transformation in the other direction. As an example

consider a 1-form with coordinates X, for which

X = e a

Xa , Xa = e a X .

On the other hand, contravariant indices are transformed as

X =Xae a , Xa =Xe a .

We can now introduce spinors (x) in our theory, which transform as scalars under the

general space-time coordinate transformations, but in a spinor representationRof the localLorentz group:

(x) = R((x)) (x) .In addition, we can convert the constant matrices of the inertial frame into matrices in

the curved frame by the action of the vielbein:

(x) =e a

(x)a = {, } = 2g.The choice of the locally inertial frame ya is of course only unique up to Lorentz transfor-

mations, i.e. the tetrad is only defined up to a rotation in the Lorentz indices e a e b ab .The algebra so(1, 3) of the local Lorentz transformation group SO(1, 3) is given by the gen-

erators Mab, which are antisymmetric in their indices and satisfy the commutation relations

[Mab, Mcd] = 2a[c Md]b 2b[c Md]a . (3.9)The action ofMab on vectors is given by

MabXc = 2c[aXb] ,

while on spinors it is

Mab= 12ab ,

where ab stands for [ab] and we have dropped the explicit spinor indices , as we will

usually do from now on. On the other hand, the space-time metric gstays invariant under

so(1, 3) transformations, so that these will automatically preserve the Einstein-Hilbert action.

In order to couple matter to our theory, we will also need the Lorentz covariant derivatives.

They are defined by

D + 12 ab Mab . (3.10)The ab are a set of gauge fields; there is one for each generator ofso(1, 3), correspondingly

they are antisymmetric in the Lorentz indices a, b. Together, they define an object called the

spin connection . It is determined by requiring that the tetrad is covariantly constant (the

so-called vielbein postulate):2

0 = De a e a = e a + a b e b e a .

2In our notation, here and in the following the covariant derivative D = D() always includes the spin

connection, but not the Christoffel symbols.

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An equivalent way to write this is

D[e a

] =

1

2Ta. (3.11)

This constitutes a set of algebraic equations for which has a unique solution expressing

the spin connection in terms of the vielbein and the torsion. For vanishing torsion, the

equations reduce to

D[e a

] = 0 , (3.12)

of which the solution is given by

ab =

ab[e] =1

2ec

abc bca cab

, (3.13)

where abc= e a e

b (ec ec) are the so-called objects of anholonomity. In the presence

of torsion, the solution of (3.11) is given by ab =

ab[e]+ Kab, where the contorsionKa

b

has been expressed in terms ofTaafter equation (3.3) above. The spin connection also

defines a curvature tensor with coefficients

Rab[] = 2

[

ab]

+ 2 ac[

b

]c . (3.14)

It can be shown that this curvature is in fact nothing but the ordinary space-time curvature

where two space-time indices have been converted to Lorentz indices, i.e.

R ab =R

e a e

b .

3.2 The Palatini action

We shall now apply the vielbein formalism to the formulation of general relativity. This is done

by rewriting the Einstein-Hilbert Lagrangian (3.5) directly in terms of vielbein quantities:

LEH[e] = 14

|g|R= 14 |e| e a e b R ab . (3.15)

Note that|e| stands for the determinant of the tetrad on the right hand side, while on theleft hand side the argument e is a short hand notation to denote dependence ofLon the fulltetrad e a . The equations of motion for this Lagrangian are the Einstein equations (3.6). In

the vielbein formalism, these second order equations can be expressed quite nicely in terms

of a set of two equations of first order. This goes under the name of Palatini (or first order)

formulation. It is achieved by considering the spin connection to be a priori independent

of the tetrad, described by the Palatini Lagrangian

LP[e, ] = 14 |e| e a e b R ab [] . (3.16)

There are now two field equations: The first one comes from the variation with respect to

the vielbein, LP/e, and the second from the variation with respect to the spin connection,LP/. Explicitly, the variation yields

LP = 12|e|

R a 1

2e a R

e a

3

2|e| (De a ) e [a e

b

e

c]

bc , (3.17)

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as we will show in a moment. Variation w.r.t. ab thus gives precisely (3.12) and is solved

by (3.13). Upon inserting this solution = [e] into the first equation from (3.17), we

recover the Einstein equations (3.6). In the classical theory, the Palatini formulation is hence

equivalent to the standard second order formulation of general relativity.

It is instructive to note that the relationLEH[e] =LP[e, ]|=[e] between the Einstein-Hilbert and the Palatini action provides a very simple way to also compute the variation ofthe former. Namely

LEH[e] =

LPe a

=

[e]+

LPbc

=

[e]

bc

e a

e a . (3.18)

However, the second term vanishes identically, because LP/ is proportional to (3.12)and thus zero for =

[e]. I.e. the variation ofLEH is simply given by the first term of

(3.17) and thus by Einsteins equations (3.6). A similar observation will greatly facilitate the

computation of a supersymmetric extension of Einstein gravity in the following.

Variation of the Palatini action. Let us derive in detail the variation of the Palatini

action (3.17). We first observe that the Palatini action (3.16), can be recast in the form

LP[e, ] = 116abcde c e d R ab [] (3.19)To see this, first note that

abcdefcd= 4[a[e b]

f], (3.20)

where one can determine the numerical factor by tracing over the index pairs (a, e) and (b, f),

on both sides. The tensor density e is obtained from eabcd by

=abcde a e b e

c e

d |e| , (3.21)

where the determinant shows up since we are dealing with a tensor densityrather than with

a tensor (as a consequence is just a number). Plugging (3.21) into (3.19) and making

use of (3.20) readily yields (3.16).

The equivalent form (3.19) is a convenient starting point for calculating its variation with

respect to the spin connection. Since only shows up in the expression for the Riemann

tensor, we are interested in R ab , for which we have

R ab = 2D[ ab

] .

This can easily be seen, since according to (3.14), R ab starts with the non-covariant term

2[ ab

] , followed by a term of the form , which must complete the non-covariantfirst part in order to yield a covariant quantity in total. We therefore find

LP = 116abcde c e d 2D ab ,where we dropped the antisymmetric bracket since already takes care of the antisym-

metrisation. Integration by parts leads us to

LP = 14abcde c De

d

ab ,

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where we made use of the antisymmetrisation by the tensor densities again. Using the

identity abcdec =6|e|e[aebed] (which follows in complete analogy to (3.21)), we

find

LP= 32 |e|

De d

e[a

ebed]

ab .

On the other hand, variation of the Palatini action w.r.t. the tetrad gives rise to

LP = 14R ab [] (|e| e a e b )= 14 |e| R ab [] (2e a e b e a e b (e c e c ))= 12 |e| R b [] (ab 12e b e a ) e a = 12 |e| (R a [] 12e a R) e a ,(3.22)

where we have used that (detA) = (detA) Tr[A1A] for any matrixA. Together, we arrive

at (3.17) for the variation of the Palatini action.

3.3 The supersymmetric action

We now come back to the reason why we introduced the vielbein formalism, which was the

inclusion of spinor fields and in particular of the gravitino in our theory. WhileLEH (orLP)can be considered as the kinetic term for gravity, we should now add a kinetic term for the

gauge field . Such a term has in fact been known since the 1940ies, when Rarita and

Schwinger published a proposition for the kinetic term of a spin- 32 field [8]:

LRS= 125D. (3.23)

Originally, Rarita and Schwinger wrote this equation in the context of quantum field theory

on flat Minkowski space. With (3.23) we have already done the necessary to covariantly

couple this field to a dynamical metric. In particular, we have used the Lorentz covariant

derivative D from (3.10), and introduced curved(coordinate-dependent) gamma matrices

according to

= ea a, (3.24)

such that

{a, b} = ab {, } = , (3.25)

with the flat Minkowski metric ab and the curved dynamical metric g.3 Just addingLRSto the Einstein-Hilbert Lagrangian will however not yet yield a supersymmetric theory. In

order to guarantee supersymmetry, we must also allow further interaction terms which contain

higher powers of . Our ansatz should thus be a Lagrangian of the form

L[e, ] = LEH[e] + LRS[e, ] + L4 [e, ] , (3.26)3As a note of warning, when we discussed global supersymmetry in section 2, we used coordinatesx for

flat Minkowski space. With hindsight, we should have denoted coordinates, the constant gamma-matrices,

etc. by flat indices a,b, . . . . It is only now that we have coupled the theory to gravity, that the difference

becomes important.

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where the last part only contains terms of order at least 4 in the gravitino. However, to

determine the actual form ofL we must first understand how the fields transform under asupersymmetry variation.

Our strategy shall be the following. We will first present the transformation laws for the

fields under a supersymmetry transformation, and verify them by checking the supersymme-

try algebra to lowest order in the fermions. After that, we shall come back to the Lagrangian(3.26) and determine its precise form such that it is supersymmetric.

To lowest order in the fermions, the supersymmetry transformation laws of the vielbein

and the gravitino are given by

e a

= ia , = D . (3.27)

At least heuristically, they seem of a good form, as the bosonic vielbein transforms into its

presumed superpartner and as which is supposed to play the role of a gauge field of

local supersymmetry indeed transforms as D, the derivative of the local parameter. More

precisely, expanding the metric around Minkowski space, these transformation laws preciselyreproduce what we found in the perturbative Noether analysis of (2.26), (2.32).

To establish the proposed transformations we will have to check that they satisfy the

supersymmetry algebra. In particular, the commutator of two such supersymmetry trans-

formations should give rise to a suitable combination of local symmetry transformations of

the theory, in a way similar to (2.24). Consider the commutator of two supersymmetry

transformations on the vielbein,

[1 , 2 ]e a

= 2[12]e a

= 2(i1a) 1(i2a)

= i1

aD

2

i2

aD

1

= i2D(2a1 1a2)

= iD(2a1) ,

where we have used the property a =a of Majorana spinors , in the last line.In order to understand this result, define the space-time vector field :=i2

1, such that

[1 , 2 ]e a

= D(e a )

= De a + e

a

. (3.28)

In the last equation we have used that D is the covariant derivative with respect to only the

Lorentz indices, which means that it operates as a mere space-time derivative on . Taking

into account (3.12), we can exchange the lower indices in the first term of the last line in

(3.28), such that by writing out the covariant derivative we are finally left with

[1 , 2 ]e a

= e

a + e

a

I

+ a be b

II

. (3.29)

We recognise partIas being the infinitesimal action of a diffeomorphism on e a , and part I I

as an infinitesimal internal Lorentz transformation e a

with ab = a b. We have thus

shown that the transformations (3.27) close onto the vielbein as

[1 , 2 ]e a

=+ , (3.30)

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into a combination of local symmetries: diffeomorphism and Lorentz transformation.

Proceeding, we should next apply the commutator of two supersymmetries to the grav-

itino. This however involves already fermionic terms of higher order (e.g. variation of the

connection term inD2 in2 gives rise to terms in (12) which are of the same fermion

order as the variation of possible higher order terms in 2 that we are neglecting

for the moment). Furthermore, it turns out that closure of the supersymmetry algebra onthe fermionic fields will require these fields to obey their first order equations of motion

meaning that the supersymmetry algebra only closes on-shell. This involves the exact form

of the Lagrangian, so let us first turn to the Lagrangian; we shall come back to this point in

a short comment in section 3.4.

We hence go back to the Lagrangian (3.26), and try to determine its constituents such

that it becomes supersymmetric. To do this, we will make use of a trick and formally include

the higher powers in the fermionic fieldsL4 into the first two parts, without changing theirgeneral form. This can be achieved by properly adapting the spin connection. Our starting

point is the Lagrangian

L0 LEH+ LRS , (3.31)which depends on the vielbein and the gravitino field. As we have done in the case of the

Palatini action, we may write this in the explicit form

L0[e,, [e]] = 14 |e| e a e b R ab [[e]] + 12

5D, (3.32)

with[e] from (3.13). Computing the variation of this Lagrangian under supersymmetry, we

will thus find schematically

L0 = L0e =

[e]e +

L0 =

[e] +

L0 =

[e]

[e]

e e . (3.33)

It would greatly simplify the calculation, if we could drop the last term by the same argument

used at the end of section 3.2, using the fact that L0/ was identically zero for = [e].Unfortunately, this is no longer true. As L0/ receives additional contributions from theRarita-Schwinger term, it follows that variation ofL0 w.r.t. the spin connection gives rise tothe equation

D[e a

] = i2 a, (3.34)instead of (3.12). As we have seen for (3.11) above, the solution of (3.34) is given by the

modified spin connection

ab

= ab

[e, ] =

ab

[e] + Ka

b

, with Ka

b

= i(

[a

b]

+ 1

2

a

b

), (3.35)

which depends quadratically on the gravitino . Our ansatz for the supergravity Lagrangian

will thus be

L = L0[e,,[e, ]] = 14 |e| e a e b R ab [] + 125D, (3.36)

where is given by (3.35). We also use the notation D= D() to indicate that the Lorentz

connection in this covariant derivative is built by . The Lagrangian (3.36) thus differs from

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(3.32) by quartic terms in . Correspondingly, we modify the supersymmetry transformation

rules (3.27) to

e a

= ia , = D , (3.37)

including additional cubic terms in the variation of the gravitino. We will now show that(3.36) already gives the full supersymmetric Lagrangian to all orders in the fermions.

Schematically, the variation of (3.36) is given by

L = L0e

=[e,]

e +L0

=[e,]

+L0

=[e,]

[e, ]e

e +[e, ]

,(3.38)

and now the entire last term vanishes, asL0/ is proportional to (3.34) and thus identicallyzero for = [e, ]. We can thus in the following computation consistently neglect all

contributions due to explicit variation w.r.t. the spin connection.

Using (3.17), we find that variation of the Einstein-Hilbert term then gives rise to

LEH = 12 |e|R a 12e a R (e a )= i2 |e|

R a 12e a R

a , (3.39)

where Ra = R[]

a denotes the Ricci tensor obtained by contracting the curvature (3.14)

of the modified spin connection .

On the other hand, varying LRS while again neglecting the terms from variation w.r.t.the spin connection yields

LRS = ( 125D)

= 1

2 D5D+ 5DD + ()5D .

Since the Lagrangian has to stay invariant only up to a total derivative, we can shift the

derivative D in the first term to the right by partial integration. Hence

LRS = 12

5D[D] 5D[D] (D[])5D+ ()5D

+ (total derivatives) , (3.40)

where we have used the possibility to antisymmetrise in various indices in the bracket, as

they are contracted with the totally antisymmetric tensor density. In the third term in

the bracket, the covariant derivative of the gamma matrix D[] = aD[e

a] is quadratic

in the fermions due to (3.34). Together with the two fermionic fields and , this term isthus at least quartic in the fermions. Likewise, the last term in (3.40) is at least quartic in

the fermions. Denoting these two quartic terms by(4) LRS and making use of [D, D] =

14R

ab ab, we find

LRS= 1165ab+ 5ab R ab + (4) LRS , (3.41)

where we have relabeled a few of the indices in the first terms. We observe that

5ab = ab5

= ab 5+ 2e[ab]5 ,

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and since ab = ie d dabc

c5 and (5)2 = 11, we have

5ab = ie d dabc

c + 2e[ab]5 .

Plugging this back into the variation (3.41) and keeping in mind that c =c forMajorana spinors, we are thus led to

LRS= i8e d dabcc R ab + (4) LRS 18(e[ab]5+e[ab]5)R ab .

Since we also have 5a= 5a for Majorana spinors, the last two terms in this expression

mutually cancel. If we furthermore make use of the identity e d dabc= 6|e|e[aebec],

we finally obtain

LRS= i8e d dabcc R ab = 6i8|e|e[aebec](c)R ab + (4) LRS= 3i4|e|([)R ] + (4) LRS= i

2 |e|(a)R a 12e a R + (4) LRS .

The first term precisely cancels the variation of the Einstein-Hilbert part (3.39).

We have thus shown that up to total derivatives the variation of the full LagrangianLfrom (3.26) reduces to the last two terms of (3.40)

L = (4) LRS = 12(D[])5D+ ()5D

, (3.42)

quartic in the fermions. With (3.34) we find

(4) LRS = i2 12(

a)(a5D) ( a5D)(a) . (3.43)A priori, the form of the two terms is rather different. In order to bring them into the same

form, we will need some so-called Fierz identities:

)( = 14()I 14(5)5 14(a)a + 14(a5)a5+ 18(ab)ab , (3.44)

where the l.h.s. is to be understood as a matrix to be contracted with further spinors from

the left and the right. (The easiest way to prove this identity is to note that the 16 matrices

{I, 5, a, a5, ab} Bform a basis of the 4 4 matrices which is orthogonal with respectto Tr(AB).) Applying (3.44) to the second term of (3.43) (with = D and = ),

and noting that due to the symmetry properties of Majorana spinors the bilinear [] for

Bis non-vanishing only for =a and =ab, we find i2 ( a5D)(a) = i4 ( a5b5a)(b5D)

i8 ( a5bc5a)(bc5D) .

Using that aba =2b and abca = 0, the second term vanishes while the first term

precisely cancels the first term of (3.43). Together, we have thus shown that the Lagrangian

(3.36) is invariant up to total derivatives under the supersymetry transformations (3.37). It

constitutes the sought-after minimal supersymmetric extension of Einstein gravity in four

dimensions.

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3.4 Results

Let us collect and complete our results, and simplify the notation. The full supersymmetric

Lagrangian is given byL =L0[e,,[e, ]] from (3.36). By properly adapting the spinconnection we got rid of all explicit quartic fermion interaction terms. Equation (3.34)

corresponds precisely to the equation

L0/ = 0 and is identically solved by given in

equation (3.35) as (schematically)

= [e, ] = [e] + K ,

depending on the tetrad and the gravitino. Equivalently, the full Lagrangian L can be broughtinto the form (3.26) in which the first terms depend on the standard spin connection[e], and

the quartic interaction terms are explicit. They can in fact be retrieved rather conveniently

by formally expandingL0[e ,,] around = . SinceL0 is no more than quadratic in ,this expansion stops after two terms; we find (again schematically)

L0[e ,,] = L0[e ,,]= K

L0[e , ,]

= +1

2K

2

L0[e , ,]= .Now the term linear in K vanishes, because L0/ = 0 at = , as we have already usedabove. Moreover, the first term on the right-hand side is precisely the full supersymmetric

Lagrangian (3.36). The last term finally has only contributions from the term in the cur-

vatureRab[] in the Einstein-Hilbert term. Together, this yields for the full supersymmetric

Lagrangian

L = L0[e,, [e]] 14 |e|

K aca K bb c+ K

abcKcab

, (3.45)

where Kabc given in (3.35) explicitly describes the quartic fermion interactions. One can see

that the fermions are subject to an interaction of current-current type.

Let us mention two more things before we conclude this chapter. The first one is that

with the full supersymmetry rules (3.37)

e a

= ia ,=D ,

(3.46)

containing cubic fermion terms in the variation of the gravitino, the computation of the super-

symmetry algebra leading to (3.30) needs to be revisited. Extending our previous calculation,

the commutator of two supersymmetry transformations on the tetrad can be shown to be

[1 , 2 ]e

a

=

D(

e

a

)= e a + e

a

+ a be b

+ i

a ,

for =i21 as in (3.29), where the last term is due to the non-vanishing torsion (3.34).

Again, the result is given by a combination of local transformations; the first two terms being

an infinitesimal diffeomorphism, the third term an internal Lorentz transformation (now with

ab = a b), while the last term represents an additional supersymmetry transformation

with parameter =. If on the other hand one applies the commutator of supersym-metry transformations on the gravitino, one will find that the only way to interpret the result

as a combination of local transformations involves the application of the equations of motion

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of the gravitino, i.e. the supersymmetry algebra closes only on-shell (some details for the

computation can be found in [2]). This is a generic feature of supersymmetric theories.

Finally, a remark concerning nomenclature: the straightforward although very tedious way

to prove supersymmetry of the full Lagrangian would amount to starting from its explicit

form (3.45) and check its invariance under (3.46) including lengthy terms up to order 5 due

to the explicit dependence of the spin connection on the tetrad. Historically, this is how four-dimensional supergravity was first constructed by Freedman, van Nieuwenhuizen and Ferrara

in 1976 [9]. Nowadays, this is referred to as the second order formalism. Soon after,

Deser and Zumino proved supersymmetry by using an analogue of the Palatini formalism,

treating the spin connection as an independent field. This first order formalism simplifies

the computation but requires to find the correct supersymmetry transformation law for the

independent spin connection. The method outlined in these lectures is the most common used

these days and goes under the name of the 1.5th order formalism. It combines features of

both of the older approaches. Morally, we stay second order in the sense that only tetrad and

gravitino are independent fields in the Lagrangian. From the first order formalism we borrow

however the observation that the equation obtained by formal variation of the LagrangianL/vanishes identically for = . As we have seen, this greatly simplifies the computationof the higher order fermion terms.

4 Matter couplings in D = 4,N = 1 supergravityIn the last chapter, we have discussed how to construct minimalN= 1 supergravity in D = 4dimensions, i.e. how to construct the supersymmetric couplings for theN= 1 supergravitymultiplet (g,

). The aim of this section is to sketch how we can implement further matter

inN= 1 supergravity (for details see [10, 11, 5]). Besides the supergravity multiplet, we mayconsider chiral multiplets (

i

, i

), consisting of a spin-1/2 Weyl fermion i

and a complexspin-0 scalar i for each i, and vector multiplets (M, AM ) containing one spin-1 gauge field

AM and a spin-1/2 Majorana fermionM for eachM. Both are representations of theN = 1

supersymmetry algebra and contain four degrees of freedom.

4.1 The structure of the bosonic sector

Let us first discuss the general form of the couplings of the bosonic sector in four-dimensional

supergravity, i.e. the interactions of the metric g, the scalar fields i and the (abelian)

gauge fieldsAM . Restricting the dynamics to two-derivative terms, these couplings are rather

constrained by diffeomorphism and gauge invariance which allows for the following terms:

the gravitational sector, described by the Einstein-Hilbert term

LEH= 14|e|R , (4.1)

which was extensively discussed in the last chapter.

the scalar sectorLscalar= 1

2|e| g jk Gjk (i) |e| V(i), (4.2)

where Gjk (i) is the metric on the scalar target space and V(i) is a scalar potential.

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the vector sector

Lvector = 14

|e| gg FFI(i) 1

8FF

R(i), (4.3)

described by a standard Maxwell term and a topological term (which does not depend

on the metric) in terms of the abelian field strength F

A

A

. The matrices

I(i) andR(i) may depend on the scalar fields. IfR is constant, the secondterm is a total derivative and can be considered as a generalization of the usual instanton

term.

Minimal couplings between scalar fields and vector fields can be introduced upon gaug-ing part of the global symmetries of the scalar Lagrangian (4.2), i.e. introducing non-

trivial gauge charges and covariant derivatives according to

D = At , (4.4)

with the generators t of the gauge algebra acting on the scalar fields via Killing vector

fields of the scalar target space

t i ki(). (4.5)

Indeed, it is a straightforward calculation to check that (4.5) is a symmetry of the

kinetic part of (4.2), provided k i is a Killing vector, i.e. satisfies

ikj+ jki = 0 LkGij = 0. (4.6)

The gauge algebra is given by the algebra of Killing vectors

kjjk

i

k

jjki = f

ki, (4.7)

with structure constants f. Accordingly, the abelian field strengthsFin (4.3) are

then covariantized to the standard Yang-Mills form

F A A f AA . (4.8)

In supersymmetric theories, the introduction of minimal couplings in the vector/scalar

sector also induces corrections in the fermionic sector, modification of the supersymme-

try transformation rules, and furthermore a contribution to the scalar potential. This

procedure of assigning non-trivial gauge charges to fields in the matter sector is usually

referred to as gauging of the theory.

The data that need to be specified in order to fix the bosonic Lagrangian thus are the

scalar dependent matrices Gjk (i),I(i),R(i), the potential V(i) and the minimal

couplings (4.5). The fermionic couplings of the Lagrangian are then entirely determined by

supersymmetry, without any freedom left. In turn, the bosonic data cannot be chosen arbi-

trarily, but are constrained by supersymmetry, i.e. by demanding that the bosonic Lagrangian

LEH+ Lscalar+ Lvector can be completed by fermionic couplings to a Lagrangian invariantunder local supersymmetry.

The construction of matter coupled supergravities is facilitated by powerful techniques

such as superspace techniques or the superconformal tensor calculus. We will not enter in

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the details of these approaches, but rather give a brief summary of some of the important

results on the resulting structures of matter couplings. As we shall describe in the next

section,N = 1 supersymmetry requires the scalar target space to be a K ahler manifold,and the combinationN R+ iI of the vector kinetic matrices to be a holomorphicfunction of the complex scalar fields i. Also the scalar potential is of very particular and

restricted form. Within extended supergravity, i.e. theories withN > 1 supercharges, thegeneric structure of the Lagrangian remains the same. However, the conditions on Gjk (i),IMN(i), andRMN(i) become much more restrictive. We come back to this in section 5below.

4.2 N = 1 and Kahler geometryThe essential restriction fromN= 1 supersymmetry on the scalar geometry states that thescalar target space should be a complex Kahler manifold. A good working definition of such

a manifold is the following: for a particular set of complex coordinates{i,}, the metric isobtained as the second derivative of a real function, the so-called Kahler potentialK(, ) as

Gij = 0 = G, Gi = iK(,). (4.9)

As an immediate consequence, the Christoffel symbols and the Riemann tensor have only

particular non-vanishing components, given by

kij , k , Rkl . (4.10)

The same geometrical structure is required for the possible scalar couplings in globally su-

persymmetric theories.4 In that case, the most natural way of understanding the appearance

of the Kahler potential comes from the construction of supersymmetric actions via integrals

over full superspace of functions of chiral superfields

S =

d4x

d4 K(,), (4.11)

whose integrand gives rise to terms like iK(,) i upon Taylor expansion on the

fermionic coordinates .

A mathematically more satisfying (coordinate-free) definition of a Kahler manifold simply

requires the existence of a covariantly complex structureJ2 = I, and hermitean metric. Inthe coordinates of (4.9), the complex structure takes the form

Jij = i ji ,

J =

i . (4.12)

Let us finally observe, that the form of (4.9) shows that the target space metric is invariant

under the following change of the Kahler potential

K(,) K(,) + f() + f(), (4.13)

for any holomorphic function f, referred to as Kahler transformations.

Let us consider two examples of Kahler manifolds of complex dimension n:

4Strictly speaking, local supersymmetry imposes an additional global condition on the geometry, the man-

ifold should be Kahler-Hodge [12].

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The homogenous Kahler potential

K(,) = ||2 i = Gi = ij , (4.14)

simply describes flat space Cn .

The Kahler potentialK(,) = 2 ln

1 + ||2 = Gi = 2

1 + ||2

ij i

1 + ||2

, (4.15)

describes the Fubini-Study metric on complex projective space CPn = SU(n + 1)/U(n) .

This is an Einstein space with

Ri = 2(n + 1)

2 Gi (4.16)

Let us finally note, thatN = 1 supersymmetry requires the couplings in the vector sector(4.3) to be compatible with the complex geometry of the scalar target space in the sense thatthe combination

N R(,) + iI(,), (4.17)

must be a holomorphic function on the Kahler manifold

N = 0. (4.18)

4.3 The scalar potential

Apart from the kinetic part defined by the Kahler geometry we have discussed, the scalar

sector (4.2) carries the scalar potential V(,) that we shall describe now. InN= 1 super-gravity, the potential has two different contributions

V(, ) = VF(,) + VD(,), (4.19)

referred to as F-term and D-term contributions, respectively. We describe them separately.

TheF-term contribution to the scalar potential is defined in terms of a (freely chosen)

holomorphic function W(), the so-called superpotential. The potential is of the form

VF(,) = e

KGi DiW DW 3|W|2 , (4.20)with the Kahler covariant derivatives

DiW = iW+ iK W . (4.21)

We note, that this is an indefinite potential, implying that the resulting supergravity theories

can in principle support AdS, dS, and Minkowski geometries. In contrast, recall that in glob-

ally supersymmetric theories, the scalar potential is always positive definite. In particular, a

constant superpotentialW =(in absence of scalar field fields) defines an extension ofN = 1supergravity by a negative cosmological constant. Let us also note, that the introduction of

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a superpotential also gives rise to additional Yukawa couplings in the fermionic sector and

modified fermionic transformation rules, such as

,L = DL+ i

2 e

K/2 W R+ . . . . (4.22)

Here, it turns out to be convenient to define the chiral projections

L = 1

2

I + 5

, R =

1

2

I 5 , etc., (4.23)

which map Majorana into Weyl spinors, which naturally couple to the fermions of the chiral

multiplets.

The D-term contribution to the scalar potential have a very different origin, they

arise from assigning non-trivial gauge charges to the scalar fields according to the general

discussion after (4.4). In order to realize the action of a non-trivial gauge group Gg on the

scalar fields, this group has to be embedded into the isometry group of the scalar target

space.N = 1 supergravity imposes further constraints, in order to allow for supersymmetriccouplings, the generators of the gauge group must be identified with holomorphic Killing

vectors on the Kahler manifold, i.e. Killing vectors that also preserve the complex structure

(4.12). In explicit complex coordinates, this means that the Killing vector condition (4.6)

ikj+ jki = 0, ik+ ki = 0, (4.24)

is amended by further demanding that

ik = 0 ik = 0, (4.25)

where in the last step we have used that ik = 0 = ik . We note, that (4.25) in fact impliesthe first equation of (4.24), while the second equation of (4.24) implies that locally the Killing

vector can be expressed as the gradient of a real function

ki = iiP , (4.26)

the so-called associated moment map. In term ofP, the conditions (4.24), (4.25) reduce tothe equation

ijP(, ) = 0. (4.27)

To realize a non-trivial gauge group, we will have to associate a holomorphic Killing vectorki, i.e. a moment mapP to every generator t of the gauge group. We note that themoment maps are defined by (4.26) up to constant shifts

P P+ c . (4.28)

There is a canonical choice of moment maps which transform in the adjoint representation

of the gauge algebra:

kjjki+ k

k

i ( ) = f ki, (4.29)

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with the structure constants from (4.7). This fixes the shift (4.28) except for thosePassociated to abelian factors in the gauge group. The remaining free parameters c are

compatible with supersymmetry and gauge symmetry and referred to as Fayet-Iliopoulos

parameters. They can be useful in order to shape the form of the scalar potential. Having

introduced the minimal couplings (4.4) further implies corrections in the fermionic sector and

the supersymmetry transformation rules in order to preserve supersymmetry. Eventually,supersymmetry even requires a further modification of the bosonic sector by adding the D-

term contribution

VD = 1

2

I1 PP, (4.30)to the scalar potential. The moment maps appearing in the potential are those subject

to (4.29). Note that unlike the F-term contribution (4.20) to the potential, the D-term

contribution (4.30) is positive definite.

What we have seen in this discussion is a structure typical in supergravity. Introduction

of non-trivial gauge charges induces additional couplings in the fermionic sector which in turn

imply a scalar potential in the bosonic sector. This is referred to as gauging supergravity,

see [4] for a review. ForN >1 supergravities and in higher dimensions D >4 this is the onlyknown mechanism which allows the introduction of a scalar potential. In particular, there is

no analogue of the F-terms (4.20) and a superpotential. In contrast to theD-term potential

(4.30) however, in these cases the potential induced by gauging is generically indefinite, i.e.

allows for AdS and dS solutions.

4.4 AdS supergravity

This is probably the simplest extension of pure D = 4,N= 1 supergravity, by introducinga (negative!) cosmological constant to the action. See [13].

5 Extended supergravity in D = 4 dimensions

5.1 Multiplet structure forN > 1In the last section we have discussed the couplings of chiral and vector multiplets to minimal

N = 1 supergravity. In principle, there is anotherN = 1 multiplet that one may consider,the so-called gravitino multiplet which consists of a spin-32 field and a spin-1 field A. Itscoupling is slightly more subtle due to a simple reason: just as the two degrees of freedom of

the massless vector are described by a standard U(1) gauge theory, the additional massless

gravitino must be described by introducing yet another local supersymmetry. In other words,

coupling the gravitino multiplet to theN= 1 supergravity multiplet gives rise to a theory,which has two local supersymmetries and is thus referred to asN = 2 supergravity [14].Accordingly, multiplets now organize under theN = 2 supersymmetry algebra in D = 4dimensions. TheN = 2 supergravity multiplet comprises the twoN= 1 multiplets mentionedabove and thus includes one spin-2 field, two spin- 32 fields and a spin-1 field A. Analogously

to the discussion of the last section, this theory may be enlarged by further coupling matter

inN= 2 multiplets.

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2 32 1 12 0 12 1 32 2N = 2 1 2 1 sugra

1 2 1 vector

1 2 1 hyper

N = 4 1 4 6 4 1 sugra

1 4 6 4 1 vectorN = 8 1 8 28 56 70 56 28 8 1 sugra

Table 1: The various multiplets forN = 2, 4, 8 supersymmetry. denotes the helicity of thestates, i.e. their charge under the SO(2) little group in D = 4 dimensions.

Upon coupling more gravitino multiplets one arrives at theories with larger extended

supersymmetry, whose structures and field contents are more and more constrained. The

particle content of the most common multiplets with extended supersymmetry (N = 2, 4, 8)are summarized in table 1. Note that while we have listed here the irreducible representations

of the super-Poincare group, we must add the CPT conjugates with flipped helicities to obtainthe physical field content. The only CPT self-conjugate multiplets are theN = 4 vectormultiplet and theN = 8 supergravity multiplet. In addition there are multiplets for othervalues ofN, involvingN = 3,N= 5 orN= 6 supercharges. We restrict the discussion toN 8 multiplets since representations of supersymmetry algebras withN >8 superchargescontain states with helicities larger than two. No consistent interacting theories involving a

finite number of such fields are known.

As mentioned above, the constraints on supergravity theories with extended supersym-

metry are very severe. ForN= 2, the scalar geometry is of special Kahler and quaternionicKahler type for vector and hyper-multiplets, respectively. For

N 3, the scalar geometry is

necessarily a symmetric space, which we discuss subsequently.

5.2 D = 4,N = 2 supergravityInN = 2 supergravity, scalar fields have their origin from nV vector multiplets and nHhypermultiplets, respectively, c.f. table 1. Accordingly, the scalar target space has the form of

a direct product MSKMQKwith the vector multiplet scalars describing a (2nV)-dimensionalSpecial Kahler manifold, and the hyper-scalars describing a (4nH)-dimensional quaternionic

manifold. We refer to [15, 16, 5].

5.3 Symmetric spacesAs mentioned above, forN 3, the scalar geometry in four dimensions is necessarily asymmetric space, i.e. the scalars are described as coordinates of a coset space G /K. More

precisely, the scalar target space ofN = 4 supergravity has to be of the formSO(6, n)

SO(6) SO(n) SL(2)

SO(2) , (5.1)

with some positive integern. The dimension of this manifold is 6n+2 and thus coincides with

the number of scalar fields expected for n vector multiplets and one supergravity multiplet

(including the CPT conjugated multiplet) withN= 4 supercharges, c.f. table 1.

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The (ungauged)N= 8 supergravity in D = 4 dimensions is unique and its field contentcan only consist of a single supergravity multiplet. The corresponding scalar target space is

given by the coset-space5

E7(7)SU(8)

. (5.2)

Because of dim E7(7)= 133 and dim SU(8) = 63, the target space dimension is 70 in agreementwith the number of scalars in the theory, c.f. table 1.

In this section, we shall discuss the structure of coset spaces G /K, which also describe

the target spaces of higher-dimensional extended supergravities. In general G is the global

symmetry group of the theory, and K is its maximal compact subgroup. A convenient formu-

lation of this sigma-model has the scalar fields parametrize a G-valued matrix V (evaluatedin some fundamental representation of G) and makes use of the left-invariant current

J = V1 V g LieG. (5.3)

In order to accommodate the coset space structure, J is decomposed according to

J = Q+ P , Q k, P p, (5.4)

where k Lie K andp denotes its complement, i.e. g= k p, orthogonal w.r.t. the Cartan-Killing form. The scalar Lagrangian is given by

Lscalar = 12e Tr (PP). (5.5)

It is invariant under global G and local K transformations acting as

V = V Vk(x), g, k(x) k, (5.6)

on the scalar matrix V. Under these symmetries the currentsQ andP transform accordingto

Q = k+ [ k, Q], P = [ k, P], (5.7)

showing that the composite connection Qbehaves as a gauge field under K. As such it plays

the role of a connection in the covariant derivatives of the fermion fields which transform

linearly under the local K symmetry, e.g.

Di i

1

4

ab ab i (Q)ki k , (5.8)

for the gravitinos i, etc. Likewise, one defines DV V Q = P for the scalarmatrixV. The current P on the other hand transforms in a linear representation of K,builds the K-invariant kinetic term (5.5) and may be used to construct K-invariant fermionic

interaction terms in the Lagrangian.

The two symmetries (5.6) extend to the entire supergravity field content and play a

crucial role in establishing the full supersymmetric action. They will furthermore be of vital

importance in organizing the construction of the gauged theories described in the next section.

5By E7(7) we denote here a particular real form of the complex exceptional group E 7, whose compact part

is given by SU(8).

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The global g transformations may be expanded as = t into a basis of generators tsatisfying standard Lie-algebra commutation relations

[t, t] = f t , (5.9)

with structure constants f.

The local K symmetry is not a gauge symmetry associated with propagating gauge fields(the role of the gauge field is played by the composite connectionQ), but simply takes care of

the redundancy in parametrizing the coset space G/K. It is indispensable for the description

of the fermionic sector, with the fermionic fields transforming in linear representations under

K. In particular, the scalar matrixV transforming as (5.6) can be employed to describecouplings between bosonic and fermionic fields, transforming under G and K, respectively.

To make this more explicit, it is useful to express (5.6) in indices as

VMN = (t)MKVKN VMK kKN , (5.10)

with G-generators (t)MK

and the underlined indicesK , N referring to their transformationbehavior under the subgroup K. The matrixVMN allows to construct couplings of e.g. abosonic field strengthFM transforming in the associated fundamental representation of G to

the fermionic fields according to (schematically)

FM VMN ( )N , etc. , (5.11)

where ( )N denotes the projection of the fermionic bilinear onto some K-subrepresentation

in the corresponding tensor product of K-representations.

It is often convenient to fix the local K symmetry by adopting a particular form of the

matrix

V, i.e. choosing a particular set of coset representatives. In this case, any global

G-transformation in (5.6) needs to be accompanied by a compensating K-transformation

V = V Vk , (5.12)

where k depends on (and onV) in order to restore the particular gauge choice, i.e. topreserve the chosen set of coset representatives. This defines a non-linear representation of

G on the (dimG dim K) coordinates of the coset space, i.e. on the physical scalar fields.Likewise, it provides a non-linear realization of the group G on the fermion fields via the

compensating transformation k. Two prominent gauge fixings are the following:

unitary gauge: in which the matrix

V is taken of the form

V = exp {a Ya} , (5.13)

where the non-compact generators Yaspan the spacep. In this gauge, thea transform

in a linear representation of K G, thus global K-invariance of the Lagrangian remainsmanifest. The currentP = P

aY

a takes the formPa =a + . . . , where dots refer to

higher order contributions. This shows that the kinetic term (5.5) is manifestly ghost-

free with Gab() ab+ . . . . It is here that the importance of K being the maximalcompact subgroup of G shows up.

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triangular gauge: in which the matrixV is taken of the form

V = exp {m Nm} exp

h

, (5.14)

where = 1, . . . , rank G, labels a set of Cartan generators h of g and the Nm form

a set of nilpotent generators such that the algebra spanned by{h, Nm} constitutes aBorel subalgebra ofg. With suitable choice of the Borel subalgebra, it is in this gaugethat a possible higher-dimensional origin of the theories becomes the most transparent.

The grading associated with the chosen Borel subalgebra is related to the charges of the

fields under rescaling of the volume of the internal compactification manifold. See [17]

for a systematic discussion for the maximal theories and their eleven-dimensional origin.

E.g. as mentioned above, the scalar sector of the maximal (N = 8) supergravity inD = 4 space-time dimensions is described by the coset space E7(7)/SU(8). This theory can

be obtained by dimensionally reducing D = 11 supergravity on a torus T7. The eleven-

dimensional origin of the fields can be identified in the triangular gauge associated with the

GL(7) grading of E7(7)

. A type IIB origin of the fields on the other hand is identified in the

triangular gauge associated with a particular GL(6) SL(2) grading. For the half-maximal(N= 4) supergravity in D = 4 dimensions, the scalar sector is described by the coset space

G/K = SL(2)/SO(2) SO(6, 6+ n)

(SO(6)SO(6+n)), (5.15)

c.f. (5.1). This theory can be obtained by dimensional reduction of the ten-dimensional type-I

theory withn vector multiplets, upon reduction on a torusT6. The ten-dimensional origin of

the fields is identified in the triangular gauge associated with the GL(6) grading of SO(6 , 6).

In order to construct the full supersymmetric action of the theory, it is most convenient to

keep the local K gauge freedom. When discussing only the bosonic sector of the theory, it is

often functional to formulate the theory in terms of manifestly K-invariant objects. E.g. thescalar fields can equivalently be described in terms of the positive definite symmetric scalar

matrixMdefined by

M V VT , (5.16)

where is a constant K-invariant positive definite matrix (e.g. for the coset space SL(N)/SO(N),

withVin the fundamental representation, is simply the identity matrix). The matrixMis manifestly K-invariant and transforms under G as

M = M + M T , (5.17)

while the Lagrangian (5.5) takes the form

Lscalar = 18Tr (M M1). (5.18)

To finish this section, let us evaluate the general formulas for the simplest non-trivial

coset-space SL(2)/SO(2) which appears in the matter sector of several supergravity theories.

With the sl(2) generators given by

h=

1 0

0 1

, e=

0 1

0 0

, f=

0 0

1 0

, (5.19)

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the matrixVin triangular gauge (5.14) is given as

V = eCe e h =

1 C

0 1

e 0

0 e

. (5.20)

Evaluating the non-linear realization (5.12) of SL(2) on these coset coordinates C, leads to

h = 1, h C= 2C , e C= 1, f = C , fC= e4 C2 . (5.21)

This shows thathacts as a scaling symmetry on the fields, whereas eacts as a shift symmetry

onC, andfis realized non-linearly. This toy example exhibits already all the generic features

of the global G-symmetries that we will meet in more generality in section ?? below.

The matrixM for this model can be computed from (5.16) with = I2, and is mostcompactly expressed in terms of a complex scalar field =C+ ie2, giving rise to

M = 1

||2

1 , (5.22)

while the kinetic term (5.5) takes the form

e1 Lscalar = 14e4 C C = 1

4()2 . (5.23)

It is manifestly invariant under the scaling and shift symmetries of (5.21), whereas invari-

ance under the non-linear action of fis not obvious and sometimes referred to as a hidden

symmetry. In terms of, the action of a finite SL(2) group transformation can be given in

the compact form

a+ bc+ d

, for exp() = a b

c d

SL(2). (5.24)6 Supergravity in higher dimensions

The motivation to study supergravity theories in dimensions D > 4, and their dimensional

reduction is two-fold. On the one hand supergravity appears as the low energy effective action

for fundamental theories such as string theories, which generically live in higher dimensions.

In this top-down approach one has to consider the dimensional reduction of these theories

in order to derive from the fundamental theories a meaningful four-dimensional theories

supposed to describe our universe. On the other hand from a purely four-dimensional pointof view, the dimensional reduction of higher-dimensional supergravities can be considered as

a powerful technique in order to construct extended D = 4 supergravity theories.

6.1 Spinors in higher dimensions

In order to discuss extended supergravities we have to know the possible types of spinors

in higher space-time dimensions, see e.g. [18] for a reference. In this section we consider

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D-dimensional Minkowski space-time with signature = diag(+1, 1, . . . , 1). The spino-rial representation of the Lorentz group can be obtained from a representation of the D-

dimensional Clifford algebra associated with the metric ab. The generators of this algebra,

called Gamma matrices a, satisfy the anticommutation relations

{a, b

}= 2 ab . (6.1)

The spinorial representation of theD-dimensional Lorentz algebra SO(1, D 1) can then beconstructed using antisymmetric products of gamma matrices; it is straightforward to verify

that the Clifford algebra (6.1) implies that the generators

Mab 12

ab, ab= [ab] , (6.2)

satisy the Lorentz algebra (3.9). Finite Lorentz transformations are given by exponentiation

R() = exp1

2ab Mab .

It can be shown that in even dimensions, the Clifford algebra has a unique (up to similarity

transformations) irreducible representationCD which has 2[D/2] complex dimensions. Itselements are referred to as Dirac spinors. An explicit construction can be given recursively:

denoting by j the Gamma-matrices in (D 2)-dimensional space-time, the ensemble

i

0 jj 0

, i

0

0

, i

0

0

, (6.3)

satisfies the algebra (6.1) in D-dimensional space-time.

An important object is the following combination of Gamma matrices

= iD2+101 D1 , (6.4)

which generalizes the 5 in D = 4 dimensions. The Clifford algebra implies that it satisfies

relations

{ , a} = 0, 2 = . (6.5)

This has two important consequences:

The ensemble (, a) of matrices defines a representation of the Clifford algebra in

D+ 1 dimensions. This shows that in also odd dimensions, the Clifford algebra (6.1)has an irreducible representationCD+1 CD of 2[D/2] complex dimensions. In fact, inodd dimensions there are two inequivalent representationsCD+1,C D+1 of the Cliffordalgebra, related by , a a, which can not be achieved by a similaritytransformation. Via (6.2) this defines a (unique and irreducible) 2[D/2]-dimensional

representation of the Lorentz group in odd dimensions.

In even dimensions, the operator commutes with all generators of the Lorentz alge-bra: [, Mab] = 0. This implies that as a representation of the Lorentz algebra, the

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D W M pM MW dimension (real)

2 13 24 45 8

6 87 168 169 16

10 1611 32

Table 2: Possible types of spinors in D-dimensional Minkowski space-time. Weyl, Majorana,

pseudo-Majorana and Majorana-Weyl are denoted by W, M, pM and MW respectively.

representationCD is reducible and may be decomposed into its irreducible parts byvirtue of the projectors

P 12

i

. (6.6)

The corresponding 2[D/2]1-dimensional chiral spinors are calledWeyl spinors.

Let us now consider the reality properties of the spinors. It follows from (6.1) that also

the complex conjugated Gamma matrices a fulfill the same Clifford Algebra

{a,

b

}= 2ab .

Because of the uniqueness of the representations of the Clifford algebra discussed above, this

implies that a has to be related to a (up to a sign) by a similarity transformation

a = BaB1 .

This suggests to impose on the spinors on of the following reality conditions

= B; and = B+ ,

which are referred to as Majoranaand pseudo-Majorana, respectively. Obviously, these con-

ditions only make sense provided that

BB= 1,

which only holds in certain space-time dimensions. A closer analysis shows that (pseudo)

Majorana spinors exits in dimensions D 0, 1, 2, 3, 4 mod 8. They possess half of thedegrees of freedom of a Dirac spinor. In even dimensions, we may further ask if these reality

conditions are compatible with the chiral split (6.6) into Weyl spinors. It turns out that this

is only the case in dimensions D 2 mod 8.All possible types of spinors in D 11 dimensional Minkowski space-time are given in

table 2. Based on these structures, supergravity theories can be constructed in dimensions

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D spinors N fields nB

11 M 1 ea, , B 128

10 MW (1,1) ea, +, , C , B, A, +, , 128

(2,0) ea, 2+, C

(+) , 2B, 2, 2 128

(1,0) ea, +, B, , 64A, + 8

9 pM 2 ea, 2, C, 2B, 3A, 4, 3 128

1 ea, , B, A, , 56

A, , 8

8 pM 2 ea, 2, C, 3B, 6A, 6, 7 1281 e

a, , B, 2A, , 48A, , 2 8

7 sM 4 ea, 4, 5B, 10A, 16, 14 128

2 ea, 2, B, 3A, 2, 40A, , 3 8

6 sMW (2,2) ea, 4+, 4, 5B, 16A, 20+, 20, 25 128

(2,1) ea, 4+, 2, 5B

(+) , B

() , 8A, 10+, 4, 5 64

(1,1) ea, 2+, 2, B, 4A, 2+, 2, 32A, 2+, 2, 4 8

(2,0) ea, 4+, 5B

(+) 24

B() , 4, 5 8

(1,0) ea, 2+, B

(+) 12

A, 2+ 4

B() , 2, 4

2

, 4 4

5 s pM 8 ea, 8, 27A, 48, 42 128

6 ea, 6, 15A, 20, 14 64

4 ea, 4, 6A, 4, 24

A, 4, 5 8

2 ea, 2, A 8

A, 2, 42, 4 4

4 M 8 ea, 8, 28A, 56, 70 128

6 ea, 6, 16A, 26, 30 64

5 ea, 5, 10A, 11, 10 32

4 ea, 4, 6A, 4, 2 16A, 4, 6 8

3 ea, 3, 3A, 8

A, 4, 6 8

2 ea, 2, A 4

A, 2, 42, 4 4

1 ea, 2

A,

2, 2 2

Table 3: Supermultiplets in different space-time dimensions. The red background indicates the

supergravity multiplets. The last column gives the number of bosonic degrees of freedom.

D 11. Similar to what we have seen in four dimensions, an analysis of the representations ofthe supersymmetry algebras in the various dimensions shows that apart from the supergravity

multiplet for a given number of supercharges, in general there exist also matter multiplets,

that can consistently be coupled to supergravity. We give a table of the various multiplets

in table 3. The highest-dimensional supergravity theory is the unique eleven-dimensional

theory, constructed 1978 by Cremmer, Julia and Scherk [19], which we shall discuss in some

more detail now.

6.2 Eleven-dimensional supergravity

The highest-dimensional supergravity theory lives in eleven space-time dimensions and was

constructed by Cremmer, Julia and Scherk [19]. Its field content is given by the lowest mass-

less representation of the supersymmetry algebra. For massless states which fulfillPP = 0

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we can set without loss of generality P0 = P10 and all other components to zero. The

supersymmetry algebra then turns into

{Q, Q} = 2P

0

= 2P0

1 + 100

, (6.7)

where the Q are the 32 independent real supercharges in D = 11 dimensional Minkowski

space-time. As the r.h.s. of (6.7) describes a projector of half-maximal rank in spinor space,it follows that only 16 out of the 32 supercharges act non-trivially on massless states and

satisfy the Clifford algebra (6.1). From the discussion of the previous section, we then know

that there is a 28 = 256-dimensional irreducible representation C16 of this algebra which thengives the field content of the eleven-dimensional theory. Under SO(16), this representation

decomposes into 128s+ 128c chiral spinors (6.6). It is important to stress that in contrast

to the discussion of the last section, the 16 here has no meaning as a number of space-time

dimensions, but just denotes the number of supercharges; we are just exploiting the fact, that

the same abstract algebra (6.1) appears in a rather different context again. In order to find

the space-time interpretation of the 256 physical states, we note that the 16 supercharges form

the physical degrees of freedom of a space-time spinor in eleven dimensions, i.e. they buildan irreducible 16-dimensional representation of the little group SO(9) under which massless

states are organized in eleven dimensions.

We thus consider the embedding SO(9)SO(16) of the little group into SO(16), underwhich the relevant representations decompose as [20]

16 16,128s 44 84,128c 128. (6.8)

What is the meaning of these SO(9) representations? Recall, that a massless vector in elevendimensions has 9 degrees of freedom and transforms in the fundamental representation of

SO(9). The 44 corresponds to the symmetric traceless product of two vectors: these are

the degrees of freedom of a massless spin-2 field, the graviton. The 84 on the other hand

describes the three-fold completely antisymmetric tensor product of vectors:93

= 84 . The

bosonic field content of eleven-dimensional supergravity thus is given by the metric g and

a totally anti-symmetric three-form field A. The fermionic field content of the theory is

the irreducible 128 of SO(9) which is the Gamma-traceless vector-spinor 9 16 16 = 128.Accordingly, this describes the degrees of freedom of a massless spin-3/2 field, the gravitino.

The full eleven-dimensional supergravity multiplet thus is given by (g, , A). A similar

argument shows why supergravity theories do not exist beyond eleven dimensions: repeatingthe same analysis for say a twelve-dimensional space-time yields a minimal field content that

includes fields with spin larger than two. No consistent interacting theory for such fields can

be constructed.

The action of eleven-dimensional supergravity is given by the eleven-dimensional analogue

of (3.45)

L0[e, ] = 14|e| R 1

4|e|D, (6.9)

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supplemented by a kinetic, a topological term and a fermionic interaction term for the three-

form field A

LA[A,e,] = 196

|e|FF

2

6912 FFA

2

384 |e| + 12 F , (6.10)with the abelian field strength F = 4[A ], and quartic fermion terms. Most other

supergravities can be obtained by Kaluza-Klein reduction from this theory.

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