General Matter Coupled N=2 Supergravity

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UCLA/96/TEP/3

POLFIS-TH 02/96 NSF-ITP-96-11

CERN-TH/95-350 hep-th/xx9603004

General Matter Coupled N=2 Sup ergravity

1 2 y 2;4

L. Andrianop oli , M. Bertolini , A. Ceresole

2 35 6

R. D'Auria ,S.Ferrara and P.Fre

Dipartimento di Fisica, Universita di Genova, via Do decaneso 33, I-16146 Genova, Italy

Dipartimento di Fisica, Politecnico di Torino,

Corso Duca degli Abruzzi 24, I-10129 Torino, Italy

CERN, CH 1211 Geneva 23, Switzerland

Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

DepartmentofPhysics, University of California, Los Angeles, CA 90024, USA

Dipartimento di Fisica Teorica, UniversitadiTorino, via P. Giuria, 1 I-10125 Torino, Italy

Abstract

The general form of N = 2 sup ergravity coupled to an arbitrary number of vector multiplets

and hyp ermultiplets, with a generic gauging of the scalar manifold isometries is given. This ex-

tends the results already available in the literature in that we use a co ordinate indep endent and

manifestly symplectic covariant formalism which allows to cover theories dicult to formulate

within sup erspace or tensor calculus approach. We provide the complete lagrangian and sup er-

symmetry variations with all fermionic terms, and the form of the scalar p otential for arbitrary

quaternionic manifolds and sp ecial geometry, not necessarily in sp ecial co ordinates. Our results

can b e used to explore prop erties of theories admitting N = 2 sup ergravityaslow energy limit.

Fellowby Ansaldo Ricerche srl, C.so Perrone 24, I-16152 Genova

Supp orted in part by DOE grant DE-FGO3-91ER40662 Task C., EEC Science Program SC1*CT92-

0789, NSF grant no. PHY94-07194, and INFN.

1 Intro duction

Impressive results over the last year on non p erturbative prop erties of N = 2 sup ersym-

metric Yang-Mills theories[1, 2 ] and their extension to string theory[3, 4, 5, 6 ] through the

notion of string-string duality[7 ], have used the deep underlying mathematical structure

of these theories and its relation to algebraic geometry.

In the case of N =2vector multiplets, describing the e ectiveinteractions in the

Ab elian (Coulomb) phase of a sp ontaneously broken gauge theory, Seib erg and Witten

[1] have shown that p ositivity of the metric on the underlying mo duli space identi es the

geometrical data of the e ective N = 2 rigid theory with the p erio ds of a particular torus.

In the coupling to gravityitwas conjectured by some of the present authors [3, 4] and

later con rmed by heterotic-Typ e I I duality[8, 9, 10, 11 ], that the very same argument

based on p ositivity of the vector multiplet kinetic metric identi es the corresp onding

geometrical data of the e ective N = 2 sup ergravity with the p erio ds of Calabi-Yau

threefolds.

On the other hand, when matter is added, the underlying geometrical structure

is much richer, since N = 2 matter hyp ermultiplets are asso ciated with quaternionic

geometry[12, 13 , 14 ], and charged hyp ermultiplets are naturally asso ciated with the gaug-

ing of triholomorphic isometries of these quaternionic manifolds [15].

It is the aim of this pap er to complete the general form of the N = 2 sup ergravity

lagrangian coupled to an arbitrary numberofvector multiplets and hyp ermultiplets in

presence of a general gauging of the isometries of b oth the vector multiplets and hyp er-

multiplets scalar manifolds. Actually this extends results already obtained years ago by

some of us [15], that in turn extended previous work by Bagger and Witten on ungauged

general quaternionic manifolds coupled to N = 2 sup ergravity[12 ], by de Wit, Lauwers

and Van Pro eyen on gauged sp ecial geometry and gauged quaternionic manifolds obtained

by quaternionic quotient in the tensor calculus framework [16], and by Castellani, D'Auria

and Ferrara on covariant formulation of sp ecial geometry for matter coupled sup ergravity

[17 ].

This pap er rstly provides in a geometrical setting the full lagrangian with all the

fermionic terms and the sup ersymmetry variations. Secondly, it uses a co ordinate in-

dep endent and manifestly symplectic covariant formalism which in particular do es not

require the use of a prep otential function F (X ). Whether a prep otential F (X ) exists or

not dep ends on the choice of a symplectic gauge[4 ]. Moreover, some physically interesting

cases are precisely instances where F (X ) do es not exist[4].

Of particular relevance is the fact that we exhibit a scalar p otential for arbitrary

quaternionic geometries and for sp ecial geometry not necessarily in sp ecial co ordiantes.

This allows us to go b eyond what is obtainable with the tensor calculus (or sup erspace)

approach. Among many applications, our results allow the study of general conditions for

sp ontaneous sup ersymmetry breaking in a manner analogous to what was done for N =1

matter coupled sup ergravity [18 ]. Many examples of sup ersymmetry breaking studied in

the past are then repro duced in a uni ed framework.

Recently the p ower of using simple geometrical formulae for the scalar p otential was

exploited while studying the breaking of half sup ersymmetries in a particular simple

mo del, using a symplectic basis where F (X ) is not de ned[19]. The metho d has p oten-

tial applications in string theory to study non p erturbative phenomena such as conifold 1

transitions [20], p-forms condensation [21] and Fayet-Iliop oulos terms [19, 22 ].

Although the sup ersymmetric Lagrangian and the transformation rules lo ok quite

involved, all the couplings, the mass matrices and the vacuum energy are completely

xed and organized in terms of few geometrical data, such as the choice of a gauge group

G, and of a sp ecial Kahler SK(n ) and of a Quaternionic manifold Q(n ) describing

V H

the self-interactions of the n vector and n hyp ermultiplets resp ectively, whose direct

V H

pro duct yields the full scalar manifold of the theory

MSK(n ) Q(n ) : (1.1)

V H

If G is non-ab elian, it must b e a subgroup of the isometry group of the scalar manifold

SK(n ) with a blo ck diagonal immersion in the symplectic group Sp(2n +2;IR)of

V V

electric-magnetic duality rotations.

An expanded version of this pap er, with particular attention to the geometrical prop-

erties of the scalar manifolds, the rigidly sup ersymmetric version and further related issues

is given in [23].

2 Resume and Glossary of Sp ecial and Quaternionic

Geometry

Here we collect some useful formulae for sp ecial and quaternionic geometry, following

closely the conventions of [24]. The n complex scalar elds z of N =2vector multiplets

are co ordinates of a sp ecial Kahler manifold, that is a Kahler-Ho dge manifold SK(n )

with the additional constraint on the curvature

? ? ? ? ? ? ? ? ?

R = g g + g g C C g ; (2.1)

ij kl ij kl il kj ik p j l p

? ?

where g = @ @ K is the Kahler metric, K is the Kahler p otential and C is a com-

ij i j ik p

pletely symmetric covariantly holomorphic tensor. We remind that the Levi-Civita con-

nection one form and the Riemann tensor are given by

i i k i il i i

? ?

= dz ; = g @ g ;R = @ : (2.2)

j kl k

j kj kj jk l jl

AKahler-Ho dge manifold has the prop erty that there is a U (1) bundle L whose rst

Chern class coincides with the Kahler class. This means that lo cally the U (1) connection

Q can b e written as

i i

Q = (@ Kdz @ Kdz ) : (2.3)

i i

The covariant derivative of a generic eld , that under a Kahler transformation K !

i i

K + f + f transforms as ! exp[ (pf + p f )] is given by

j j k j

D = @ + + @ K ;

i i i

j j j

? ? ?

D @ = @ K : (2.4) +

i i i

p = p). Note that has weight(p; p). Since (In the following we always have

? ?

C is covariantly holomorphic and has weight p = 2, it satis es D C =(@

ik p q ik p q

@ K)C =0.

q ik p 2

A more intrinsic and useful de nition of a sp ecial Kahler manifold can b e given by

constructing a at 2n + 2-dimensional symplectic bundle over the Kahler -Ho dge manifold

whose generic sections (with weight p =1)

V =(L ;M ) = 0;:::;n ; (2.5)

are covariantly holomorphic

? ? ?

D V =(@ @ K)V =0 (2.6)

i i i

and satisfy the further condition

i=i(L M M L )=1 ; (2.7)

0 11

where <; > denotes a symplectic inner pro duct with metric chosen to b e .

11 0

De ning U = D V =(f ;h ), and intro ducing a symmetric three-tensor C by

i i i ij k

D U = iC g U ; (2.8)

i j ij k k

one can show that the symplectic connection

D V = U

i i

D U = iC g U

i j ij k k

? ?

D U = g V

i j ij

D V = 0 (2.9)

is at, provided the constraint (2.1) is veri ed. Furthermore, the Kahler p otential can

b e computed as a symplectic invariant from eq. (2.7). Indeed, intro ducing also the

holomorphic sections

K=2 K=2

= e V = e (L ;M )= (X ;F )

@ = 0 (2.10)

eq. (2.7) gives

K = ln i< ; >=ln i(X F X F ) : (2.11)

From eqs. (2.9), (2.7) wehave

= 0 ! X @F @X F =0 ; (2.12)

i i i

V;U > = 0 ; <

L = ! K = ln 2(X Im N X ) ; (2.13) Im N L

where the complex symmetric (n +1)(n + 1) matrix N is de ned through the

V V

relations

; h = N f : (2.14) M = N L

i 3

The Kahler metric and Yukawa couplings C can b e written in a manifestly symplectic

ij k

invariant form as

? ?

g = i< U ;U >= 2f Im N f ; (2.15)

ij i j

i j

C = : (2.16)

ij k i j k

It is also useful to de ne

1 ij

(Im N ) L L ; (2.17) U f g f =

i j

which is the inverse relation of eq. (2.15). Under co ordinate transformations, the sections

transform as

f (z)

=e S ; (2.18)

A B

where S = is an elementof Sp(2n +2;IR),

C D

T T T T T T

A D C B =11 ; A C C A=B DD B =0 ; (2.19)

f (z )

and the factor e is a U (1) Kahler transformation. We also note that

f f e

N (X; F )=(C+DN(X; F ))(A + B N (X; F )) ; (2.20)

a relation that simply derives from its de nition, eq. (2.14) .

Note that under Kahler transformations K ! K + f + f and ! e . Since

X ! X e , this means that we can regard, at least lo cally, the X as homogeneous

co ordinates on SK(n )[25], provided the matrix

a a 0

e (z )=@(X =X ) a =1;:::;n (2.21)

i V

is invertible[4]. In this case, wemay set

F = F (X ) (2.22)

and then eq. (2.12) implies the integrability condition

@F @F (X ) @F

=0 ! F = (2.23)

@X @X @X

with

X @ F =2F: (2.24)

F (X (z )) is the prep otential of N = 2 sup ergravityvector multiplet couplings [25] and

\sp ecial co ordinates" corresp ond to a co ordinate choice for which

0 a a

: (2.25) = @ (X =X )= e

i i

0 i i

This means X =1, X =z. Since F = X F , under symplectic transformations the

prep otential transforms as

1 1

T T T

e f

F (X )=F(X)+X (C B) F + X (C A) X + F (D B ) F ; (2.26)

2 2 4

where

@ F

X =(A+BF)X ; F = F = : (2.27)

@X @X

a 0 2 a

In terms of the sp ecial co ordinates t = , one has F (X )=(X )f(t), and the

Kahler p otential and the metric are expressed by

h i

t) = ln i 2f 2f +(t t )(f + f ) K (t;

G = @ @ K (t; t) : (2.28)

ab b

Eq. (2.27) shows that the tranformation X ! X can b e actually singular, thus im-

plying the non existence of the prep otential F (X ), dep ending on the choice of sym-

plectic gauge[4 ]. On the other hand, some physically interesting cases, such as the

N =2!N = 1 sup ersymmetry breaking [19], are precisely instances where F (X )

do es not exist. On the contrary the prep otential F (X ) seems to b e a necessary ingredient

in the tensor calculus constructions of N = 2 theories that for this reason are not com-

pletely general. This happ ens b ecause tensor calculus uses sp ecial co ordinates from the

very start.

Next we turn to the hyp ermultiplet sector of an N = 2 theory. N =2hyp ermultiplets

are eld representations of N = 2 sup ersymmetry which contain a pair of left-handed

fermions and a quadruple of real scalars. N = 2 sup ergravity requires that the 4n

scalars q of n hyp ermultiplets b e the co ordinates of a quaternionic manifold[12 ].

Sup ersymmetry requires the existence of a principal SU (2){bundle SU that plays

for hyp ermultiplets the same role played by the the line{bundle L in the case of vector

multiplets.

A quaternionic manifold is a 4n -dimensional real manifold endowed with a metric h:

2 u v

ds = h (q )dq dq ; u; v =1;:::;4n (2.29)

uv H

and three complex structures J ; (x =1;2;3) that satisfy the quaternionic algebra

x y xy xy z z

J J = 11+ J (2.30)

and such that the metric is hermitian with resp ect to the three complex structures:

x x

h (J X;J Y)=h(X;Y) (x=1;2;3) (2.31)

where X; Y are generic tangentvectors. The triplet of two-forms K

x x u v x x w

K = K dq ^ dq ; K = h (J ) (2.32)

uv uv v

that provides the generalization of the concept of Kahler form o ccurring in the complex

case, is covariantly closed with resp ect to an SU (2) ' Sp(2) connection !

x x xy z y z

rK dK + ! ^ K =0 (2.33)

with curvature given by

x x xy z y z x

d! + ! ^ ! = K (2.34)

2 5

where is a real parameter related to the scale of the quaternionic manifold. Sup ersym-

metry, together with appropriate normalizations for the kinetic terms in the lagrangian

xes it to the value = 1. Intro ducing a physical normalization one can set = M ,

M b eing the Planck mass, so that the limit of rigid sup ersymmetry M ! +1 can b e

P P

identi ed with ! 0[23 ]. In that case eq. (2.34) de nes a at SU (2) connection relevant

to the hyp erKahler manifolds of rigid sup ersymmetry, treated extensively in [23]).

The holonomy group of a generic quaternionic manifold is in Sp(2) Sp(2n ). In-

tro ducing at indices fA; B ; C =1;2g ;f ; ; =1; ::; 2n g that run, resp ectively,in

the fundamental representations of SU (2) and Sp(2n ; IR), we can intro duce a vielb ein

1-form

A A u

U = U (q )dq (2.35)

such that

A B

h = U U C (2.36)

uv AB

u v

where C = C , C = 11, = , = 11 are, resp ectively, the at Sp(2n )

AB BA H

and Sp(2) SU (2) invariant metrics.

The vielb ein U satis es the metric p ostulate, i.e. it is covariantly closed with resp ect

to the SU (2)-connection ! and to some Sp(2n ; IR)-Lie Algebra valued connection =

of the holonomy group:

A A x 1 A B A

rU dU + C =0 (2.37) ! ( ) ^U + ^U

x B

where ( ) are the standard Pauli matrices. Furthermore, b ecause of the realityofh,

U satis es the reality condition:

B A

C U (2.38) U (U ) =

A AB

A stronger version of eq. (2.36) is

A B A B AB

(U U + U U )C = h

u v v u

A B A B

(U U + U U ) = h C : (2.39)

AB uv

u v v u

Wehave also the inverse vielb ein U de ned by the equation

u A u

U U = : (2.40)

A v v

Let us consider the Riemann tensor

A B x 1 AB AB

R = ( ) C +IR (2.41)

uv uv uv

is the eld strength of the Sp(2n ) connection: where IR

u v

d + ^ C IR =IR dq ^ dq : (2.42)

The and IR curvature satisfy the following relations

x x u v x

= U U = iC ( )

A ;B uv A B

B A B A B A

U )+U U U IR = U (U C C ; (2.43)

AB AB

v u u v v uv u

where is a completely symmetric tensor. The previous equations imply that the

quaternionic manifold is an Einstein space with Ricci tensor given by

R = (2 + n )h : (2.44)

uv H uv 6

3 The Gauging

The problem of gauging matter coupled N = 2 sup ergravity theories consists in identifying

the gauge group G as a subgroup, at most of dimension n + 1 of the isometries of the

pro duct space

M = SK(n ) Q(n ) : (3.1)

V H

Here we shall mainly consider two cases even if more general situations are p ossible. The

rst is when the gauge group G is non ab elian, the second is when it is the ab elian group

n +1

G = U (1) . In the rst case sup ersymmetry requires that G b e a subgroup of the

isometries of M, since the scalars (more precisely, the sections L )must b elong to the

adjoint representation of G. In such case the hyp ermultiplet space will generically split

into[26 ]

X X

P P

n = n R + n R (3.2)

H i i

l l

P P

where R and R are a set of irreducible representations of G and R denote pseudoreal

l l

representations.

In the ab elian case, the sp ecial manifold is not required to haveany isometry and if

the hyp ermultiplets are charged with resp ect to the n +1 U(1)'s, then the Q manifold

should at least have n + 1 ab elian isometries.

The gauging pro ceeds byintro ducing n + 1 Killing vectors generically acting on M

i i i

z ! z + k (z )

u u u

q ! q + k (q ) : (3.3)

i u

The Kahler and quaternionic structures of the factors of M imply that k ;k can b e

determined in terms of \Killing prep otentials" which generalize the so called \D-terms"

of N = 1 sup ersymmetric gauge theories. The analysis of such prep otentials was carried

out in ref. [15], and wenow brie y summarize the main results.

For the SK manifold, the Killing prep otential is a real function P satisfying

i ij

k = ig @ P ; (3.4)

with inverse formula

? ?

i i i i

? ?

iP = (k @ K k @ K )= k @ K = k @ K: (3.5)

i i i i

Formulae (3.4), (3.5) are the results of N = 1 sup ergravity.For sp ecial geometry wehave

the stronger constraint that the gauge group, in the non ab elian case, should b e imb edded

in the symplectic group Sp(2n +2;IR), and thus one must have

i i

@ V + k @ V = T V + f V (3.6) L V k

i i

for some T 2 Sp(2n +2;IR) Lie algebra, and f (z ) corresp onding to an in nitesimal

Kahler transformation ( L is the Lie derivative acting on the symplectic section V ). We

consider here the case where f = 0 and G is a subgroup of the classical isometries of SK

whichhave a diagonal emb edding in the symplectic group. Since f = 0, this means that

L K = 0 and P satis es eq. (3.5). Moreover wehave

k U =T V +iP V (3.7)

and taking the symplectic scalar pro duct with V we get

P = < V;T V>=e < ;T >: (3.8)

By using the prop erty

f 0

T = ; (3.9)

0 f

we nally get the explicit expression

P = e (F f X + F f X ) (3.10)

in terms of the holomorphic sections = (X ;F ).

For quaternionic manifolds, eq. (3.4) is replaced by

u x x x xy z y z

k = r P = (@ P + ! P ) ; (3.11)

v v

uv v

where P is a triplet of real zero-form prep otentials. Eq. (3.11) can b e solved for the

Killing vectors in terms of the prep otentials as follows

vw x x tu u

h (r P )h ; (3.12) k =

x=1

where h is the inverse quaternionic metric. If the gauge group G has structure constants

i u

f the Killing vectors k ;k satisfy the following relations

? ?

j j

i i

ig (k k k k )=f P ;

x u v xy z z x

K k k P P = f P : (3.13)

2 2

Eq. (3.13) can b e derived from the group relations of the Killing vectors

[k ;k ]=f k (3.14)

together with their relation to the prep otential functions.

An imp ortant observation comes from the p ossible existence of Fayet- Iliop oulos terms

in N = 2 sup ergravity. This corresp onds to a constant shift in the prep otential functions

P ! P + C

x x x

P ! P + : (3.15)

It is imp ortant to observe that in N = 2 sp ecial geometry the Killing vectors and prep o-

tentials satisfy the relation

i i

k L = k L = P L = P L =0 (3.16)

This implies that C =0.However, a similar relation do es not exist for the quaternionic

Killing vectors so that a F-I term in that case is p ossible, sub ject to the constraint (3.13).

This implies that in a pure ab elian theory with only neutral hyp ermultiplets we can still

have 6= 0 provided

xy z x

=0 (3.17)

holds. Mo dels of this sort, breaking N =2 ! N = 0 with vanishing cosmological

constant, were constructed in ref. [27] and will b e discussed in the last section.

The gauging pro cedure can now b e p erformed through the following steps [28 ]. One

rst de nes gauge covariant di erentials

i i i

rz = dz + gA k (z)

? ? ?

i i i

rz = dz + gA k (z)

u u u

rq = dq + gA k (q) : (3.18)

Secondly, one gauges the comp osite connections by mo difying them by means of killing

vectors and prep otential functions.

i i k i i k i

dz ! = rz + gA @ k

j jk j jk

i i

? ?

i i i i 0

? ?

Q (@Kdz @ Kdz ) ! Q = (@ Krz @ Krz )+gA P

i i i i

2 2

x x u x x u x

! ! dq ! ! = ! rq + gA P

u u

u u v u A

dq ! = rq + gA @ k U U ; (3.19)

u u vA

where for simplicitywehave assumed a single coupling constant. If there are many

coupling constants corresp onding to various factors of the gauge group the formulas are

obviously mo di ed. Corresp ondingly, the gauged curvatures are:

i i ` k i

R = R r z ^rz + gF @ k

j j` k

i j 0

b b

KdQ = ig r z ^rz + gF P

x x u v x

= rq ^rq + gF P

u v v uj A

IR = IR rq ^rq + gA @ k U U : (3.20)

v jA

4 The Complete N=2 Sup ergravity Theory

We are nally ready to write the sup ersymmetric invariant action and sup ersymmetry

transformation rules for a completely general N = 2 sup ergravity.

Such a theory includes

1. the gravitational multiplet

a A 0

(V ; ; ;A ) (4.1)

describ ed by the vielb ein one-form V ,(a=0;1;2;3) (together with the spin con-

ab A

nection one-form ! ), the SU (2) doublet of gravitino one-forms ; (A =1;2

and the upp er or lower p osition of the index denotes right, resp ectively left chirality,

A 0

namely = = 1), and the graviphoton one-form A

5 A 5

2. n vector multiplets

I iA i i

(A ; ; ;z ) (4.2)

containing a gauge b oson one-form A (I =1;:::;n ), a doublet of gauginos (zero-

iA i

form spinors) , of left and rightchirality resp ectively, and a complex scalar

i i

eld (zero-form) z (i =; 1;:::;n ). The scalar elds z can b e regarded as arbitrary

co ordinates on the sp ecial manifold SK of complex dimension n .

V 9

3. n hypermultiplets

( ; ;q ) (4.3)

formed by a doublet of zero-form spinors, that is the hyp erinos ( =1;:::;2n

and here the lower or upp er p osition of the index denotes left, resp ectively right

chirality), and four real scalar elds q (u =1;:::;4n ), that can b e regarded as

arbitrary co ordinates of the quaternionic manifold Q, of real dimension 4n .As

already mentioned, any quaternionic manifold has a holonomy group:

Hol (Q) SU (2) Sp(2n ; IR) (4.4)

and the index of the hyp erinos transforms in the fundamental representation of

Sp(2n ; IR).

The de nition of curvatures in the gravitational sector is given by:

a a a b a A

T dV ! ^ V i ^

i 1

ab B

! ^ + Q ^ + ! ^ r d

ab A A B A A A

4 2

1 i

A A ab A A A B A

d ! ^ Q ^ + ! ^ r

4 2

ab ab a cb

R d! ! ^ ! ; (4.5)

x B A AC D B

! ( ) and ! = ! . In all the ab ove formulae the pull{back where ! =

x DB

A B C A

on space{time through the maps

i u

z : M !SK ; q : M !Q (4.6)

4 4

is obviously understo o d. In this way the comp osite connections b ecome one-forms on

space-time.

In the vector multiplet sector the curvatures and covariant derivatives are:

i i i

rz dz + gA k (z)

? ? ?

i i i

r z dz + gA k (z)

i 1

ab iA iA i jA A iB iA iA

b b

! Q + + ! ^ r d

j B

4 2

1 i

? ? ? ? ? ?

i i ab i i i B i

b b

r d ! + Q + + ! ^

A A A A j A B

4 2

AB B

F dA + gf A ^ A + L ^ + L ^ (4.7)

B AB

where L = e X is the rst half (electric) of the symplectic section intro duced in

equation (2.5). (The second part M of such symplectic sections would app ear in the

magnetic eld strengths if we did intro duce them.)

Finally, in the hyp ermultiplet sector the covariant derivatives are:

v A v v A A

(q) rq U dq + gA k U U

v v

1 i

b b

r d ! Q +

4 2

i 1

b b

! + Q + (4.8) r d

4 2 10

with

b b b b

C ; C (4.9)

Note that the Kahler weights of all spinor elds are given by the co ecients of iQ in

the de nition of their curvatures and covariant derivatives.

Our next task is to write down the N = 2 space-time lagrangian and the sup ersymme-

try transformation laws of the elds. The metho d employed for this construction is based

on the geometrical approach, review in [29], and a more detailed derivation is given in the

app endices A and B of [23 ]. Actually, one solves the Bianchi identities in N = 2 sup er-

space and then constructs the rheonomic sup erspace Lagrangian in suchaway that the

sup erspace curvatures and covariant derivatives given by the solution of the Bianchi iden-

tities are repro duced by the variational equations of motion derived from the lagrangian.

After this pro cedure is completed the space-time lagrangian is immediately retrieved by

restricting the sup erspace p-forms to space-time. The resulting action can b e split in the

following way:

h i

4 0

S = gd x L + L + L ;

k 4f

inv

L = L + L ;

k P auli

kin

inv non inv

L = L + L ;

4f 4f

z; q) ; (4.10) L = L V (z;

mass

inv

where L consists of the true kinetic terms as well as Pauli-like terms containing the

kin

derivatives of the scalar elds. The mo di cations due to the gauging are contained not

only in L but also in the gauged covariant derivatives in the rest of the lagrangian. We

collect the various terms of (4.10) in the table b elow.

N=2 Supergravity lagrangian

inv i j u v A

L = R + g r z r z + h r q r q + p

ij uv A A

kin

2 g

iA j

i g r + r r + r

A A

+ + j iA

+ i F + g r F N F N F z

u u A j A

r q r q 2U + g r z +2U +h:c:

ij A A

u u A

(4.11)

A i

B AB

(Im N ) [4L L = F 4if +

P auli AB

i A

+ r f L C ]+h:c: (4.12)

i AB

inv A B

L = g 2

ij A

4f B B

2 g 11

iA jC

B kD

C + h:c:

ij k AC BD

A iA i

B B

+2g 2

A B A B

1 3

iA j

lB k

? ? ? ? ? ?

+ R + g g g g

ij lk ik lj ij lk

A B

4 2

1 1

t s AB

+ g + R U U C

ts A B

4 2

jA kC

mB lD

r C + h: c:

m jkl AC BD

A A A B j

+ g +2 + C + h:c: (4.13)

ij AB

A B A

n h

A C

non inv B D

L = (Im N ) 2L L

AB CD

A i

8i L f

i A

? ?

i j

AB CD

2f f

? ?

D j

i j A B C

k` B jD

+ L f g C

ij k AB CD

m iA

k` jB

+ g C f f

? ?

ij k B

m ` A

L L C

+ iL f C

i A B

iA kC

k r ns jB lD

C C g g f f

ij k lmn AB CD

r s

L r f C

i AB

i o

+h:c: (4.14) + L L C C

A j

B iAB A

L = g 2S +ig W +2iN

mass AB ij

A B A

iB `B

+ h.c. (4.15) + M + M + M

iA`B

h i

i u v x x 2 ij x x

V(z; g k k +4h k k P P 3 z; q ) = g L L + g f f L L P P :(4.16)

ij uv

i j

1 i

where F = (F F ) and (:::) denotes the self dual part of the fermion

2 2

bilinears. The mass{matrices are given by:

C x

S = ( ) P L

AB x BC

iAB AB i B CA x ij

W = k L +i( ) P g f

C j

u A A

k = 2 U L N

v ]

A B [u

M = U U " r k L

u v

M = 4 U k f

iB Bu i 12

1 x

M = " g k f +i( ) P r f (4.17)

AB ij x `

iAj`B

` i

The coupling constant g in L is just a symb olic notation to remind that these terms

are entirely due to the gauging and vanish in the ungauged theory, where also all gauged

covariant derivatives reduce to ordinary ones. Note that in general there is not a single

coupling constant, but rather there are as many indep endent coupling constants as the

numb er of factors in the gauge group. The normalization of the kinetic term for the

quaternions dep ends on the scale of the quaternionic manifold, app earing in eq. (2.34),

for whichwehavechosen the value = 1.

Furthermore, using the geometric approach, the form of the sup ersymmetry transfor-

mation laws is also easily deduced from the solution of the Bianchi identities in sup erspace

[23 ]byinterpreting the sup ersymmetry variations as sup erspace lie derivatives. One gets

Supergravity transformation rules of the Fermi elds

iB i

= D @ K @ K

A A A i B i

B v CD C

! U C +

B D

Av C

B 0 B

+ A + A

A A

h i

+ B

+ i gS + (T + U ) (4.18)

AB AB

jB j

iA iA B

@ K = @ K

j B j

A v CD iB C

! U C +

Bv C

kB iA

i jA i A

+i r z

B A

i AB iAB

+G + D (4.19)

B B

v AB A

= U C +

v A

iB i

+ @ K @ K

i B i

B u BC A B A

+i U r q (4.20) C C + gN

A C AB

Supergravity transformation rules of the Bose elds

a a A a

= i V i (4.21)

A A

AB B

L +2L A = 2

B AB

A 13

iA i

B AB

+ i f +if (4.22)

AB B

i i A

z = (4.23)

i A

z = (4.24)

u u A AB

(4.25) q = U + C

where wehave:

Supergravity values of the auxiliary elds

? ?

k k

`B B `C

A = (4.26) g

k `

A A C

? ?

i 1 i

k k

0 B

`B B C` B

A = g (4.27)

k `

A A A C A

4 2 4

1 1

F + r f (4.28) T = 2i ( Im N ) L C L

i AB

8 4

1 1

+ + AB

T = 2i ( Im N ) L F + r f C L (4.29)

j A B

8 4

U = C (4.30)

U = C (4.31)

i ij `B

G = g (Im N ) F + r f f

k AB

` j

(4.32) C L

? ? ?

i + i j + ` AB

G = g f (Im N ) F + r f

j ` A B

C L (4.33)

? ?

ij iAB ` AC BD iAB

? ? ?

g C D = + W (4.34)

j k `

C D

In eqs. (4.28), (4.29), (4.32), (4.33) wehave denoted by F the sup ercovariant eld

strength de ned by:

A iA i

B AB B AB

+ L i f i f : F = F + L

AB B AB

[ [ B]

i ] i A

(4.35) 14

Let us make some observation ab out the structure of the Lagrangian and of the transfor-

mation laws.

i) We note that all the terms of the lagrangian are given in terms of purely geo-

metric ob jects p ertaining to the sp ecial and quaternionic geometries. Furthermore the

Lagrangian do es not rely on the existence of a prep otential function F = F (X ) and it is

valid for anychoice of the quaternionic manifold.

ii) The lagrangian is not invariant under symplectic duality transformations. However,

in absence of gauging (g = 0), if we restrict the lagrangian to con gurations where the

vectors are on shell, it b ecomes symplectic invariant[30 , 4 ]. This allows us to x the terms

non inv

app earing in L in a way indep endent from sup ersymmetry arguments.

iii) We note that the eld strengths F originally intro duced in the Lagrangian are

the free gauge eld strengths. The interacting eld strengths which are sup ersymmetry

eigenstates are de ned as the ob jects app earing in the transformation laws of the graviti-

nos and gauginos elds, resp ectively, namely the b osonic part of T and G de ned in

eq.s (4.28), (4.32).

5 Comments on the scalar p otential

A general Ward identity[31] of N -extended sup ergravity establishes the following formulae

for the scalar p otential V () of the theory (in appropriate normalizations for the generic

fermionic shifts )

a B b A

Z 3M M = V () A; B =1;:::;N (5.36)

ab A AC

where is the extra contribution, due to the gauging, to the spin sup ersymmetry

variations of the scalar vev's, Z is the (scalar dep endent) kinetic term normalization

and M is the (scalar dep endent) gravitino mass matrix. Since in the case at hand

(N = 2) all terms in question are expressed in terms of Killing vectors and prep otentials,

contracted with the symplectic sections, we will b e able to derive a completely geometrical

formula for V (z; z; q ). The relevant terms in the fermionic transformation rules are

= ig S ;

A AB

iA iAB

= gW ;

= gN : (5.37)

In our normalization the previous Ward identity gives

x x i u v

L L +(U 3L L )P P : (5.38) V =(g k k +4h k k )

ij uv

with U is de ned in (2.17). Ab ove, the rst two terms are related to the gauging of

isometries of SK Q.For an ab elian group, the rst term is absent. The negative term

is the gravitino mass contribution, while the one in U is the gaugino shift contribution

due to the quaternionic prep otential.

Eq. (5.38) can b e rewritten in a suggestive form as

x x

L L +(U 3L L )(P P P P ) ; (5.39) V =(k ;k )

where

1 0 1 0

k 0 g 0

C B C B

i i u

k (5.40) (k ;k )=(k ; k ; k ) g 0 0

A @ A @

i j

k 0 0 2h

is the scalar pro duct of the Killing vector and wehave used eqs. (3.4),(3.16) . P are the

quaternionic (triplet) prep otentials and U ;L are sp ecial geometry data.

In a theory with only ab elian vectors, the p otential may still b e non-zero due to

Fayet-Iliop oulos terms:

x x xy z z

P = (constant); =0 : (5.41)

In this case

x x

V (z; z)= (U 3L L ) : (5.42)

Examples with V (z; z) = 0 but non-vanishing gravitino mass (with N = 2 sup ersymmetry

broken to N =0)were given in [27], then generalizing to N = 2 the no scale mo dels of

N = 1 sup ergravity [34]. These mo dels were obtained by taking a =( ;0;0) . In this

case the expression

00 0

V = U 3 L (5.43) L

reduces to

ij K

V =(@Kg @ K 3)e (5.44)

i j

which is the N = 1 sup ergravity p otential, with solution ( V = 0) the cubic holomorphic

prep otential

A B C

X X X

F (X )= d A =1;:::;n : (5.45)

AB C

SU (1;1) SO(2;n)

Another solution is obtained by taking the coset in the SO(2;n)

U (1) SO(n)

symmetric parametrization of the symplectic sections (X ;F = SX ; X X =

0 ; =(1;1;1;:::;1)) where a prep otential F do es not exist. In this case

U 3L L = (5.46)

i(S S )

where wehave used the fact that

: (5.47) S)( + )+S ; = N =(S

1=2

(X X )

The identity (5.46) allows one to prove that the tree level p otential of an arbitrary heterotic

string compacti cation (including orbifolds with twisted hyp ermultiplets) is semi-p ositive

de nite provided we don't gauge the graviphoton and the gravidilaton vectors (i:e: P =0

for = 0; 1, P 6=0 for=2;:::;n ). On the other hand, it also proves that tree level

sup ergravity breaking may only o ccurr if P 6=0 for=0;1. This instance is related to

mo dels with Scherk-Schwarz mechanism studied in the literature [32, 33 ].

Avanishing p otential can b e obtained if =( ;0;0) with

=0 : (5.48)

p+2

In this case wemay also consider the gauge group to b e U (1) G(n p) and intro duce

=(;:::; ; 0;:::;0) such that = 0 where is the SO(2;p) Lorentzian

0 p+1

metric. The p otential is now:

i x x

V = k g k L L ; (U 3L L )P P =0 (5.49)

where k L = 0 for p + 1. The gravitino have equal mass

K=2

j m j' e j X j (5.50)

3=2

with =0,=0;:::;p +1.

It is amusing to note that the gravitino mass, as a function of the O (2;p)=O (2) O (p)

mo duli and of the F-I terms, just coincides with the central charge formula for the level

N = 1 in heterotic string (H-monop oles), if the F-I terms are identi ed with the O (2;p)

lattice electric charges.

Note that, b ecause of the sp ecial form of the gauged Q, ! ,we see that whenever

P 6= 0 the gravitino is charged with resp ect to the U (1) factor and whenever P 6= 0 the

gravitino is charged with resp ect to the SU (2) factor of the U (1) SU (2) automorphism

group of the sup ersymmetry algebra. In the case of U (1) gauge elds with non-vanishing

F-I terms =(0;0; ) the gauge eld A = A gauge a U (1) subgroup of SU (2)

p L

susy algebra.

Mo dels with breaking of N =2 to N = 1 [19] necessarily require k not to b e zero.

The minimal mo del where this happ ens with V = 0 is the one based on

SU (1; 1) SO(4; 1)

SK Q = ; (5.51)

U (1) SO(4)

where a U (1) U (1) isometry of Q is gauged. In this case the vanishing of V requires a

comp ensation of the ; variations with the gravitino contribution

u v x x x x

4k k h + U P P =3 L L P P : (5.52)

The mo duli space of vacua satisfying (5.52) is a four dimensional subspace of (5.51).

One maywonder where are the explicit mass terms for hyp ermultiplets. In N =2

sup ergravity, since the hyp ermultiplet mass is a central charge, which is gauged, such

term corresp onds to the gauging of a U (1) charge. This is b est seen if we consider the

case where no vector multiplets (and then gauginos) are present. In this case L = L =1

and the p otential b ecomes

u v x x

V =4h k k 3P P (5.53)

is the Killing vector of a U (1) symmetry of Q, gauged by the graviphoton and where k

SO(4;1)

P is the asso ciated prep otential. For this repro duces the Zachos mo del [35 ]. The

SO(4)

gauged U (1) in this mo del is contained in SU (2) which commutes with the symmetry

SU (2) in the decomp osition of SO(4) = SU (2) SU (2). This mo del has a lo cal

L L R

minimum at vanishing hyp ermultiplet vev at which U (1) is unbroken, and the extrema

(at u = 1) (maxima) which break U (1). The extremal mo del is when b oth n = n =0.

H V

Still wemayhave a pure F-I term

V = 3 =(; 0; 0) (5.54)

This corresp onds to the gauging of a U (1) SU (2) and gravitinos havecharged coupling.

This mo del corresp onds to anti-De Sitter N = 2 sup ergravity [36 ]. 17

Acknowledgements

A.C. and S. F. would like to thank the Institute for Theoretical Physics at U. C. Santa

Barbara for its kind hospitality and where part of this work was completed.

References

[1] N. Seib erg and E. Witten, Nucl. Phys. B426 (1994) 19; Nucl. Phys. B431 (1994) 484.

[2] A. Klemm, W. Lerche, S. Theisen and S. Yankielovicz, Phys. Lett. B344 (1995) 169;

P. Argyres and A. Faraggi, Phys. Rev. Lett. 74 (1995) 3931.

[3] A. Ceresole, R. D'Auria and S. Ferrara, Phys. Lett. 339B (1994) 71, hep-th/9408036.

[4] A. Ceresole, R. D'Auria, S. Ferrara and A. Van Pro eyen, Nucl. Phys. B444 (1995)

92, hep-th/9502072.

[5] C. M. Hull and P.K. Townsend, Nucl. Phys. B438 (1995) 109, hep-th/9410167.

[6] E. Witten, Nucl. Phys. B443 (1995) 85, hep-th/9503124 .

[7] M. J. Du , Nucl. Phys. B442 (1995) 47; for a recent review see also \Elec-

tric/Magnetic Duality and its Stringy Origins", hep-th/9509106.

[8] S. Ferrara, J. A. Harvey, A. Strominger and C. Vafa, Phys. Lett. B361 (1995) 59,

hep-th/9505162.

[9] S. Kachru and C. Vafa, Nucl. Phys. B450 (1995) 69, hep-th/9505105.

[10] M. Billo, A. Ceresole, R. D'Auria, S. Ferrara, P.Fre, T. Regge, P. Soriani and A.

Van Pro eyen, \A Search for non-p erturbative Dualities of Lo cal N=2 Yang-Mills

Theories from Calabi-Yau Threefolds", preprint hep-th/9506075, to app ear on Class.

and Quantum Grav.

[11] A. Klemm, W. Lerche and P.Mayr, Phys. Lett. B357 (1995) 313, hep-th/9506112.

[12] J. Bagger and E. Witten Nucl. Phys. B222 (1983) 1.

[13] N. J. Hitchin, A. Karlhede, U. Lindstrom and M. Ro cek, \Hyp erKahler Metrics and

Sup ersymmetry", Commun. math. Phys. 108 (1987) 535.

[14] K. Galicki, Comm. Math. Phys. 108 (1987) 117.

[15] R. D'Auria, S. Ferrara and P.Fre, Nucl. Phys. B359 (1991) 705.

[16] B. de Wit, P.G.Lauwers and A. Van Pro eyen Nucl. Phys. B255 (1985) 569.

[17] L. Castellani, R. D'Auria and S. Ferrara, Phys. Lett. 241B (1990) 57; Class. Quantum

Grav. 7 (1990) 1767.

[18] E. Cremmer, S. Ferrara, L. Girardello and A. Van Pro eyen, Nucl. Phys. B212 (1983)

413. 18

[19] S. Ferrara, L. Girardello and M. Porrati, \Minimal Higgs Branch for the Breaking

of Half of the Sup ersymmetry in N=2 Sup ergravity", preprint CERN-TH/95-268,

hep-th/9510074.

[20] B. Greene, D. Morrison and A. Strominger, Nucl. Phys. B451 (1995) 109, hep-

th/9504145.

[21] J. Polchinski and A. Strominger, \New vacua for Typ e I I String Theory", preprint

hep-th/9510227.

[22] I. Antoniadis, H. Partouche and T. R. Taylor, \Sp ontaneous Breaking of N=2 Global

Sup ersymmetry", preprint hep-th/9512006.

[23] L. Andrianop oli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P.Fre and T.

Magri, \N=2 Sup ergravity and N=2 Yang-Mills Theory on fully General Scalar Man-

ifolds", in preparation.

[24] A. Ceresole, R. D'Auria and S. Ferrara, \The Symplectic Structure of N=2 Sup er-

gravity and its Central Extension", Pro ceedings of the ICTP Trieste Conference

on \S-Duality and Mirror Symmetry", Miramare, Trieste, Italy, June 1995, hep-

th/9509160.

[25] B. de Wit, P.G. Lauwers, R. Philipp e, Su S.Q. and A. Van Pro eyen, Phys. Lett. 134B

(1984) 37; S. J. Gates, Nucl. Phys. B238 (1984) 349; B. de Wit and A. Van Pro eyen,

Nucl. Phys. B245 (1984) 89.

[26] J. P. Derendinger, S. Ferrara and A. Masiero, Phys. Lett. 143B (1984) 133.

[27] E. Cremmer, C. Kounnas, A. Van Pro eyen, J.P. Derendinger, S. Ferrara, B. De Wit

and L. Girardello, Nucl. Phys. B250 (1985) 385.

[28] B. de Wit and H. Nicolai, Nucl. Phys. B208 (1982) 323.

[29] L. Castellani, R. D'Auria and P.Fre, \Sup ergravity and Sup erstring Theory: a

Geometric Persp ective", World Scienti c (1991) , Volumes 1, 2, 3.

[30] S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B121 (1977) 393.

[31] S. Ferrara and L. Maiani, Pro c. V Silarg Symp., World Scienti c (1986); S. Cecotti,

L. Girardello and M. Porrati, Nucl. Phys. B268 (1986) 295; A. Ceresole, R. D'Auria,

S. Ferrara, P.Fre and E. Maina, Nucl. Phys. B268 (1986) 317.

[32] S. Ferrara, C. Kounnas, M. Porrati and F. Zwirner, Nucl. Phys. B318 (1989) 75.

[33] I. Antoniadis, Phys. Lett. B246 (1990) 377.

[34] E. Cremmer, C. Kounnas, S. Ferrara and D. Nanop oulos, Phys. Lett. 133B (1983)

61.

[35] C. Zachos Phys. Lett. 76B (1978) 329.

[36] D.Z. Freedman, A. Das Nucl. Phys. B120 (1977) 221 19