Real special geometry

28 July 1994 PHYSICS LETTERS B

PhysicsLettersB 333 (1994) 62-67

ELSEVIER

Real special geometry* M . B e r t o l i n i a, A. C e r e s o l e b'c, R. D ' A u r i a b'c, S. F e r r a r a d a Dipartimento di Fisica Teorica, Universitd di Torino, Via P. Giuria 1, 10125 Torino, Italy b Dipartimento di Fisica, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino, Italy c INFN, Sezione di Torino, Italy d CERN, 1211 Geneva 23, Switzerland

Received 22 April 1994 Editor: L. Alvarez-Gaum6

Abstract

We give a coordinate-free description of real manifolds occurring in certain four-dimensional supergravity theories with antisymmetric tensor fields. The relevance of the linear multiplets in the compactification of string and five-brane theories is also discussed.

In four-dimensional supergravity theories scalar fields are usually embedded in linear [ 1 ] or chiral multiplets, depending on whether antisymmetric tensor fields are present or not. Examples of theories with antisymmetric tensor fields naturally occur in Kaluza-Klein compactifications of certain D-dimensional supergravity theories with D > 6, or in the low-energy limit of some string and p-brane compactifications. In string theory, classical examples of linear multiplets (L, Bgv, X) are the dilaton multiplet in heterotic strings and the Kahler class moduli multiplets in "dual theories" such as five-brane compactifications [2-4]. However, while massless chiral multiplets with an associated continuous Peccei-Quinn symmetry are classically equivalent to linear multiplets through a "duality transformation" [5-7], this may not be so at the quantum level, due to the violation of the Peccei-Quinn symmetry by quantum effects. These comprehend both a t string corrections, i.e. quantum effects on the world-sheet sigma-model, and non-perturbative effects in the string coupling constant, such as Yang-Mills and gravitational instantons [ 8-10]. On the other hand, in supersymmetric compactifications, while a ' corrections are expected to be relevant only for configurations for which the moduli v.e.v.'s are of the order of the string scale, the breaking of the space-time axion symmetry may be relevant at a much lower scale, especially if we expect that it plays a r61e in the supersymmetry breaking mechanism and in a non-trivial effective dilaton potential, which stabilizes the dilaton field. Therefore it is of interest, also for physical applications, to treat the moduli fields as "classical" and the dilaton field as having a non-trivial superpotential. This is the natural choice in the framework of five-brane theories [2] where the K~ihler moduli are associated to linear multiplets [3,4] and the dilaton (with its pseudoscalar partner) to a chiral multiplet. * Supported in part by DOE grants DE-AC0381-ER50050and DOE-AT03-88ER40384,TaskE. ElsevierScienceB.V. SSDI 0370-2693(94)00707-E

M. Bertolini et al. / Physics Letters B 333 (1994) 62-67

63

In this letter we show that under such circumstances there is a new sigma-model geometry associated to the Kahler class moduli fields yi, which we call "real special geometry". Indeed, for particular couplings o f the moduli fields (but not in general), it is related to the special K~ihler geometry of Calabi-Yau compactifications [ 11,12] . As special geometry, in a coordinate-free description, real geometry is characterized by a condition on the curvature tensor (1)

Rijkl = cnt![k Clln~j ,

where Cijk (i = 1 . . . . . n) is a completely symmetric tensor. In a particular set of coordinates L I ( y ) , it turns out that CIJK -~ 010JOX F ( L t) ,

(2)

G I j -~ o310j F ( L l ) ,

where F ( L 1) is a real function of the scalar fields and GIj is the metric tensor. If CIjK is constant, then this geometry is related to the geometry occurring in five-dimensional supergravity or to the special geometry of Calabi-Yau moduli spaces with CIJK = d l J K ,

F =dIJK(T'4-'F)I(T'-b

(3)

T)J(T'-k T) K ,

where dlJK are the intersection numbers of the Calabi-Yan manifold, T / are the moduli fields of the Kahler class ( I = 1 . . . . . hi.j) and F is its "volume". If the ClJ K are not constant, the geometry is not related to special Kahler geometry, but it is still described by the above curvature condition. We now turn to the derivation of real special geometry. The procedure followed below is closely analogous to the derivation of a coordinate-free description of special geometry in D = 4, N = 2 supergravity [ 12,13]. t We introduce a set of n self-interacting linear multiplets ( L t ( y ) , X t, Bu,,), I = 1 . . . . . n, where L / ( y ) are scalar fields, functions of the coordinates y i ( x # ) of the sigma-model manifold .Mn (x ~ being the fourdimensional space-time coordinates); X~(x) and B t ~ ( x ) are the dilatino and axion fields. We then promote the space-time multiplet to a superspace multiplet and introduce the supervielbein basis on superspace ( V a, ¢¢), where V a, a = 1 . . . . . 4 is the usual vielbein and q~ is the gravitino one-form. For our present purposes it is sufficient to work in global supersymmetry. In this case, the vielbein V a and the space-time spin connection satisfy the zero curvature conditions T a ==-D V a - i ~ y " ~ = O,

D V a = d V a - toab A V b ,

R ab = d w ab - toac A 09Cb .

(4)

We also introduce on .A4n a basis of "internal" supervielbein E a = E a dy i such that EA=E AV a+q'x

A,

(5)

where the supercovariant field strength E a is defined as EA=(EA_~

tzX A ) V aIz, E • a = E A Oyi 0~

(6)

and X a is related to X t by some set of covariant vectors f l a ( y ) : XI = flaxa

'

xA = fA xI

(fA = (fla)--l)

.

(7)

Together with E A, which may be thought of as the curvature of the yi fields, we define the curvatures o f X a and B I as follows: ~7X m ~ d x a - ~r~AB)(B,

H / =- dB t + iL/(y)ffff ~laXItV a ,

where l~ a is the spin connection on .Mn and H / is a three-form on superspace.

(8)

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M. Bertolini et al. / Physics Letters B 333 (1994) 62-67

The parametrizations of V X a and H t are easily found from the Bianchi identities V E a = d H t = 0, and they read vxA----~axAVa'~-

I~A : -B C ~l.~ BC~,X X +'~5xBy5xC)xIt+

[-1/,,1 ,cA [ ~ " a J I + 2l i E Aa - - 41 C BAc X- B Y5~/aX C Y5] ' ~ a x I t '

H t = nlabcVaVbV c q- f l a f ( a ~ / a b X t r v a v b ,

(9)

with f l a = EiAOiL l ,

(10)

C A c = f AIXTBf I C

(11)

and

symmetric in BC. We note that the further Bianchi identity V2X A ~ RA X B ,

(12)

where R A is the curvature two-form, is a true identity since the linear multiplet gives an off-shell representation of supersymmetry without auxiliary fields [ 1 ]. For this reason, no constraint on the geometry can be extracted from (12). However, geometric constraints can be most naturally obtained by constructing the superspace Lagrangian in the geometric framework [ 13], which, in contrast to tensor calculus techniques [ 14], gives a Lagrangian that is covariant under reparametrizations of the scalar fields manifold. Applying the set of rules of [ 13], one finds /2 = const, x {/~Aa(EA -- ,~a~

_ ff(A.xIt.)vbvcVdeabcd

_ 1 ~,A ¢,A, ,a • ,b, ,c • ,d ~lZ 115 i v V V V eabcd -- i ( f ( a " y a V X a. d - f ( A • a V X A ' ) v b v c v d • a b c d 3 ~Ftsh~Ic [ n J - f s a (f(aeab~. + f(A'rab~" ) v a v b] V c

+ ~1h~clh~cJ F H V a V b V c V d eabcd + 3 i E t ( f ( a y a b ~ . - ff(a'~/abX~')vavb _ 3 i f l a H 1(f(.axlt. _ f(A.~.) _ 6CABc fAHIf(B'TaXC. V a + 2iCaBc(f(C'ydx~. + ,~CTd~') (f(AxB" + f(A'xB')vavbvceabcd .~_U ABCD f(a. XC. f(B x D V a V b V c V d Eabcd } ,

( 13 )

where we have used chiral formalism for a generic spinor field A, 1 + T5

A.- --a, 2

A'=

Y5

1 -

2

A.

(14)

Above, /~a and hct are auxiliary first-order fields, which are identified through their equations of motion with the physical components (along the vielbein) of/~a and H t = E A,

h'a = h'a - ,o d /-/'bcd '

(15)

Fig is a function of the scalar fields L t ( y ( x ) ) , which appears in the kinetic term of the axion fields, ftA = 8aBfBt, and UAnCD is a four-index tensor so far undetermined. The superspace equations of motion along the outer directions (i.e. projected on p-forms containing at least one ~ ) yield that the scalar field functions satisfy 8AB = F l y f l a f ~ ,

CABC totally symmetric,

(16)

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as well as all the numerical coefficients in (13). The form of the tensor UABCD, being a term proportional to the space-time volume element, cannot be retrieved by an outer projection, but rather it is determined by a supersymmetry transformation on the Lagrangian in a particular sector. One finds 3 M UABCD = RABCD -- ~ C B c C A D M "t- ~ D C A B c •

(17)

Using these results, we can take the restriction of (13) to space-time and obtain the four-dimensional supersymmetric Lagrangian for the component fields E = const v/-~{EA~E A/z + 2i(~(A'~x A + ) ( : ~ X

A')

-- "~FIjhl IuhJ~ _ 2Eag(f(Aqt.U + ~(A.att.lZ) + iF/jhlfJA [Xa.*.u - x'AaIQIz -- ()(:'~l~u*.r' -- /~A"~/.t~u~I'r~)] . A B c X . - A X. A (f(C'ylz~kt.l.* -~- 2f'~lz~xt'lz) -- 1 " ~ II C -S D } , gUABcDX- A "XC X.X.

(18)

where components along the vielbein of the various (super)covariant field strengths have been substituted using the parametrizations (5) and (9). Let us exploit the consequences of Eqs. (16). By differentiating (Oc = E~O/oyi), Eq. (16) and using (11) we find 0 = OcFIjflAfJB =

+ 2FIJVcflAfJB

aFIJ f l A f J B f K c + 2 F I J C Dc A f 1 D f JB

aLr

OFts ~1 ,eJ ,el( - ~ J a J B J C -4- 2CA8c.

(19)

On the other hand, the first of Eqs. (16) also implies

OFij -aL -r

02F1 03F OLJOLX OLtOLJOLx ,

(20)

hence

CASC =

1 ,93F / s r 20L1"O-'~-OLgJf A f B f C"

(21)

By covariant differentiation on this equation and using again (11) we also find ~ D C ABc

1 ~, .el .eJ gK ,eL 3 L" t-,L ,el ,eJ ,eK = - - ~ I " I J K L J A J B J C J D -- ~rlJK~'~ A D J L J B J C

(22)

(with FH... =-- aF/OtOJ...), and thus V [ D C A ] B C -~ O.

(23)

From the definition (11) of CABC one easily deduces

0 = V[DCAIBC = RnCOA -- C~[DCAIcM,

(24)

and finally we obtain the geometric constraint on the curvature M

RABCD = C AIcCDIsM.

(25)

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M. Bertolini et a l . / Physics Letters B 333 (1994) 6 2 ~ 7

Until now we have used the fiat vielbein indices A, B . . . . in the internal manifold .A4n. If we take coordinate indices i , j . . . . . then in special coordinates yi __ L t ( y ) we have f l i = OiL I = t~li

(26)

and thus Eqs. (16) and (21) become OZF gH - OLIOLJ ,

CHK -

1

03F

20LtOL~OL K

(27)

while the constraint on the curvature becomes RHKL = Ft~i KFL]JM

(28)

FtjK being the Levi-Civita connection. It is worthwhile to observe that the above equation is exactly the same as that found in Ref. [ 15] in the construction of the sigma model of the scalar fields in D = 5 supergravity coupled to N -- 2 supermultiplets. There, Eq. (28) is obtained as a condition on the curvature of an (n - 1)-dimensional hypersurface F -- const. embedded in an n-dimensional Riemannian space AT/n and choosing a particular coordinate system. When this theory is dimensionally reduced to D = 4, a new scalar field o- appears (from the fifth component of the vielbein) and the equation for F becomes F = const, o-. By varying tr, all the space A74n is covered so that it can be identified with .A4n, while the previously constrained function F becomes actually free and can be identified with our function F. We also remark that we have found the geometric characterization of .A4n using only the supersymmetric self-interaction of the linear multiplets without coupling to supergravity, that is, in a global supersymmetric approach. One could wonder whether such characterization would change in the presence of supergravity, as it happens in special K~ihler geometry [ 12]. There, the curvature constraint in the absence of supergravity (29)

ei3k 7 = - - C i k m f 3Thgmh

changes to (30)

Ri3~r = gogk[ + gifg~3 - CikmC3i~ gm~

when supergravity is turned on. In the above formulae, Cijk =- e K W i j k ( Z ) , where Wijk are the holomorphic Yukawa couplings and K(z, ~) is the KSJaler potential. However, in the case under investigation, the constraint (25) is not changed in the presence of supergravity. The change in special Kahler geometry is due to the fact that the K~ihler manifold of the moduli fields in the globally supersymmetric case becomes a Kahler-Hodge manifold in the presence of supergravity, i.e. it acquires the structure of a holomorphic U( 1 )-bundle with U( 1 ) connection given by OiK(z, Z). It is such U(1) gauging that triggers the presence of the extra terms in (30) with respect to (29). In the present case there is no superimposed U( 1 ) bundle structure and thus the constraint (25) remains unchanged. This has been verified [16] by the explicit coupling of supergravity (in the new minimal framework) to the Lagrangian (13). Finally, it is interesting to see what is the characterization of the special coordinates y i = L t with respect to which formulas (27) hold. It is easy to show that different sets of special coordinates are related by a duality transformation, that is by a Legendre transformation with generating function F ( L ) [4,14]. In fact, let OF LI --~ L t =- OL t ,

P(L)= L t -OF ~i

- F.

(31 )

Then L t = ~ZT, oP and therefore OZF OLt oLt 1 FH =-- oUoL-----7 - o L s - ( ~ - ~ ) - = ( F - I ) t s ,

(32)

M. Bertolini et al. / Physics Letters B 333 (1994) 62-67

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in a g r e e m e n t w i t h the transformation law o f the metric gtJ =- F t j in the special coordinates ~gLt c~LJ gKL = glJ c~LK cgLL •

(33)

N o t e that the transformation (31 ) also i m p l i e s ~2F c~iLt ~

~ c ~ L

j ,

(34)

that is ft i ~

FljfJi .

(35)

In this respect, Eq. ( 3 1 ) is the a n a l o g u e o f S p ( 2 n + 2) transformations relating different sets o f special c o o r d i n a t e s in special K~ihler g e o m e t r y [ 17]. A p o s s i b l e generalization o f this framework, left to future work, is to i n c l u d e chiral and vector multiplets and the c o u p l i n g to supergravity, w h i c h is o f course needed in any study o f five-brane compactifications. It is a p l e a s u r e to thank L. A n d r i a n o p o l i and C. Vicari for their help with the c o m p u t a t i o n s d u r i n g the early stages o f this work.

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