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Kähler geometry of the space of N=2 superconformal field theories
Volume 235, number 3,4
PHYSICS LETTERS B
1 February 1990
K)~.IILER G E O M E T R Y O F T H E S P A C E O F N = 2 S U P E R C O N F O R M A L F I E L D T I t E O R I E S Vipul P E R I W A L t Institute for Theoretical Physics, University of CaliJornia, Santa Barbara, CA 93106, USA and Andrew S T R O M I N G E R 2 Department of Physics, University of California, Santa Barbara, CA 93106, USA Received 20 September 1989; revised manuscript received 30 October 1989
The moduli space, ./#, of (0, 2) superconformal field theories with integral U(1) charges is studied, adapting results from previous work on string theory and the Calabi-Yau moduli space. It is shown that Jr' is K~ihlerand that the Kahler potential is the norm of the unique top weight right-chiral primary state. Furthermore, the operator which creates this state is shown to be a holomorphic section of a vector bundle on Jr'. Some aspects of the duality transformation which flips the U ( 1) charge are discussed in this context.
1. Introduction and summary Given a c = 9 , N = (0, 2) superconformal field theory, one can construct a four dimensional string compactification with N = 1 spacetime supersymmetry [ I-6 ]. These are phenomenologically the most interesting solutions o f string theory. It is the purpose of this paper to study the local geometry o f the moduli space, ~/, o f these N = 2 superconformal theories for general values ofc. (In general, ..¢/may have a number o f c o m p o n e n t s o f different d i m e n s i o n and a priori, there is no reason to expect that .J# is even connected.) O u r results will be derived and phrased in the language o f conformal field theory, but the analysis is heavily influenced by insights gained from string theory and closely related studies o f the C a l a b i - Y a u moduli space. The geometry and topology of.,¢[ is o f interest both for string theory and for the classification program in conformal field theory. While a complete description (e.g., an explicit u n i f o r m i z a t i o n ) of.J# is an ambitious goal, general properties o f J / c a n be fruitfully t [email protected]/[email protected] "- [email protected]
investigated. Indeed, recent studies of the special case o f (2, 2) supersymmetry have revealed a rich and intriguing structure [ 7-12 ]. The geometry o f J / n e a r nodes [10 ] is relevant to the problem o f transitions between string vacua characterized by differing topological invariants. The related problem of supers y m m e t r y breaking and the selection o f a preferred string vacuum involves the study o f line bundles over ,/bt;[.
.J/[ also provides, along with its covering space, the p r o p e r framework for study of the fascinating duality transformations which generalize the R , - , 1 / R duality transformation o f toroidal compactifications to general string vacua (see e.g., refs. [ 13-17,7,18] ). These transformations form the m o d u l a r group associated to .AI. At long distances, a four dimensional string compactifieation is described by a supersymmetric field theory. An effective way to study ..¢/for the special case c = 9 is in terms of this low energy field theory [8,1 1,9]. This field theory contains massless scalar " m o d u l i fields" that are coordinates on .1#. The kinetic term o f these fields defines a metric, ~, on .J{/. Remarkably, it turns out that this low energy field theory is entirely d e t e r m i n e d (at least for (2, 2) the-
0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
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ories) by a locally holomorphic function .~ on .,/4' [ 19 ]. In particular, the metric f# is dctcrmined from .~.
,As explained in ref. [ 12 ], just as the notion o f a K~ihlcr potential exists only in the context of complex coordinates, the function .N exists only in the context of certain .special coordinates on .,¢/. In those special coordinates, the derivatives o f . ~ arc the components of a section £2 of a vector bundle on ~q. Properties of this section imply the existence o f . ~ in special coordinates (just as properties of a metric can imply lhe existence of a K~ihler potential in complex coordinates), but the section -(2 can be defined independently of special coordinates. Thus a central problem in the study of c = 9 , (2, 2) theories is to understand and detcrmine ~. Considerable progress has bccn made toward determining .Y or f2 locally in the weak-coupling limit for which the conformal field theory can be dcscribed as a sigma model with a Calabi-Yau target space. (This corresponds to a subset of'.,,//.) It was shown in refs. [ 2 0 - 2 2 ] that the cntirc low cnergy field theory (in particular .~ or £2.) can be computed from topological data of the Calabi-Yau space. Through a series o f non-renormalization arguments [ 2,23,24,9 ], it is now known that most of those results are valid to all orders in perturbation theory and some of them are cxact non-perturbatively. One would, of course, like to understand the gcometry of the full moduli space, ,¢l, without restriction to the weak-coupling limit. It appears that the time is now ripe to tackle this problem. A priori, the parameters of the low energy field theory or the correlations of the conformal field t h c o r y are functions of the real coordinates on ..//. It is quite striking that the essential information is contained within a single holomorphic function or section on ,,/~'. Ultimately, this simplification is due to spacetime supersymmetry, a powerful symmetry which is hidden in thc usual formulations of conformal ficld theory. This holomorphic structure is obviously a key to understanding the geometry of.~//and N = 2 conforreal ficld theories in general. The situation is reminiscent of the study of conformal field theory correlations as functions on the moduli space, M, of the Ricmann surface on which they are defined. A priori, the corrclators are real functions on M. However, it was realized [25] several years ago that they arc 262
I February. 1990
composed of simpler building blocks, called conformal blocks, which vary holomorphically on M. We suspect that N = 2 conformal field theories are similarly composed of building blocks which vary holomorphically on ./,¢. In this paper one such holomorphic object, the top weight "' chiral primary field .(2+, will be discussed. It is our belief that this field defines globally a holomorphic section of the bundle o f c o h o m o l o g y classes o f chiral fields over .,# which can be identified with the aforementioned section £2. The results of this paper are consistent with this conjecture. A first step in studying the geometry of,.,//is to obtain conformal field thcoretic expressions for the various geometrical objcct on .h'. An important example is the Zamolodchikov formula for the metric [ 26 ]: ',~z:( Z ~, 2 x) = ~ ( M , (0)'k/r( 1 ) ) z ,
( 1)
where .,'v/zare truly marginal moduli field vertex operators, Z K are coordinates on .,g and the subscript Z on the expectation value denotes that the correlation function is evaluated in the conformal field theory associated to the point Z K. This is a natural metric on ,,,/./and coincides for c = 9 with that of the low energy field theory of string compact i fication ~2. N-- 1 spacetime supersymmet~3" of the low energy field theory (which tbllows from N = 2 world sheet supersymmetry) requires for c = 9 that f,e be Kfihler. This suggests that for any value o f c f¢ will be K~ihler. In section 2 we will verify this by showing that for a general (0, 2) conformal field theory with integral U( 1 ) charges and c > 3 the K~ihlcr potential is given by
.g(Z K, 2 K) = I n (-(2* (0)-(2- (1) ) z ,
(2)
where-Q+ (-(2-) is the top weight (0, c/6) chiral (antichiral) field which exists in every (0, 2) conformal field theory. (This formula is not restricted to c = 9. ) We will further show, in a sense to be made precise, that -(2+ (£2-) varies holomorphically (antiholomorphically) on ..1t. e Jc may then be interpreted as the norm o f a holomorphic section of the vector bun*~ A note regarding the term "top weight" "top" means that the weight of this field is the largest weight of any primary field in the theory; this terminology follows that used for differential forms. ,2 One way to see this is to compare the derivative of ~qwith the string formula for the three point thnction.
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die on ,i/whose fibers are isomorphic to the conformal field theory Hilbert space. Such a representation for c* was already known to exist for the special case of weakly coupled (2, 2) theories [27,28,10].
2. The K/ihler potential
Before proceeding to the general case, we first consider (2, 2) theories in the weak-coupling limit where they can be described by sigma models with complex dimension ? - c / 3 Calabi-Yau target spaces. [Integral U( 1 ) charges implies ~ is an integer.] The top weight chiral primary operator then corresponds to the top rank holomorphic ?-form -(2 which exists on every' Calabi-Yau space. This operator may be expressed
£2.+~( z ) -
ie(e+ 1)/2
?!
f2,...i,(X)q/"(z)...~#e(z).
(3)
X g is a complex coordinate on the Calabi-Yau space with world sheet superpartner ¢/~,z is a complex world sheet coordinate and the subscript a denotes that the quantity is defined in sigma model perturbation theory. This is simply the usual expression for the top rank form with the differentials dX ~replaced by world shcct fermions ~,'(z). Similarly the top weight antichiral primary field is
D ; (z) -
ie(e+l)/2
~.
_
-
F,
O,,...s(X)~, (z)...~'e(z).
(5)
(All matrix elements in this paper are connected and normalized.) To leading order in sigma model perturbation theory this is, up to constants which can be absorbed by a K~ihler transformation ,g~ = I n f d2eX ([2 ^ ~ ) - l n
f d2c~X( J ^ J ^ J ) ,
the moduli space ,,¢¢-,,t of complex structures, and a choice of J specifies a point on the moduli space ,~l,J of K~ihler structures. (6) is thus a function on .~=,ffgLiXJg2., and is known [27,28,10] to be the K~ihler form which reproduces the desired metric [21,3], up to a sign. In fact, the signs in (6) give a negative metric on ,fig2,. and a positive metric on .lg~,~. This negative sign will bc discussed further after the calculation for the general conformal field theory has been described. The target space geometry for weakly coupled (0, 2) theories has been discussed in refs. [2,3]. It is not in general K~ihler, but it has been shown [ 3 ] that a holomorphic ?-form nevertheless always exists. Little is known about this interesting class of manifolds, but we shall see from the following conformal field theoretic arguments that their moduli space should be K~ihler. To proceed to the general case, denote by M i ( z d ) the weight (1, 1) moduli fields of a (0, 2) superconformal field theory which generate supersymmctrypreser~'ing deformations. The index I is tangent to ,~/. lnvariancc of the perturbed action under supersymmerry transformations implies that such operators must obey [G+1/2, Mz] = O P t . The graded Jacobi identity for G +_1/2, G_-,/2 and Mz is
3([G+,/z,PF]+[G-,/e,F'~']-M,)=O.
(7)
This implies M~ = M + + MF , with (4)
Consider the expression ,./{i = In (f2 + ( 0 ) £ 2 j ( 1 ) ) z .
I February 1990
(6)
where .I is the K~ihler form on the Calabi-Yau space. The second integral is the volume of the Calabi-Yau space. This appears because we have divided out the norm of the vacuum in (6), which contains a factor of the volume arising from the integration over X zero modes. A particular choice of£2 specifics a point on
mi =[G-'-t/>Pf ] .
(8)
Using the fact that P+ (PF) is a chiral (antichiral) primary field, it is easily checked that i ( M + - M F ) is also a moduli field. [A chiral (antichiral) field commutes with G +_1/', ( G - i/2). ] M]- ( M F ) generates holomorphic (antiholomorphic) deformations of the (0, 2) theory, where here and henceforth the complex structure on ,/t/ is used to distinguish between barred and unbarred indices. These deformations arc described in tcrms of the connected correlation functions of the theory according to thc formula (~1 ""~n )Z \
= (exp (_f
d2x ( Z L'VU + Z qvU )
~,...~,) , (9)
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in addition to possible renormalizations of the operators 0;. Ever3; (0, 2) superconformal field theory has a charge q=?, weight ( 0 , ? / 2 ) chiral primary field o + (z) and a charge - ? weight (0, g/2) antichiral primary field g2- (z). (2 + is the top weight chiral primary field. In the special case of (2, 2) Calabi-Yau models, O + is related to the top rank holomorphic (0, ?)-form which exists on every Calabi-Yau manifold via eq. (4). This form is known to vary holomorphically on the Calabi-Yau moduli space [28,27], i.e., its integrals over a fixed set o f f cycles are holomorphic [27 ]. An analogous statement may bc shown for the general (0, 2) conformal field theory. Namely, the integral of g2+,
+
g2(2_e)/2 =
IZ
g2+ (z) ,
(10)
commutes with the antiholomorphic moduli fields Mr, while g2~3-e)/2 commutes with M +"
['Q?Z-E)/2, M ;
] =0.
(12)
where fr is a ( 1 , 0 ) field. It then follows from [g2~?z, -Q+ ] = 1 where g2/)z is the L o = -?/2 mode of 12-, that
[Oh2, R,] =ft.
where we have used the fact that g2~)2 commutes with both G - 1/2 and I ' ~ . The former is the statement that g2- is antichiral primaD' and the latter follows from the absence of h = ½( 1 - ? ) charged fields. Comparing (13) with (14) we find that f r = 0, and finally, + :F [-Q-~-2-e)/2,M ]=0.
(15)
This equation can also be easily shown to hold for the regulated forms of M -+ to be introduced shortly, except for the marginal case g= 1, for which a more detailed analysis than is given here is required. Now consider the two point function: eer(u, v) - (-(2+ ( u ) I 2 - ( v ) ) z .
(16)
We will show that .~(0, 1 ) is the K~ihler potential for the Zamolodchikov metric on moduli space. The metric derived from .~"(u, v) is
~u(u, v)= O/O~,,~(u, v) = d 2zt d 2~z (M[(zl,g~)My(z2,~,2)
( 11)
It follows that -Q+ (~'2-) does not need to be renormalized under holomorphic (antiholomorphic) deformations of the moduli, g2+ ( ~ - ) may thus be defined as a holomorphic (antiholomorphic) section of the bundle of operators over ,.~//~3. + To prove ( 11 ), define Rr= [.c2 -~2-e)/2, Mi- ]. Rris a charge ? weight ( 1, ? / 2 ) primary, field. Since it has 2h=q, it is a chiral primar? field. Using the fact that the boson of the U ( 1 ) charge decouples from the rest of the theory, R r m a y bc expressed as
Rr=Ln +,
1 February 1990
(13)
On the other hand, the c o m m u t a t o r may be directly evaluated using the Jacobi identity and the definition of R;: One has
X
g2+(u)~-(v))(u-v) ~,
(17)
plus a term involving two matrix elements linear in M which vanishes. This expression is defined by first replacing M + with the nearly marginal, supersymmetric operators of the form M +-(k)
= [G+I/2,
P~ exp(ikX) ]
= [m ±(0) +ik'~'+P T ] e x p ( i k . X ) ,
(18)
where X(z, ~), qJ(z) is a ? = 1, N = (0, 2) supermultipiet with ~,+- = [G---i/2, X]. k is then taken to zero at the end of the calculation. One might have also considered the possibility that finite renormalizations oft'2 should be included in the expression (17) for perturbations of the correlation function (16) under deformations of the moduli. This is partially a matter of definition of s'2 as a function on ~¢, and a choice of K~ihler gauge. Our " m i n i m a l " definition, which does not include such renormalizations, is consistent because g2 remains holomorphic in perturbed correlation functions
[0~2, Rr] = [g2~)2, [12~%_a)/2, [G-l~2, e ~ ] ] ] = 0 , (14)
O~ l~2+ (u)O,(v, ~)... I d2z'~I]-(z, ~)...) =O ,
~3 The results of ref. [ 12 ] suggc~stthat the chiral cohomology class (with respect to G_+1.2) o f w h i c h g2 + is an e l e m e n t can be defined as a h o l o m o r p h i c section of the finite d i m e n s i o n a l bundie o f c o h o m o l o g y classes of chiral operators.
264
(19) for k-*0. This identifies g2 as (a complex multiple o f )
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the top rank chiral field. For non-zero k, fe/zjmay be evaluated by differentiating with respect to a and r7 and then integrating. Using 0~
1 - zr~2~(u-z) , /,/--Z
(20)
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contour (26) may be reduced to the Zamolodchikov formula. For general values of?, note that the contour integral of ze-2.Q + (z) at infinity vanishes in correlation functions since g2+ is weight ?/2. This leads to the identity
o = ( u - v ) 2 - e ~-~ll~ dz (z-v)e-2
and ( 18 ) with (17), one finds
~,,0~%s(u, v)= ~< [o- ~3_~/> ml (u, a; k)] X[g2?-z_e)/2, M y ( v , O ; - k ) ] > ( u - v )
e.
(21)
It is crucial here that [12~:~_e)/2, M - V ] = 0 even for k#0. Defining
x <~+ (z)Pi- (u, a)[~272_e~/2, PY(v, 0) 1> ~---< [n(+2_g)/2, P]- (b/, a) ] [-Q~2-g')/2, eJ" (u,/7)] >
+(v-u)2-~fl'F(u,a)[Jo, PJ(v,O)l) .
(27)
[~Q+ -¢2-e)/2, Pl- (u, t ~ ) ] = S + ,
(22)
The U ( l ) charge o f P + is + I. The two point function (26) may thus be rewritten
[~2~2_e)/~, 1"3 (v, 0) ] =S)- ,
(23)
r.#,.r= - ~ ( l'i- (0)PI ( 1 ) > .
This is related to the Zamolodchikov formula by world sheet supersymmetry. By deforming contours of G -+ one obtains finally
and using
<,,+(u)~,-(v) > _ -
1
--
~l--P
(28)
'
0~0sln <~+ (0).o- (1) >
( e x p ( i k . X ) (u, a) e x p ( - i k . X ) (v, 0)>
='(M~-(0)My(I)>.
1 -lu_vl2k~,
(29)
(24) 3. Duality
one has k2(-) e
8 ( u - v) ~ - e + ~ 2 ( a - 0) k2
× .
(25)
Integrating with respect to a and tT, letting k2--,0 and setting u = 0 and v = 1, one has .%j= ~ ( S , + (0)$5- ( 1 ) > .
(26)
The integration constant is set to zero by conformal invariance. The equivalence of this to the Zamolodchikov formula for 3 = 3 can be seen by using the result of refs. [29,15] that the ( 1, 1 ) operators S a r e string vertex operators for auxiliary fields which lie in the same spacetime supcrmultiplet as the moduli fields. Spacetime supersymmetry then requires that their metric equals that of the moduli fields. More explicitly, S may be written as two contour integrals of the spacetime supercurrent around M and by deforming the
The metric (29) is positive-definite, in apparent contradiction to the sigma model result, eq. (6), which leads to a negative metric for the (2, 1 ) form and a positive metric for the ( 1, 1 ) forms. The resolution of this apparent sign discrepancy is quite interesting. The fact that the conformal field theory calculation could not possibly reproduce the sigma model result can be seen without detailed calculation. The conformal field theory' analysis was valid for a general (0, 2 ) model, and thus used only right-moving supersymmerry. (2, 1) and (1, 1) fields can be distinguished only in (2, 2) theories. They differ in the relative signs of the left and right charges. Our calculation was blind to the left charge, so the metric for all the moduli had to have the same sign. Even in a (2, 2) theory the labclling of (2, 1 ) and ( 1, 1 ) fields is a matter of convention. Thcre is a Z2 duality symmetry which interchanges them [ 30 ] :
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/-I~* - H ,
PHYSICS LETTERS B
(30)
where H is the right-moving U ( 1 ) boson. The sigma model calculation was able to give different signs for (2, 1 ) and ( 1, 1 ) fields because the prescription for finite renormalizations o f (2~ under motion in ,,# broke the s y m m e t r y (30). In particular, the c o m p o n e n t s .Q,..~ are renormalized under perturbations generated by (2, 1 ) fields but not under those generated by ( 1, 1 ) fields. Our definition of,Q + as a function on ,Jr' thus does not correspond exactly to the o p e r a t o r g2+ in the weak-coupling limit. ,Q+, in fact, reduces to a different " d u a l i z e d " operator as follows: consider the operator iete+ z)/2 ~¢~+ ~
(~
~t~'~tl"'id~l/il...~li e ,
(31)
normalized so that (*Off (0)-(2-+ ( 1 ) > z = 1 ,
sentation, the target space is a different manifold
X~,X,
In (-Q+ (0)-Qg ( 1 ) ) z I x
= - I n <.f2~+ (O).Qg ( 1 ) > z l . x ,
~2+ ~* *g2f .
This might have been expected since G+~/2, which corresponds to O, is interchanged with G : ~/2, which corresponds to *0-. Thus cohomology is dualized,
This appears to be consistent with (38).
(33)
This definition does not involve a metric. It is easy to see that (*0-)2=0, so that cohomo]ogy classes may be defined. * ~ + is an element of such a eohomology class, and (32) depends only on the respective cohomology classes of*~2~- and ~ + . Furthermore, the K~ihler potential
(34)
leads to m i n u s the metric o f (6). There are two more operators that can be defined in this m a n n e r by separately dualizing on either the ,~//2.~ or .,16,~ pieces o f these moduli spaces. One o f these will lead to a positive-definite metric on ,/{=J[2a ×.~/6.~. This is the o p e r a t o r which coincides with Q+ in the weak-coupling limit. The reason such dualized operators arise can possibly be understood as follows. U n d e r the duality transformation (30) the (2; 1 ) forms, or elements o f II ~( T ) interchange with the ( 1, 1 ) forms H ~( T * ) :
II~ ( T ) ~ t l t
(T *) .
(35)
If the dualized field theory has a sigma model repre266
(38)
(32)
(*0- 7)
* . ~ = In < .~2~- (0)*£22 ( 1 ) > z = - ~
(37)
where the left-hand side is defined with respect to sigma model perturbation theory, on X a n d the righthand side is defined with respect to sigma model perturbation theory, on *X. C o m p a r i n g with (34), one is led to the conclusion that
O~* *O*.
T [ab,...l
(36)
with opposite Euler character. O f course, no explicit construction o f such a dualized manifold has been given, so this paragraph should be regarded as speculative. (35) implies that the K~ihler potential as defined in ( 6 ) changes sign under (30):
at every point on moduli space. The object *,Q" '~' is an a n t i s y m m e t r i c rank ( c o n t r a v a r i a n t tensor density o f weight 1. There is a natural dual cohomology theory defined for such objects. Define *0- by tb,.a...] = 0,,
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( 39 )
Acknowledgement We are grateful to L. Dixon and S. Ferrara for useful conversations. A.S. was supported in part by DOE grant 8-484062-2500-3 and the A.P. Sloan F o u n d a tion. This research was s u p p o r t e d in part by the National Science F o u n d a t i o n under G r a n t No. PHY8217853, s u p p l e m e n t e d by funds from the National Aeronautics and Space Administration, at the University o f California at Santa Barbara.
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[7]S. Cecotti, S. Ferrara and L. Girardello, Phys. Left. B 213 ( 1988 ) 443; Geometry oftype II super,strings and the moduli of superconformal field theories, Intern. J. Mod. Phys. 4 ( 1989 ) 2475. [8] N. Sciberg, Nucl. Phys. B 303 (1988) 286. [ 9 ] L.J. Dixon, V. Kaplunovsky and J. Louis, On effective field theories describing (2, 2) vacua of the heterotic string, SLAC-PU B-4959 (April 1989 ). [10IP. Candelas, P.S. Green and T. Hiibsch, Rolling among Calabi-Yau vacua, UTTG-10-89 (April 1989), in: Trieste Proc., eds. M. Green, A. Polyakov and A. Strominger, to appear; Connected Calabi-Yau compactifications (Other worlds are just around the corner), UTTG-28-88 (January 1989). [ I I ] S . Ferrara and A. Strominger, N = 2 space-time supersymmetry and Calabi-Yau moduli space, CERN preprint TH 5291/89 (February. 1989), in: Proc. Strings '89 Workshop (College Station, TX, March 1989), to appear. [ 12] A. Strominger, Special geometry, to appear. [13]K. Kikkawa, and M. Yamasaki, Phys. Lett. B 149 (1984) 357; N. Sakai and I. Senda, Prog. Theor. Phys. Suppl. 85 ( 1985 ) 228. [ 14 ] V.P. Nair, A. Shapere, A. Strominger and F. Wilczek, Nuel. Phys. B 287 (1987) 402. [ 15 ] M. Dine, P. Huet and N. Seiberg, Large and small radius in string theory, IAS preprint IASSNS-HEP-88/54 (January 1989). [16] S. Ferrara, D. Liist and S. Theisen, World sheet versus spectrum symmetries in heterotie and type 11 superstrings, CERN preprint TH-5341/89 (March 1989). [ 17 ] S. Ferrara, D. Liist, A. Shapere and S. Theisen, Phys. Lett. B 225 (1989) 363.
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[ 18 ] C. Vafa, Quantum symmetries of string vacua, preprint HUTP-89 / A021 (May 1989 ) ; String vacua and orbifoldizcd L-G models, preprint HUTP-89/A018 (April 1989). [ 19] B. de Wit, P.G. Lauwers and A. Van Proeyen, Nucl. Phys. B 255 (1985) 569. [20] A. Strominger and E. Witten, Commun. Math. Phys. 101 (1985) 341. [21 ] A. Strominger, Phys. Rev. Lctt. 55 ( 1985 ) 2547. [ 22 ] Examples are worked out by A. Strominger, in: Unified string theories, eds. M.B. Green and D.J. Gross (World Scientific, Singapore, 1986 ); P. Candelas, Nucl. Phys. B 298 (1988) 458. [231 M. Dine and N. Seiberg, Phys. Rev. Lett. 57 (1986) 2625; Nucl. Phys. B 301 (1988) 357; B 306 (1988) 137. [ 24 ] J. Distler and B. Greene, Nucl. Phys. B 304 (1988) 1; B 309 (1988) 295. [25] J.L Cardy, Nucl. Phys. B 270 [FS16] (1986) 186; A.A. Belavin and V.G. Knizhnik, Sov. Phys. JETP 64 (1986) 214; Phys. Left. B 168 (1986) 201; D.H. Friedan and S.H. Shenker, Nucl. Phys. B 281 (1987) 509. [26] A.B. Zamolodchikov, JETP Left. 43 (1986) 730. [ 27 ] R. Bryant and P. Griffiths, in: Arithmetic and geometry, Vol. II, eds. M. Artin and J. Tate (Birkh/iuser, Basel, 1983) p. 77. [28 ] G. Tian, in: Mathematical aspects of string theory, ed. S.-T. Yau (World Scientific, Singapore, 1987) p. 629. [29 ] J.J. Atick, L.J. Dixon and A. Sen, Nucl. Phys. B 292 ( 1987 ) 109. [ 30] L.J. Dixon, preprint PUPT-1074 (October 1987 ); lectures given at the 1987 ICTP Summer Workshop in High energy physics and cosmology (Trieste, Italy, June-August 1987).
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K)~.IILER G E O M E T R Y O F T H E S P A C E O F N = 2 S U P E R C O N F O R M A L F I E L D T I t E O R I E S Vipul P E R I W A L t Institute for Theoretical Physics, University of CaliJornia, Santa Barbara, CA 93106, USA and Andrew S T R O M I N G E R 2 Department of Physics, University of California, Santa Barbara, CA 93106, USA Received 20 September 1989; revised manuscript received 30 October 1989
The moduli space, ./#, of (0, 2) superconformal field theories with integral U(1) charges is studied, adapting results from previous work on string theory and the Calabi-Yau moduli space. It is shown that Jr' is K~ihlerand that the Kahler potential is the norm of the unique top weight right-chiral primary state. Furthermore, the operator which creates this state is shown to be a holomorphic section of a vector bundle on Jr'. Some aspects of the duality transformation which flips the U ( 1) charge are discussed in this context.
1. Introduction and summary Given a c = 9 , N = (0, 2) superconformal field theory, one can construct a four dimensional string compactification with N = 1 spacetime supersymmetry [ I-6 ]. These are phenomenologically the most interesting solutions o f string theory. It is the purpose of this paper to study the local geometry o f the moduli space, ~/, o f these N = 2 superconformal theories for general values ofc. (In general, ..¢/may have a number o f c o m p o n e n t s o f different d i m e n s i o n and a priori, there is no reason to expect that .J# is even connected.) O u r results will be derived and phrased in the language o f conformal field theory, but the analysis is heavily influenced by insights gained from string theory and closely related studies o f the C a l a b i - Y a u moduli space. The geometry and topology of.,¢[ is o f interest both for string theory and for the classification program in conformal field theory. While a complete description (e.g., an explicit u n i f o r m i z a t i o n ) of.J# is an ambitious goal, general properties o f J / c a n be fruitfully t [email protected]/[email protected] "- [email protected]
investigated. Indeed, recent studies of the special case o f (2, 2) supersymmetry have revealed a rich and intriguing structure [ 7-12 ]. The geometry o f J / n e a r nodes [10 ] is relevant to the problem o f transitions between string vacua characterized by differing topological invariants. The related problem of supers y m m e t r y breaking and the selection o f a preferred string vacuum involves the study o f line bundles over ,/bt;[.
.J/[ also provides, along with its covering space, the p r o p e r framework for study of the fascinating duality transformations which generalize the R , - , 1 / R duality transformation o f toroidal compactifications to general string vacua (see e.g., refs. [ 13-17,7,18] ). These transformations form the m o d u l a r group associated to .AI. At long distances, a four dimensional string compactifieation is described by a supersymmetric field theory. An effective way to study ..¢/for the special case c = 9 is in terms of this low energy field theory [8,1 1,9]. This field theory contains massless scalar " m o d u l i fields" that are coordinates on .1#. The kinetic term o f these fields defines a metric, ~, on .J{/. Remarkably, it turns out that this low energy field theory is entirely d e t e r m i n e d (at least for (2, 2) the-
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ories) by a locally holomorphic function .~ on .,/4' [ 19 ]. In particular, the metric f# is dctcrmined from .~.
,As explained in ref. [ 12 ], just as the notion o f a K~ihlcr potential exists only in the context of complex coordinates, the function .N exists only in the context of certain .special coordinates on .,¢/. In those special coordinates, the derivatives o f . ~ arc the components of a section £2 of a vector bundle on ~q. Properties of this section imply the existence o f . ~ in special coordinates (just as properties of a metric can imply lhe existence of a K~ihler potential in complex coordinates), but the section -(2 can be defined independently of special coordinates. Thus a central problem in the study of c = 9 , (2, 2) theories is to understand and detcrmine ~. Considerable progress has bccn made toward determining .Y or f2 locally in the weak-coupling limit for which the conformal field theory can be dcscribed as a sigma model with a Calabi-Yau target space. (This corresponds to a subset of'.,,//.) It was shown in refs. [ 2 0 - 2 2 ] that the cntirc low cnergy field theory (in particular .~ or £2.) can be computed from topological data of the Calabi-Yau space. Through a series o f non-renormalization arguments [ 2,23,24,9 ], it is now known that most of those results are valid to all orders in perturbation theory and some of them are cxact non-perturbatively. One would, of course, like to understand the gcometry of the full moduli space, ,¢l, without restriction to the weak-coupling limit. It appears that the time is now ripe to tackle this problem. A priori, the parameters of the low energy field theory or the correlations of the conformal field t h c o r y are functions of the real coordinates on ..//. It is quite striking that the essential information is contained within a single holomorphic function or section on ,,/~'. Ultimately, this simplification is due to spacetime supersymmetry, a powerful symmetry which is hidden in thc usual formulations of conformal ficld theory. This holomorphic structure is obviously a key to understanding the geometry of.~//and N = 2 conforreal ficld theories in general. The situation is reminiscent of the study of conformal field theory correlations as functions on the moduli space, M, of the Ricmann surface on which they are defined. A priori, the corrclators are real functions on M. However, it was realized [25] several years ago that they arc 262
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composed of simpler building blocks, called conformal blocks, which vary holomorphically on M. We suspect that N = 2 conformal field theories are similarly composed of building blocks which vary holomorphically on ./,¢. In this paper one such holomorphic object, the top weight "' chiral primary field .(2+, will be discussed. It is our belief that this field defines globally a holomorphic section of the bundle o f c o h o m o l o g y classes o f chiral fields over .,# which can be identified with the aforementioned section £2. The results of this paper are consistent with this conjecture. A first step in studying the geometry of,.,//is to obtain conformal field thcoretic expressions for the various geometrical objcct on .h'. An important example is the Zamolodchikov formula for the metric [ 26 ]: ',~z:( Z ~, 2 x) = ~ ( M , (0)'k/r( 1 ) ) z ,
( 1)
where .,'v/zare truly marginal moduli field vertex operators, Z K are coordinates on .,g and the subscript Z on the expectation value denotes that the correlation function is evaluated in the conformal field theory associated to the point Z K. This is a natural metric on ,,,/./and coincides for c = 9 with that of the low energy field theory of string compact i fication ~2. N-- 1 spacetime supersymmet~3" of the low energy field theory (which tbllows from N = 2 world sheet supersymmetry) requires for c = 9 that f,e be Kfihler. This suggests that for any value o f c f¢ will be K~ihler. In section 2 we will verify this by showing that for a general (0, 2) conformal field theory with integral U( 1 ) charges and c > 3 the K~ihlcr potential is given by
.g(Z K, 2 K) = I n (-(2* (0)-(2- (1) ) z ,
(2)
where-Q+ (-(2-) is the top weight (0, c/6) chiral (antichiral) field which exists in every (0, 2) conformal field theory. (This formula is not restricted to c = 9. ) We will further show, in a sense to be made precise, that -(2+ (£2-) varies holomorphically (antiholomorphically) on ..1t. e Jc may then be interpreted as the norm o f a holomorphic section of the vector bun*~ A note regarding the term "top weight" "top" means that the weight of this field is the largest weight of any primary field in the theory; this terminology follows that used for differential forms. ,2 One way to see this is to compare the derivative of ~qwith the string formula for the three point thnction.
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die on ,i/whose fibers are isomorphic to the conformal field theory Hilbert space. Such a representation for c* was already known to exist for the special case of weakly coupled (2, 2) theories [27,28,10].
2. The K/ihler potential
Before proceeding to the general case, we first consider (2, 2) theories in the weak-coupling limit where they can be described by sigma models with complex dimension ? - c / 3 Calabi-Yau target spaces. [Integral U( 1 ) charges implies ~ is an integer.] The top weight chiral primary operator then corresponds to the top rank holomorphic ?-form -(2 which exists on every' Calabi-Yau space. This operator may be expressed
£2.+~( z ) -
ie(e+ 1)/2
?!
f2,...i,(X)q/"(z)...~#e(z).
(3)
X g is a complex coordinate on the Calabi-Yau space with world sheet superpartner ¢/~,z is a complex world sheet coordinate and the subscript a denotes that the quantity is defined in sigma model perturbation theory. This is simply the usual expression for the top rank form with the differentials dX ~replaced by world shcct fermions ~,'(z). Similarly the top weight antichiral primary field is
D ; (z) -
ie(e+l)/2
~.
_
-
F,
O,,...s(X)~, (z)...~'e(z).
(5)
(All matrix elements in this paper are connected and normalized.) To leading order in sigma model perturbation theory this is, up to constants which can be absorbed by a K~ihler transformation ,g~ = I n f d2eX ([2 ^ ~ ) - l n
f d2c~X( J ^ J ^ J ) ,
the moduli space ,,¢¢-,,t of complex structures, and a choice of J specifies a point on the moduli space ,~l,J of K~ihler structures. (6) is thus a function on .~=,ffgLiXJg2., and is known [27,28,10] to be the K~ihler form which reproduces the desired metric [21,3], up to a sign. In fact, the signs in (6) give a negative metric on ,fig2,. and a positive metric on .lg~,~. This negative sign will bc discussed further after the calculation for the general conformal field theory has been described. The target space geometry for weakly coupled (0, 2) theories has been discussed in refs. [2,3]. It is not in general K~ihler, but it has been shown [ 3 ] that a holomorphic ?-form nevertheless always exists. Little is known about this interesting class of manifolds, but we shall see from the following conformal field theoretic arguments that their moduli space should be K~ihler. To proceed to the general case, denote by M i ( z d ) the weight (1, 1) moduli fields of a (0, 2) superconformal field theory which generate supersymmctrypreser~'ing deformations. The index I is tangent to ,~/. lnvariancc of the perturbed action under supersymmerry transformations implies that such operators must obey [G+1/2, Mz] = O P t . The graded Jacobi identity for G +_1/2, G_-,/2 and Mz is
3([G+,/z,PF]+[G-,/e,F'~']-M,)=O.
(7)
This implies M~ = M + + MF , with (4)
Consider the expression ,./{i = In (f2 + ( 0 ) £ 2 j ( 1 ) ) z .
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(6)
where .I is the K~ihler form on the Calabi-Yau space. The second integral is the volume of the Calabi-Yau space. This appears because we have divided out the norm of the vacuum in (6), which contains a factor of the volume arising from the integration over X zero modes. A particular choice of£2 specifics a point on
mi =[G-'-t/>Pf ] .
(8)
Using the fact that P+ (PF) is a chiral (antichiral) primary field, it is easily checked that i ( M + - M F ) is also a moduli field. [A chiral (antichiral) field commutes with G +_1/', ( G - i/2). ] M]- ( M F ) generates holomorphic (antiholomorphic) deformations of the (0, 2) theory, where here and henceforth the complex structure on ,/t/ is used to distinguish between barred and unbarred indices. These deformations arc described in tcrms of the connected correlation functions of the theory according to thc formula (~1 ""~n )Z \
= (exp (_f
d2x ( Z L'VU + Z qvU )
~,...~,) , (9)
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in addition to possible renormalizations of the operators 0;. Ever3; (0, 2) superconformal field theory has a charge q=?, weight ( 0 , ? / 2 ) chiral primary field o + (z) and a charge - ? weight (0, g/2) antichiral primary field g2- (z). (2 + is the top weight chiral primary field. In the special case of (2, 2) Calabi-Yau models, O + is related to the top rank holomorphic (0, ?)-form which exists on every Calabi-Yau manifold via eq. (4). This form is known to vary holomorphically on the Calabi-Yau moduli space [28,27], i.e., its integrals over a fixed set o f f cycles are holomorphic [27 ]. An analogous statement may bc shown for the general (0, 2) conformal field theory. Namely, the integral of g2+,
+
g2(2_e)/2 =
IZ
g2+ (z) ,
(10)
commutes with the antiholomorphic moduli fields Mr, while g2~3-e)/2 commutes with M +"
['Q?Z-E)/2, M ;
] =0.
(12)
where fr is a ( 1 , 0 ) field. It then follows from [g2~?z, -Q+ ] = 1 where g2/)z is the L o = -?/2 mode of 12-, that
[Oh2, R,] =ft.
where we have used the fact that g2~)2 commutes with both G - 1/2 and I ' ~ . The former is the statement that g2- is antichiral primaD' and the latter follows from the absence of h = ½( 1 - ? ) charged fields. Comparing (13) with (14) we find that f r = 0, and finally, + :F [-Q-~-2-e)/2,M ]=0.
(15)
This equation can also be easily shown to hold for the regulated forms of M -+ to be introduced shortly, except for the marginal case g= 1, for which a more detailed analysis than is given here is required. Now consider the two point function: eer(u, v) - (-(2+ ( u ) I 2 - ( v ) ) z .
(16)
We will show that .~(0, 1 ) is the K~ihler potential for the Zamolodchikov metric on moduli space. The metric derived from .~"(u, v) is
~u(u, v)= O/O~,,~(u, v) = d 2zt d 2~z (M[(zl,g~)My(z2,~,2)
( 11)
It follows that -Q+ (~'2-) does not need to be renormalized under holomorphic (antiholomorphic) deformations of the moduli, g2+ ( ~ - ) may thus be defined as a holomorphic (antiholomorphic) section of the bundle of operators over ,.~//~3. + To prove ( 11 ), define Rr= [.c2 -~2-e)/2, Mi- ]. Rris a charge ? weight ( 1, ? / 2 ) primary, field. Since it has 2h=q, it is a chiral primar? field. Using the fact that the boson of the U ( 1 ) charge decouples from the rest of the theory, R r m a y bc expressed as
Rr=Ln +,
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(13)
On the other hand, the c o m m u t a t o r may be directly evaluated using the Jacobi identity and the definition of R;: One has
X
g2+(u)~-(v))(u-v) ~,
(17)
plus a term involving two matrix elements linear in M which vanishes. This expression is defined by first replacing M + with the nearly marginal, supersymmetric operators of the form M +-(k)
= [G+I/2,
P~ exp(ikX) ]
= [m ±(0) +ik'~'+P T ] e x p ( i k . X ) ,
(18)
where X(z, ~), qJ(z) is a ? = 1, N = (0, 2) supermultipiet with ~,+- = [G---i/2, X]. k is then taken to zero at the end of the calculation. One might have also considered the possibility that finite renormalizations oft'2 should be included in the expression (17) for perturbations of the correlation function (16) under deformations of the moduli. This is partially a matter of definition of s'2 as a function on ~¢, and a choice of K~ihler gauge. Our " m i n i m a l " definition, which does not include such renormalizations, is consistent because g2 remains holomorphic in perturbed correlation functions
[0~2, Rr] = [g2~)2, [12~%_a)/2, [G-l~2, e ~ ] ] ] = 0 , (14)
O~ l~2+ (u)O,(v, ~)... I d2z'~I]-(z, ~)...) =O ,
~3 The results of ref. [ 12 ] suggc~stthat the chiral cohomology class (with respect to G_+1.2) o f w h i c h g2 + is an e l e m e n t can be defined as a h o l o m o r p h i c section of the finite d i m e n s i o n a l bundie o f c o h o m o l o g y classes of chiral operators.
264
(19) for k-*0. This identifies g2 as (a complex multiple o f )
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the top rank chiral field. For non-zero k, fe/zjmay be evaluated by differentiating with respect to a and r7 and then integrating. Using 0~
1 - zr~2~(u-z) , /,/--Z
(20)
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contour (26) may be reduced to the Zamolodchikov formula. For general values of?, note that the contour integral of ze-2.Q + (z) at infinity vanishes in correlation functions since g2+ is weight ?/2. This leads to the identity
o = ( u - v ) 2 - e ~-~ll~ dz (z-v)e-2
and ( 18 ) with (17), one finds
~,,0~%s(u, v)= ~< [o- ~3_~/> ml (u, a; k)] X[g2?-z_e)/2, M y ( v , O ; - k ) ] > ( u - v )
e.
(21)
It is crucial here that [12~:~_e)/2, M - V ] = 0 even for k#0. Defining
x <~+ (z)Pi- (u, a)[~272_e~/2, PY(v, 0) 1> ~---< [n(+2_g)/2, P]- (b/, a) ] [-Q~2-g')/2, eJ" (u,/7)] >
+(v-u)2-~fl'F(u,a)[Jo, PJ(v,O)l) .
(27)
[~Q+ -¢2-e)/2, Pl- (u, t ~ ) ] = S + ,
(22)
The U ( l ) charge o f P + is + I. The two point function (26) may thus be rewritten
[~2~2_e)/~, 1"3 (v, 0) ] =S)- ,
(23)
r.#,.r= - ~ ( l'i- (0)PI ( 1 ) > .
This is related to the Zamolodchikov formula by world sheet supersymmetry. By deforming contours of G -+ one obtains finally
and using
<,,+(u)~,-(v) > _ -
1
--
~l--P
(28)
'
0~0sln <~+ (0).o- (1) >
( e x p ( i k . X ) (u, a) e x p ( - i k . X ) (v, 0)>
='(M~-(0)My(I)>.
1 -lu_vl2k~,
(29)
(24) 3. Duality
one has k2(-) e
8 ( u - v) ~ - e + ~ 2 ( a - 0) k2
×
(25)
Integrating with respect to a and tT, letting k2--,0 and setting u = 0 and v = 1, one has .%j= ~ ( S , + (0)$5- ( 1 ) > .
(26)
The integration constant is set to zero by conformal invariance. The equivalence of this to the Zamolodchikov formula for 3 = 3 can be seen by using the result of refs. [29,15] that the ( 1, 1 ) operators S a r e string vertex operators for auxiliary fields which lie in the same spacetime supcrmultiplet as the moduli fields. Spacetime supersymmetry then requires that their metric equals that of the moduli fields. More explicitly, S may be written as two contour integrals of the spacetime supercurrent around M and by deforming the
The metric (29) is positive-definite, in apparent contradiction to the sigma model result, eq. (6), which leads to a negative metric for the (2, 1 ) form and a positive metric for the ( 1, 1 ) forms. The resolution of this apparent sign discrepancy is quite interesting. The fact that the conformal field theory calculation could not possibly reproduce the sigma model result can be seen without detailed calculation. The conformal field theory' analysis was valid for a general (0, 2 ) model, and thus used only right-moving supersymmerry. (2, 1) and (1, 1) fields can be distinguished only in (2, 2) theories. They differ in the relative signs of the left and right charges. Our calculation was blind to the left charge, so the metric for all the moduli had to have the same sign. Even in a (2, 2) theory the labclling of (2, 1 ) and ( 1, 1 ) fields is a matter of convention. Thcre is a Z2 duality symmetry which interchanges them [ 30 ] :
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/-I~* - H ,
PHYSICS LETTERS B
(30)
where H is the right-moving U ( 1 ) boson. The sigma model calculation was able to give different signs for (2, 1 ) and ( 1, 1 ) fields because the prescription for finite renormalizations o f (2~ under motion in ,,# broke the s y m m e t r y (30). In particular, the c o m p o n e n t s .Q,..~ are renormalized under perturbations generated by (2, 1 ) fields but not under those generated by ( 1, 1 ) fields. Our definition of,Q + as a function on ,Jr' thus does not correspond exactly to the o p e r a t o r g2+ in the weak-coupling limit. ,Q+, in fact, reduces to a different " d u a l i z e d " operator as follows: consider the operator iete+ z)/2 ~¢~+ ~
(~
~t~'~tl"'id~l/il...~li e ,
(31)
normalized so that (*Off (0)-(2-+ ( 1 ) > z = 1 ,
sentation, the target space is a different manifold
X~,X,
In (-Q+ (0)-Qg ( 1 ) ) z I x
= - I n <.f2~+ (O).Qg ( 1 ) > z l . x ,
~2+ ~* *g2f .
This might have been expected since G+~/2, which corresponds to O, is interchanged with G : ~/2, which corresponds to *0-. Thus cohomology is dualized,
This appears to be consistent with (38).
(33)
This definition does not involve a metric. It is easy to see that (*0-)2=0, so that cohomo]ogy classes may be defined. * ~ + is an element of such a eohomology class, and (32) depends only on the respective cohomology classes of*~2~- and ~ + . Furthermore, the K~ihler potential
(34)
leads to m i n u s the metric o f (6). There are two more operators that can be defined in this m a n n e r by separately dualizing on either the ,~//2.~ or .,16,~ pieces o f these moduli spaces. One o f these will lead to a positive-definite metric on ,/{=J[2a ×.~/6.~. This is the o p e r a t o r which coincides with Q+ in the weak-coupling limit. The reason such dualized operators arise can possibly be understood as follows. U n d e r the duality transformation (30) the (2; 1 ) forms, or elements o f II ~( T ) interchange with the ( 1, 1 ) forms H ~( T * ) :
II~ ( T ) ~ t l t
(T *) .
(35)
If the dualized field theory has a sigma model repre266
(38)
(32)
(*0- 7)
* . ~ = In < .~2~- (0)*£22 ( 1 ) > z = - ~
(37)
where the left-hand side is defined with respect to sigma model perturbation theory, on X a n d the righthand side is defined with respect to sigma model perturbation theory, on *X. C o m p a r i n g with (34), one is led to the conclusion that
O~* *O*.
T [ab,...l
(36)
with opposite Euler character. O f course, no explicit construction o f such a dualized manifold has been given, so this paragraph should be regarded as speculative. (35) implies that the K~ihler potential as defined in ( 6 ) changes sign under (30):
at every point on moduli space. The object *,Q" '~' is an a n t i s y m m e t r i c rank ( c o n t r a v a r i a n t tensor density o f weight 1. There is a natural dual cohomology theory defined for such objects. Define *0- by tb,.a...] = 0,,
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( 39 )
Acknowledgement We are grateful to L. Dixon and S. Ferrara for useful conversations. A.S. was supported in part by DOE grant 8-484062-2500-3 and the A.P. Sloan F o u n d a tion. This research was s u p p o r t e d in part by the National Science F o u n d a t i o n under G r a n t No. PHY8217853, s u p p l e m e n t e d by funds from the National Aeronautics and Space Administration, at the University o f California at Santa Barbara.
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[7]S. Cecotti, S. Ferrara and L. Girardello, Phys. Left. B 213 ( 1988 ) 443; Geometry oftype II super,strings and the moduli of superconformal field theories, Intern. J. Mod. Phys. 4 ( 1989 ) 2475. [8] N. Sciberg, Nucl. Phys. B 303 (1988) 286. [ 9 ] L.J. Dixon, V. Kaplunovsky and J. Louis, On effective field theories describing (2, 2) vacua of the heterotic string, SLAC-PU B-4959 (April 1989 ). [10IP. Candelas, P.S. Green and T. Hiibsch, Rolling among Calabi-Yau vacua, UTTG-10-89 (April 1989), in: Trieste Proc., eds. M. Green, A. Polyakov and A. Strominger, to appear; Connected Calabi-Yau compactifications (Other worlds are just around the corner), UTTG-28-88 (January 1989). [ I I ] S . Ferrara and A. Strominger, N = 2 space-time supersymmetry and Calabi-Yau moduli space, CERN preprint TH 5291/89 (February. 1989), in: Proc. Strings '89 Workshop (College Station, TX, March 1989), to appear. [ 12] A. Strominger, Special geometry, to appear. [13]K. Kikkawa, and M. Yamasaki, Phys. Lett. B 149 (1984) 357; N. Sakai and I. Senda, Prog. Theor. Phys. Suppl. 85 ( 1985 ) 228. [ 14 ] V.P. Nair, A. Shapere, A. Strominger and F. Wilczek, Nuel. Phys. B 287 (1987) 402. [ 15 ] M. Dine, P. Huet and N. Seiberg, Large and small radius in string theory, IAS preprint IASSNS-HEP-88/54 (January 1989). [16] S. Ferrara, D. Liist and S. Theisen, World sheet versus spectrum symmetries in heterotie and type 11 superstrings, CERN preprint TH-5341/89 (March 1989). [ 17 ] S. Ferrara, D. Liist, A. Shapere and S. Theisen, Phys. Lett. B 225 (1989) 363.
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[ 18 ] C. Vafa, Quantum symmetries of string vacua, preprint HUTP-89 / A021 (May 1989 ) ; String vacua and orbifoldizcd L-G models, preprint HUTP-89/A018 (April 1989). [ 19] B. de Wit, P.G. Lauwers and A. Van Proeyen, Nucl. Phys. B 255 (1985) 569. [20] A. Strominger and E. Witten, Commun. Math. Phys. 101 (1985) 341. [21 ] A. Strominger, Phys. Rev. Lctt. 55 ( 1985 ) 2547. [ 22 ] Examples are worked out by A. Strominger, in: Unified string theories, eds. M.B. Green and D.J. Gross (World Scientific, Singapore, 1986 ); P. Candelas, Nucl. Phys. B 298 (1988) 458. [231 M. Dine and N. Seiberg, Phys. Rev. Lett. 57 (1986) 2625; Nucl. Phys. B 301 (1988) 357; B 306 (1988) 137. [ 24 ] J. Distler and B. Greene, Nucl. Phys. B 304 (1988) 1; B 309 (1988) 295. [25] J.L Cardy, Nucl. Phys. B 270 [FS16] (1986) 186; A.A. Belavin and V.G. Knizhnik, Sov. Phys. JETP 64 (1986) 214; Phys. Left. B 168 (1986) 201; D.H. Friedan and S.H. Shenker, Nucl. Phys. B 281 (1987) 509. [26] A.B. Zamolodchikov, JETP Left. 43 (1986) 730. [ 27 ] R. Bryant and P. Griffiths, in: Arithmetic and geometry, Vol. II, eds. M. Artin and J. Tate (Birkh/iuser, Basel, 1983) p. 77. [28 ] G. Tian, in: Mathematical aspects of string theory, ed. S.-T. Yau (World Scientific, Singapore, 1987) p. 629. [29 ] J.J. Atick, L.J. Dixon and A. Sen, Nucl. Phys. B 292 ( 1987 ) 109. [ 30] L.J. Dixon, preprint PUPT-1074 (October 1987 ); lectures given at the 1987 ICTP Summer Workshop in High energy physics and cosmology (Trieste, Italy, June-August 1987).
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