View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server DUK-TH-94-68 IASSNS-HEP-94/23 hep-th/9404151 April, 1994 String Theory on K3 Surfaces Paul S. Aspinwall, Scho ol of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 David R. Morrison Department of Mathematics, Box 90320, Duke University, Durham, NC 27708-032 0 ABSTRACT The mo duli space of N =(4,4) string theories with a K3 target space is determined, establishing in particular that the discrete symmetry group is the full integral orthog- onal group of an even unimo dular lattice of signature (4,20). The metho d combines an analysis of the classical theory of K3 mo duli spaces with mirror symmetry. A descrip- tion of the mo duli space is also presented from the viewp oint of quantum geometry, and consequences are drawn concerning mirror symmetry for algebraic K3 surfaces. 1 Intro duction In recentyears, Calabi-Yau manifolds have received great attention in the string literature. This is mainly b ecause compacti cation on such spaces may b e used to reduce the number of dimensions in mo dels built from the intrinsically ten dimensional critical sup erstring [1]. The fo cus has largely b een on the case of complex dimension d = 3, since the corresp onding compacti cation is to four space-time dimensions. (It is also the case that, in many resp ects, within top ological eld theory on Calabi-Yau target spaces the case d = 3 plays a sp ecial r^ole [2, 3].) However, one might exp ect interesting prop erties for other values of d as well. In this pap er we analyze the case d = 2 whose sp ecial features derive from the fact that the top ology of the target must b e either a torus or a K3 surface. The classi cation of toroidal target spaces has b een known for some time [4, 5]. The lo cal geometry of the mo duli space of conformal eld theories with K3 target space top ology was understo o d shortly afterwards [6] and although some asp ects of the global form of the mo duli space have b een conjectured and studied [6, 7, 8 ], more precise statements concerning this question have remained elusive. We will use mirror symmetry to address this question of the form of the mo duli space. We nd that, given a couple of minor assumptions, combining a -mo del analysis near the eld theory limit with the study of a mirror symmetry transformation is sucient to give the precise form of the mo duli space. This pap er is not intended to provide the full mathematical exp osition of the analysis required | for that the reader is referred to [9]. Here we only give a brief summary of the metho ds used and conclusions reached by the analysis. The analysis of the mo duli space of string theories on a K3 surface di ers markedly from that of the more familiar Calabi-Yau mo duli spaces related to string compacti cations to four dimensions. Firstly, in studying the case where the target space X is a Calabi-Yau 2;0 threefold, the assumption h = 0 is usually made. Under this assumption, the deformations of complex structure of X and of the (complexi ed) Kahler form \decouple" to give the complete mo duli space a lo cal pro duct structure [10] (at least over generic p oints in the mo duli space). These twotyp es of deformations may then b e studied indep endently.For a 2;0 K3 surface wehaveh = 1 and the ab ove structure is lost. We are thus forced to analyze deformations of complex structure and deformations of Kahler form together. In another resp ect however the K3 case is simpler | the N =(2,2) sup erconformal invari- ance of a string with Calabi-Yau target space is extended to N =(4,4) sup ersymmetry in the case d = 2 [11]. This is equivalent to the geometric statement that a Calabi-Yau manifold in complex dimension two admits a hyp erkahler metric. This N =(4,4) structure serves to x the lo cal form of the mo duli space completely. This can b e contrasted with the case d =3 where the extended chiral algebra [12] has yet to provide much insightinto the classi cation of string theories. As in the case d = 3, the lo cal dimension of the mo duli space of conformal eld theories 1 can b e matched to the dimension of the mo duli space of the geometrical ob jects representing the target space if we regard the \geometrical ob ject" as including, in addition to the target space metric, the sp eci cation of a B - eld, i.e., an element of the second real (de Rham) cohomology group of the target. In the case d = 3, the B - eld naturally combined with the Kahler form to provide a complexi ed Kahler form. In the case of K3 this is clearly imp ossible | B lives in a 22-dimensional space whereas the dimension of the space of Kahler forms varies with the complex structure but is at most 20. Despite this fact we will still nd that a b eautiful structure arises when the B - eld is included in the mo duli space. At the top ological level, mirror symmetry might app ear rather trivial for a K3 surface since the mirror of a K3 surface (equipp ed with metric and B - eld) is another suchK3 surface. However, the geometric data of the mirror K3 surface thus obtained is not, in general, isomorphic to the original. Thus the mirror map acts as a non-trivial automorphism on the \classical" mo duli space of K3 surfaces (i.e., the -mo del mo duli space of metrics and B - elds). The mirror map in this context was rst studied long ago in [13] and some of the observations in that pap er lead to some of the metho ds used here. Although the purp ose of this pap er is to concentrate on the case where the target space is a K3 surface, it should b e noted that much of what follows applies to any target space 2;0 with h = 1. The only steps in the following argument which will not b e directly applicable to the determination of the mo duli space of this more general case are the explicit form of the mo duli space of Einstein metrics, and the particular self-mirror K3 theory used. In section 2 we will present the outline of the construction of the mo duli space of N =(4,4) and N =(2,2) theories. In section 3 we discuss the interpretation of the mo duli space in terms of the space of total cohomology of K3. In section 4, some asp ects of mirror symmetry on algebraic K3 surfaces are discussed. 2 The Mo duli Space Let us rst x some notations. nG=H denotes the double coset space resulting from dividing the group G by the right-action of H and left-action of . (G and H will always b e continuous a;b Lie groups and will b e discrete.) denotes the unique (up to isomorphism) even self- dual lattice in (a + b)-dimensional space with signature (a; b), when ab 6= 0. Basis vectors may b e taken such that the inner pro duct has the following form: a;b : h; i (E ) (E ) ::: H H :::; (1) = 8 8 | {z } n times 2 where b a =8n (one uses E in place of E when b a<0), E is the Cartan matrix of 8 8 8 the Lie algebra E , and H is the hyp erb olic plane: 8 ! 0 1 H = : (2) 1 0 We will use X to denote a sp eci c smo oth K3 surface. One can show that the intersection 2 3;19 pairing on H (X; Z) gives it the structure of a lattice isomorphic to (see for example a;b [14]). Let R bea(a+b)-dimensional space with an inner pro duct of signature (a; b). a;b a;b a;b a;b O (a; b) is the orthogonal group on R and O ( ) is the subgroup preserving R . Let a;b T = O (a; b)= ( O (a) O (b)) : (3) a;b a;b We can identify T as the set of space-like a-planes in R (i.e., a-planes on which the a;b inner pro duct is p ositive-de nite). T can also b e regarded as one of the two connected comp onents of the set of oriented space-like a-planes. Under this latter interpretation, we can write a;b + T = O (a; b)= (SO(a) O(b)) ; (4) + where O (a; b) is the index 2 subgroup of O (a; b) with the same connected comp onents as SO(a) O(b). It was shown in [6] using arguments from sup ergravity [15] that the required mo duli space, 4;20 M, of conformal eld theories on a K3 surface is lo cally of the form T . This is probably b est understo o d from the argument presented in [16], whichwenow review. The N =(4,4) sup erconformal algebra contains ane SU (2) algebras in b oth the left and right sectors. This symmetry acts on the marginal op erators spanning the tangent spaces of M. The existence of such a symmetry restricts the form of the holonomy group of the Zamolo dchikov metric of M.