String Theory on K3 Surfaces

Home , K3 surface, Orbifold

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:

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

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

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. The restriction is so severe, in fact, that it then follows from Berger's classi cation [17]

of holonomy groups that either M is lo cally isomorphic to a quaternionic symmetric space,

or that the holonomy is reducible. Moreover, the non- atness of the Zamolo dchikov metric

is enough to rule out the reducible holonomy case. A simple analysis of any conformal eld

theory giving rise to a K3 target space tells us that M has real dimension 80 [6, 11 ]. This,

together with the known classi cation of quaternionic symmetric spaces (cf. [18]), completes

the pro of.

One way to do this count is as follows: the space of complex structures has complex dimension 20 (real

dimension 40), the space of Kahler forms has real dimension 20, and the space of B - elds has real dimension

22. However, for each Ricci- at metric on the K3 surface there is an S of complex structures, so the overall

dimension count is40+20+22 2 = 80. 3

Let us now maketwo assumptions ab out the form of M, b oth of whichwe consider to b e

quite reasonable. Firstly we assume M is geo desically complete and secondly we assume M

to b e Hausdor . While some non-Hausdor mo duli spaces have app eared in string theory

[19], this happ ened in the context of a target space of inde nite signature. There is no reason

to b elieve that something as unpleasant as this should happ en in the case of a K3 surface.

It then follows that

4;20

M nT ; (5)

4;20

for some group acting discretely on T .Thus to complete the description of M,we only

need to nd .

We b egin our determination of by analyzing the classical form of the mo duli space of

4;20

K3 surfaces. Using the techniques of [20 ], one may decomp ose T as

4;20 3;19 3;19

T R T R ; (6)

where R is the half-line of p ositive real numb ers whichwe parameterize by . The decom-

4;20

p osition dep ends on a choice of null vector v 2 R , and the space v =Rv provides the

3;19

3;19 3;19

R on which O (3; 19) acts and on which T is based. The space T is known to b e

isomorphic to the Teichmuller space for Einstein metrics of volume one on a K3 surface (and

their orbifold limits) [21 , 22 , 23 ]. It is natural then to identify as giving the size of the K3

3;19

surface and R as the mo duli space of B - elds on the target space. These identi cations

may b e established by lo oking at the metrics on the ab ove mentioned spaces. The Zamolo d-

4;20

chikov metric on T is known to b e the left-invariant metric. From [24] this induces a

metric on each of the terms on the right hand of (6) where this should now b e viewed as

a\warp ed pro duct", i.e., the metric do es not resp ect the pro duct structure. One maynow

show (using [23]) that the metrics induced are those given precisely by the extended Weyl-

Petersson metric in the sense of [25]. Such an isometric identi cation allows us to identify

4;20

every p ointinT with an Einstein metric on a K3 surface (or orbifold metric) together

with a B - eld. This can b e taken as another version of the statement that the non-linear

-mo del is exactly conformally invariant on a K3 surface with Ricci- at metric [26, 27 , 28 ].

It also shows that the Zamolo dchikov metric and Weyl-Petersson metric coincide exactly on

the mo duli space | a fact which in general N =(2,2) theories holds only to leading order in

the large radius limit [29 ].

The mo duli space of smo oth Einstein metrics of volume one on a K3 surface X is deter-

3;19

mined in [23] to b e nT Z, where = Di (X )= Di (X ) is the group of comp onents

0 0

of the di eomorphism group of X , and Z is the space of orbifold metrics. (It is generally

b elieved that such orbifolds should b e included when considering a string target space [30 ];

As we will p oint out b elow, wemust include orbifold p oints in our mo duli space in order to ensure this

geo desic completeness. 4

3;19

if we include them, we get the geo desically complete space nT .) The discrete group

0 0

is determined in [31, 32 , 33 ] to coincide with the group

+ 3;19 3;19 +

O ( ):= O( ) \ O (3; 19); (7)

3;19 3;19

which has index 2 in O ( ). The \missing" Z in O ( )may b e generated by I acting

3;19

on R .

We also know that the non-linear -mo del on a Calabi-Yau manifold is invariant under

3;19

all translations B ! B + v , where v 2 H (X; Z). Thus we should divide the space R of

3;19

B - elds by additive translations by . (As an abstract group, these additive translations

simply form a Z .) Furthermore one may consider complex conjugation of the target space.

If one considers such a complex conjugation for a -mo del one sees that the transformation

B !B is required in addition to the conjugation of the complex structure of the target.

Such a transformation may b e represented by I and thus generates the missing Z from

3;19 3;19

ab ove. Therefore, in terms of the decomp osition (6) one sees a group O ( ) n (which

3;19

is the full space-group of ) of identi cations that should b e made on the right-hand side

4;20

and thus on T .Thus we obtain

3;19 3;19

O ( ) n : (8)

This is the maximal set of identi cations that can b e made from classical geometry.Any

further statement requires some quantum geometry. The mirror construction of [34] provides

us with such a to ol. First we need the clarify the meaning of the mirror map in the context

of N =(4,4) theories. Mirror symmetry reverses the sign of a U (1) charge derived from an

N =2 chiral algebra. Nowan N=4 chiral algebra contains an SU (2) ane subalgebra, which

is larger than the U (1) ane subalgebra of an N =2 theory. Given an N =4 theory, though,

wemaycho ose an N =2 subalgebra of the N =4 algebra by sp ecifying the corresp onding

SU (2)=U (1) of N =2 theories. U (1) SU (2). Each N =4 theory thus gives rise to an S

This leads to construction of the mo duli space of N =(2,2) theories of string on a K3, which

we denote by M , as a bre bundle

q q

: M !M; (9)

q q

2 2

with bre S S . Each p ointin M corresp onds to an N =(4,4) theory on whicha

particular N =(2,2) structure has b een chosen; the subscripts q andq denote charges with

resp ect to the left and right U (1) currents resp ectively within the N =(2,2) theory.

4;20

A (left) mirror map on the corresp onding Teichmuller space T of N =(2,2) theories is

q q

4;20 4;20 4;20

a map : T !T with the prop erty that, for all S 2T , the N =(2,2) theories at S

q q q q q q

and (S ) are isomorphic, but with a switchofcharge assignments (q; q) $ (q; q). We also 5 µ 2 µ 1

4;20

q q

-q

4;20 T

∈G

4;20

2 2 4;20

Figure 1: Trivial and nontrivial mirror maps. T is an (S S )-bundle over T as

q q

shown. A theory (shown as an op en circle) is asso ciated with a pair of p oints, one on each

sphere. In this gure is trivial and is nontrivial.

1 2

4;20

de ne right mirror maps which switch(q; q) $ (q; q). Such a map on T can b e pushed

q q

4;20

down to a map on the Teichmuller space T | since two theories which are mirrors as

4;20

N =(2,2) theories are isomorphic as N =(4,4) theories, any mirror map within T should

4;20

give rise to an element of . Note that some mirror maps on T may b e trivial on T ,

q q

in the sense that they induce the identity element of . Such maps relate pairs of p oints in

4;20

T which lie over the same p oint in the base space T . What will b e of more interest are

q q

the nontrivial mirror maps which map to a nontrivial element of . The di erence b etween

trivial and nontrivial maps is shown schematically in gure 1.

Wenow consider an example of a generically nontrivial mirror map with a xed p ointon

4;20

T . Given the Gepner mo del [35] asso ciated via [36 ] to the K3 hyp ersurface

2 3 7 42

X + X + X + X =0 (10)

0 1 2 3

we nd the mirror as an orbifold of the original in the weighted pro jective space P

f21;14;6;1g

space by the metho d of [34]. The orbifolding group thus found is trivial and so this theory

is its own mirror. The fact that this theory is self-mirror is also related to some issues

regarding Arnold's \strange duality" observed in [13]. This mirror map acts non-trivially on

4;20

the marginal op erators of this theory and thus on the tangent bundle of T . By analyzing

4;20

this action, it is p ossible to show that there is a lattice R such that the induced 6

4;20 4;20

automorphism of T lies in O ( ). In fact, there is a decomp osition

4;20 2;10 2;10

; (11)

such that acts by simply exchanging the two terms on the right-hand side. The decomp o-

2;10 3;19

sition can b e chosen so that one of the lattices in (11) is a sublattice of the lattice

3;19

on which the classical O ( ) symmetry of (8) acts.

Wehavethus found the explicit form of another generator of . A straight-forward but

3;19 3;19

somewhat involved calculation then shows that the classical symmetries, O ( ) n ,

4;20

and the ab ove Z mirror map generate the whole of O ( ). That is, wehave

4;20

O( ): (12)

4;20

But now from [37], which establishes that O ( )isa maximal discrete group acting on

4;20

T to yield a Hausdor quotient, it follows that

4;20

O ( ); (13)

completing the pro of that the mo duli space has the precise form

4;20

M = O ( )nO (4; 20)= ( O (4) O (20)) (14)

(as had b een sp eculated by Seib erg [6]).

To summarize, wehave shown that the full group of identi cations to b e made on the

Teichmuller space is generated by the following:

1. Classical identi cations made for the mo duli space of complex structures.

2. The integral B - eld shifts.

3. Complex conjugation (together with a change in sign of the B - eld).

4. The mirror map.

One should note that this is also true for the case d = 1 where the target space must b e a

torus [38].

4;20

It is interesting to note that this mo duli space M nT is precisely that which one

would obtain for toroidal compacti cations of the heterotic string down to six-dimensional

spacetime using the metho ds of [4]. This equivalence b etween the mo duli space of K3

compacti cations and toroidal compacti cations (for the relevant strings) was established

lo cally in [6] but it is unclear why they turn out to b e globally isomorphic. 7

3 The Space of Total Cohomology

The derivation of the mo duli space M in the previous section and [9] is somewhat unpleasant

and yet pro duces a b eautifully symmetric result. This is a consequence of having derived

the mo duli space using the ideas of classical geometry necessitating the intermediate step of

equation (8). Now that wehave the mo duli space wemay reinterpret it from the standp oint

of quantum geometry in a more symmetricway.

First let us quickly review the form of the mo duli space of complex structures on a

classical K3 surface. Consider a non-vanishing holomorphic 2-form on X representing

2;0

an elementof H (X). Let us sp ecify a \marking" of X , i.e., a basis , i =1;:::;22 of

3;19

H (X; Z) determining an isomorphism of that lattice with . The global Torelli theorem

[39, 40 , 21] essentially says that the complex structure of X is expressed uniquely in terms

of the p erio ds

; (15)

up to a change of basis of H (X; Z). To state this more precisely,we write = + i , where

3;19 2

R. The Ho dge-Riemann bilinear relations assert that ; 2 H (X; R)

h ; i = ^ =0

(16)

h ; i= dV k k > 0:

This implies that the oriented 2-plane ^ in H (X; R) spanned by and is space-

like, i.e., has p ositive de nite metric. In fact, sp ecifying up to multiplication by C

(which determines the Ho dge structure) is equivalent to sp ecifying the oriented 2-plane

3;19

R is given by ^ .Now the mo duli space of oriented space-like 2-planes in H (X; R)

O (3; 19)= ( SO(2) O (1; 19)) . When we mo d out by di eomorphisms, wewould exp ect to

obtain a description of the mo duli space of complex structures on a K3 surface as an op en

subset of

+ 3;19

O ( )nO (3; 19)= ( SO(2) O (1; 19)) (17)

(and wewould exp ect to get the entire space (17) if we include orbifold complex structures).

Actually, there are some technical diculties in interpreting (17) as a mo duli space, but the

interpretation is essentially correct (see [21 ] for more precise statements).

4;20

The extension of this to string theory follows immediately. The space T is one con-

4;20 4;20

nected comp onent of the set of oriented space-like 4-planes in R . The space R can

b e identi ed with the space H (X; R) of all real cohomology groups on X | the inner

pro duct is obtained by supplementing the intersection pairing on H (X; R)by a pairing on 8

0 4

H (X; R) H (X; R) which has matrix (2) on the standard generators of those two spaces

(i.e., the class of a p oint, and the class of the entire space X ). The natural lattice H (X; Z)

4;20

inside of H (X; R) R inherits an integer-valued pairing from this inner pro duct, giving

4;20

it the structure . An N =(4,4) theory on a K3 surface is thus sp eci ed uniquely by

an oriented space-like 4-plane in H (X; R) (b elonging to the correct comp onent) where the

4-plane is lo cated relative to the H (X; Z) lattice.

This picture b ecomes clearer when wegototheN=(2,2) mo duli space. Now the mo duli

2 2

space, M ,isan(S S )-bundle over M. In fact one may show that the Teichmuller

q q

space takes the form

4;20 +

T = O (4; 20)= ( SO(2) SO(2) O (20)) : (18)

q q

In the language of the previous paragraph, this is one comp onent of the space of orthogonal

4;20

pairs of oriented space-like 2-planes in R . That is, we not only sp ecify an oriented space-

like 4-plane but we giveitaninternal structure of b eing spanned bytwo orthogonal oriented

2-planes. Wemaynow sp ecify these oriented 2-planes in terms of complex vectors as in the

classical case. Namely, de ne and 0 as arbitrary elements of H (X; C ) (mo d C ) such

that

h ; i = h ; 0i = h0; 0i = h ; 0i =0;

(19)

h ; i >0; h0;0i >0:

The space of suchvectors gives precisely the Teichmuller space required. Thus wemay

identify the mo duli space of N =(2,2) theories on a K3 surface as b eing the space of all

p ossible choices of and 0 sub ject to (19) where the p ositions of these vectors are lo cated

relativeto H (X; Z). This statementmay b e considered to b e the quantum version of the

global Torelli theorem.

4;20

As explained earlier, mirror maps will act on the Teichmuller space T . In fact a

q q

nontrivial mirror map may b e identi ed with the interchange $ 0. Our picture now

b egins to lo ok very similar to that pursued in [10 ] for the case d =3. There is a slight

di erence since in the latter case one had

3;0 2;1 1;2 0;3

2 H H H H ;

(20)

0;0 1;1 2;2 3;3

0 2 H H H H :

For the K3 case however, wemay only sp ecify the general statement that b oth and 0 lie in

the full cohomology H at a generic p oint in mo duli space. The following construction maybe

4;20

used to reconcile the approaches. Let us cho ose twonull vectors v and w in H (X; Z)

such that hv; wi = 1, so that v and w span a hyp erb olic plane. Let F b e the subspace of 9

H (X; R) p erp endicular to v and w .For each oriented space-like 4-plane , we de ne

:= complex vector asso ciated to \ F;

(21)

0 := complex vector asso ciated to \ :

This amounts to cho osing a sp eci c N =(2,2) theory to representan N=(4,4) theory.We

2 2;0

maynow identify F with H (X; R). If sp eci es the H direction, (19) then implies

2;0 1;1 0;2

2 H H H ;

(22)

0;0 1;1 2;2

0 2 H H H :

The complex structure of X can thus b e determined from in the usual way. One can also

obtain the Kahler form and B - eld information from 0. The mirror map interchanges the

two.

Note that wemay equally sp ecify ! 0; 0 ! as a mirror map. Wemay comp ose

this with the previous mirror map to pro duce the map ! ; 0 ! 0. This map may

b e identi ed as the map taking (q; q) ! (q; q)intheN=(2,2) theory or equivalently

as complex conjugation of the target space. Note that such a map is usually trivial in the

mo duli space of N =(2,2) theories but that our construction, through sp eci c lab eling of the

U (1) charges, leads to a non-trivial map.

4 Mirror Symmetry for Algebraic K3 Surfaces

Now that wehave identi ed how mirror maps act on the mo duli space of conformal eld

theories on a K3 surface, we can explain several of the mirror-typ e phenomena whichhave

b een observed for algebraic K3 surfaces.

2 2

Given any subgroup M of H (X; Z) such that the cokernel H (X; Z)=M has no torsion

and the intersection form on M R has signature (1; 1), we can asso ciate the family

of \algebraic K3 surfaces of typ e M " which consists of all complex structures on the K3

for which the classes on M are all of typ e (1; 1). This construction was intro duced into

the mathematics literature [41] as part of the explanation of Arnold's \strange duality" put

forward by Pinkham and Dolgachev, and mo duli spaces of these structures were studied

in detail in [42 , 43 ]. The mo duli space of algebraic K3 surfaces of typ e M has complex

dimension 20 when M R has signature (1; 1).

We can extend this idea, and de ne the set of \CFT's of typ e M " on a K3 surface to b e

the image in the conformal eld theory mo duli space of the set of -mo dels whose complex

structure is an algebraic K3 surface of typ e M , and whose Kahler class and B - eld are taken

from the space M R. The complexi ed Kahler mo duli space has complex dimension ,so 10

the entire CFT mo duli space of typ e M has complex dimension (20 )+ = 20. Since the

construction segregates complex and Kahler deformations, the CFT mo duli space of typ e M

emb eds naturally into the N =(2,2) mo duli space M .

q q

? ?

Supp ose that the orthogonal complement M can b e written in the form M = H N

for some lattice N (orthogonal direct sum). Then a mirror map can b e de ned which

0 4

exchanges the given hyp erb olic plane H with the hyp erb olic plane H (X ) H (X ). When

is applied to the CFT mo duli space of typ e M ,itreverses the r^oles of Kahler and complex

structure deformations. It is not hard to see that the resulting mo duli space is in fact the

CFT mo duli space of typ e N | the complex structure is one for which N (rather than M )

consists of (1; 1) classes, and the Kahler structure is now taken from N R. This gives a

precise CFT interpretation of Arnold's strange duality and other related phenomena for K3

surfaces (an earlier version of whichwas given by Martinec [13]).

Finally we note a link b etween our construction of the mirror map for K3 surfaces and

Voisin's construction of mirror pairs of Calabi-Yau threefolds [44] (see also [45]). The latter

construction consists of taking an orbifold Y =(X E)=Z, where E is a torus of complex

dimension one. It was shown in [44 ] that for suitable pairs X ;X of X , Y and Y are

1 2 1 2

p;q 3p;q

mirror pairs at the level of the Ho dge numb ers, H (Y )=H (Y ); and \A-mo del" and

1 2

\B-mo del" [46 ] correlation functions. The K3 surfaces involved must b e of typ e M , with

M the lattice invariant under the Z -action. One may show that our mirror map for K3

surfaces directly establishes X and X as a mirror pair and thus it follows that Y and Y

1 2 1 2

are truly mirror manifolds at the level of conformal eld theory.

Acknowledgements

The work of P.S.A. was supp orted by NSF grant PHYS92-45317, and the work of D.R.M.

was supp orted by an American Mathematical So ciety Centennial Fellowship.

References

1. P. Candelas, G. Horowitz, A. Strominger, and E.Witten, Vacuum Con guration for Super-

strings, Nucl. Phys. B258 (1985) 46{74.

2. E. Witten, On the Structure of the Topological Phase of Two Dimensional Gravity, Nucl.

Phys. B340 (1990) 281{332.

3. M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa, Kodaira-Spencer Theory of Gravity and

Exact Results for Quantum String Amplitudes, Harvard et al 1993 preprint HUTP-93/A025,

hep-th/9309140. 11

4. K. Narain, New Heterotic String Theories in Uncompacti ed Dimensions < 10,Phys. Lett.

169B (1986) 41{46.

5. K. Narain, M. Samadi, and E. Witten, A Note on the Toroidal Compacti cation of Heterotic

String Theory, Nucl. Phys. B279 (1987) 369{379.

6. N. Seib erg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys.

303 (1988) 286{304.

7. C. Vafa, Quantum Symmetries of String Vacua,Mod.Phys. Lett. A4 (1989) 1615{1626.

8. A. Giveon and D.-J. Smit, Symmetries of the Moduli Space of (2,2) Superstring Vacua, Nucl.

Phys. B349 (1991) 168{206.

9. P. S. Aspinwall and D. R. Morrison, Mirror Symmetry and the Moduli Space of K3 Surfaces,

to app ear.

10. P. Candelas and X. de la Ossa, Moduli Space of Calabi-Yau Manifolds, Nucl. Phys. B355

(1991) 455{481.

11. T. Eguchi, H. Ooguri, A. Taormina, and S.-K. Yang, Superconformal Algebras and String

Compacti cation on Manifolds with SU (n) Holonomy, Nucl. Phys. B315 (1989) 193{221.

12. S. Odake, Extension of N =2Superconformal Algebra and Calabi-Yau Compacti cation,Mod.

Phys. Lett. A4 (1989) 557{568.

13. E. Martinec, Criticality, Catastrophes and Compacti cations, in L. Brink et al, editor, \Physics

and Mathematics of Strings", World Scienti c, 1990.

14. W. Barth, C. Peters, and A. van de Ven, Compact Complex Surfaces, Springer, 1984.

15. M. de Ro o, Matter Coupling to N =4 Supergravity, Nucl. Phys. B255 (1985) 515{531.

16. S. Cecotti, N =2Landau-Ginzburg vs. Calabi-Yau Sigma Model: Nonperturbative Aspects,

Int. J. Mo d. Phys. A6 (1991) 1749{1814.

17. M. Berger, Sur les groupes d'holonomie homogene des varietes aconnexion ane et des

varietes Riemanniennes, Bull. So c. Math. France 83 (1955) 279{330.

18. J. Wolf, Complex Homogeneous Contact Manifolds and Quaternionic Symmetric Spaces,J.

Math. Mech. 14 (1965) 1033{1047.

19. G. Mo ore, Finite in Al l Directions,Yale 1993 preprint YCTP-P12-93, hep-th/9305139.

20. A. Borel and J.-P. Serre, Corners and Arithmetic Groups, Comment. Math. Helv. 48 (1973)

436{491. 12

21. D. R. Morrison, Some Remarks on the Moduli of K3 Surfaces, in K. Ueno, editor, \Classi -

cation of Algebraic and Analytic Manifolds", volume 39 of Progress in Math., pages 303{332,

Birkhauser, 1983.

22. R. Kobayashi and A. N. To dorov, Polarized Period Map for Generalized K3 Surfaces and the

Moduli of Einstein Metrics,T^ohoku Math. J. (2) 39 (1987) 341{363.

23. M. T. Anderson, The L StructureofModuli Spaces of Einstein Metrics on 4-Manifolds,

Geom. and Funct. Analysis 2 (1992) 29{89.

24. A. Borel, Stable Real Cohomology of Arithmetic Groups, Ann. Sci. Ecole Norm. Sup. (4) 7

(1974) 235{272.

25. P. Candelas, P. Green, and T. Hubsc h, Finite Distancebetween Distinct Calabi-Yau Manifolds,

Phys. Rev. Lett. 62 (1989) 1956{1959.

26. L. Alvarez-Gaume and P. Ginsparg, Finiteness of Ricci- at Supersymmetric Nonlinear -

Models, Commun. Math. Phys. 102 (1985) 311{326.

27. C. Hull, Ultraviolet Finiteness of Supersymmetric Nonlinear Sigma Models, Nucl. Phys. B260

(1985) 182{202.

28. T. Banks and N. Seib erg, Nonperturbative In nities, Nucl. Phys. B273 (1986) 157{164.

29. P. Candelas, T. Hubsc h, and R. Schimmrigk, Relation between the Weil-Peterson and Zamolod-

chikov Metrics, Nucl. Phys. B329 (1990) 583{590.

30. L. Dixon, J. Harvey,C.Vafa, and E. Witten, Strings on Orbifolds, Nucl. Phys. B261 (1985)

678{686, and B274 (1986) 285{314.

31. T. Matumoto, On Di eomorphisms of a K3 Surface, in M. Nagata et al, editor, \Algebraic and

Top ological Theories | to the memory of Dr. Takehiko Miyaka", pages 616{621, Kinokuniya,

Tokyo, 1985.

32. C. Borcea, Di eomorphisms of a K3 Surface, Math. Ann. 275 (1986) 1{4.

33. S. K. Donaldson, Polynomial Invariants for Smooth Four-Manifolds, Top ology 29 (1990)

257{315.

34. B. Greene and M. Plesser, Duality in Calabi-Yau Moduli Space, Nucl. Phys. B338 (1990)

15{37.

35. D. Gepner, Exactly Solvable String Compacti cations on Manifolds of SU (N ) Holonomy,

Phys. Lett. 199B (1987) 380{388. 13

36. B. Greene, C. Vafa, and N. Warner, Calabi-Yau Manifolds and Renormalization Group Flows,

Nucl. Phys. B324 (1989) 371{390.

37. N. Allan, Maximality of Some Arithmetic Groups; A Note of the Arithmetic of the Orthogonal

Group, An. Acad. Brasil. Ci. 38 (1966) 223{227 and 243{244.

38. R. Dijkgraaf, E. Verlinde, and H. Verlinde, On Moduli Spaces of Conformal Field Theories with

c 1, in P. DiVecchia and J. Peterson, editors, \Persp ectives in String Theory", Cop enhagen,

1987, World Scienti c.

39. I. Piateckii-Shapiro and I. R. Shafarevich, ATorel li Theorem for Algebraic Surfaces of Type

K3, Math. USSR Izvetija 5 (1971) 547{587, translation from Russian original: Izv. Akad.

Nauk SSSR 35 (1971) 530{572.

40. D. Burns, Jr. and M. Rap op ort, On the Torel li Problem for Kahlerian K3 Surfaces, Ann. Sci.

Ecole Norm. Sup. (4) 8 (1975) 235{274.

41. H. Pinkham, Singularites exceptionnel les, la dualiteetrange d'Arnold et les surfaces K-3;

Groupe de monodromie des singularites unimodulaires exceptionnel les, C. R. Acad Sci. Paris

Ser. A 284 (1977) 615{618 and 1515{1518.

42. D. R. Morrison, On the Moduli of Todorov Surfaces, in H. Hijikata, editor, \Algebraic Geom-

etry and Commutative Algebra: in Honor of Masayoshi Nagata", pages 315{355, Academic

Press, 1988.

43. D. R. Morrison and M.-H. Saito, Cremona Transformations and Degrees of Period Maps for

K3 Surfaces with Ordinary Double Point, in T. Oda, editor, \Algebraic Geometry, Sendai

1985", volume 10 of Advanced Studies in Pure Math., pages 477{513, North-Holland, 1987.

44. C. Voisin, Miroirs et involutions sur les surfaces K3, in A. Beauville et al, editor,

\Journees de Geometrie Algebrique d'Orsay", volume 218 of Asterisque, pages 273{323, So ciete

Mathematique de France, 1993.

45. C. Borcea, K3 Surfaces with Involution and Mirror Pairs of Calabi-Yau Manifolds, Rider

College 1993 preprint.

46. E. Witten, Mirror Manifolds and Topological Field Theory, in S.-T. Yau, editor, \Essays on

Mirror Manifolds", International Press, 1992. 14