25 July 1996

PHYSICS LEl-l-ERS Ei

Physics Letters B 38 1 ( 1996) 427-436

Heterotic gauge structure of type II K3 fibrations Bruce Hunt a*1, Rolf Schimmrigk bs,2

a Fachbereich Mathematik, Universitdt, Postfach 3049, 67653 Kaiserslautem, Germany b Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

c Physikalisches Institut, Vniversitdt Bonn, Nussallee 12. 53115 Bonn, Germany

Received 25 March 1996 Editor: P.V. Landshoff

Abstract

We show that certain classes of K3 fibered Calabi-Yau manifolds derive from orbifolds of global products of K3 surfaces and particular types of curves. This observation explains why the gauge groups of the heterotic duals are determined by the structure of a single K3 surface and provides the dual heterotic picture of conifold transitions between K3 fibrations. Abstracting our construction from the special case of K3 hypersurfaces to general K3 manifolds with an appropriate automorphism, we show how to construct Calabi-Yau threefold duals for heterotic theories with arbitrary perturba& gauge groups. This generalization reveals that the previous limit on the Euler number of Calabi-Yau manifolds is an artifact of the restriction to the framework of hypersurfaces.

1. Introduction

It has been recognized recently that the agreement found in [l-3] between the perturbative structure of the prepotentials for a number of heterotic string K3 XT* vacua and certain type II Calabi-Yau back- grounds can be traced back to the K3-fiber structure of the models considered [4-71. Evidence based on the analysis of the weak coupling form of the pre- potential however is not convincing in the light of recent discussions [ 8-101 from which one learns that moduli spaces of different Calabi-Yau manifolds in- tersect in certain submanifolds. Thus weak coupling arguments would appear insufficient to identify het- erotic duals 3 . This makes it particularly important to

’ Email: hunt@mathematik.uni-kl.de. ’ E-mail: netah@avzwO2.physik.m&bonn.de. 3A detailed investigation of this problem will appear in Ref.

[Ill.

develop different tools for identifying heterotic and type II vacua which do not rely on a comparison of the perturbative couplings.

In the present paper we wish to describe a way to identify heterotic duals directly from the structure of the K3 fibrations and vice versa. Instead of analyz- ing the prepotentials we focus on the detailed geom- etry of K3 fibered Calabi-Yau manifolds which turns out to contain sufficient information to derive the het- erotic gauge structure. The basic observation is that the manifolds which have been encountered so far in the context of heteroticjtype II duality can in fact be described as orbifolds of product manifolds defined by a K3 surface and an appropriately defined curve. This shows that the essential information of the fibration is determined by a single K3 surface and thus provides an explanation of the fact that the gauge groups of the heterotic duals of K3 fibered Calabi-Yau spaces are determined by the singularity structure of K3 mani-

0370-2693/%/$12.00 Copynght 0 1996 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(96)00575-8

428 B. Hunt, R. Schimmrigk/Physics Letters B 381 (1996) 427-436

folds4. Our construction then identifies the heterotic gauge group of these theories as the invariant part of the Picard lattice of the K3 fiber with respect to the group action which gives rise to the fibration. We will also see that the combination of this result with the conifold transitions between K3 fibered Calabi-Yau manifolds introduced in [ 131, and the analysis of the origin of the gauge group in D = 6 theories presented in [ 121, provides complete control of the dual het- erotic picture of the conifold transition on the Calabi- Yau manifold.

We can then turn around this observation and start from abstractly defined orbifolds in which the fibers are not described by some weighted hypersurface, or complete intersection, as has been assumed in the most general class of fibrations presented so far. This will allow us to construct Calabi-Yau manifolds for arbi- trary gauge groups. It turns out that the theory is most easily understood in terms of elliptic fibrations of K3 surfaces in which the generic fiber is a torus. There are finitely many singular fibers, which were classi- fied by Kodaira in the sixties. This classification is related to the classification of the simple rational dou- blepoints, and through this to the classification of the simple Lie algebras. From our present standpoint we can see another reason for this coincidence - we will find K3 surfaces with elliptic fibrations (elliptic K3’s) which can be used to construct Calabi-Yau threefolds with K3 fibrations, which correspond dually to het- erotic strings. Since the unbroken gauge group of the heterotic string is Es x Es, the broken gauge group is a subgroup of this. The correspondence between the singular fibers of the K3 and the lattice of the gauge group of the heterotic string then dictates that the pos- sible singular fibers must also mirror the classification of the gauge group.

A further result we find along the way concerns the possible limits on the Hodge numbers of Calabi- Yau manifolds, i.e. the number of vector multiplets and hypermultiplets of the heterotic theory. Apply- ing our construction to other K3’s than the weighted hypersurfaces we find that the limit found in the

4 The relation in the context of N = 4 theories between the ADE

singularities of K3 and ADE gauge groups of the heterotic dual

on the torus has been explicated in [ 121. Using the adiabatic limit it was argued in [ 51 that this relation canies over to the N = 2

framework.

context of weighted hypersurfaces is not in fact a

characteristic of Calabi-Yau manifolds per se. In

[ 14 I weighted hypersurfaces PC r,l,r2,2s,s4) [ 841 and its mirror’ ~(11,41,42,498,1162,1743)[34861, have been constructed which define the ‘boundaries’ of the mir- ror plot of [ 141 with the largest absolute value of the Euler number, namely 960. This value has proved to be rather robust over the years, turning up in differ- ent constructions, such as various classes of exactly solvable string vacua based on minimal N = 2 [ 161 and Kazama-Suzuki models [ 171, the class of all Landau-Ginzburg theories [ IS], their abelian orb- ifolds [ 191, abelian orbifolds with torsion [20], as well as toric constructions [ 21,221. The fact that the space P( 1,1,12,28,84) [ 841 has the maximal Euler num- ber in the class of Calabi-Yau hypersurfaces can be traced back to the fact that the smooth fiber of the K3 fibration has an automorphism of order 42. Re- sults of Nikulin [23] show that this is not the highest possible value. There are in fact two higher values for the order of a K3 surface and we will discuss one example in this paper. Applying our construction to this example we will find a Calabi-Yau threefold with Euler number smaller than -960. This shows that the structure of the mirror plot of [ 141 is an artifact of the construction after all.

2. A class of K3 fibered Calabi-Yau manifolds

Before describing the geometry of general K3 fi- brations more abstractly we explain the essential in- gredients in the more familiar framework of weighted complete intersection manifolds. This will allow US to be fairly concrete. Thus consider the class of fibrations

~(2k,-1,2k,-1,2k*,2k3,2k4) [=I

P(1.1)

[

2 0

N B((2k,-1),(2k,-l),kz,k3,k4) (2kl - 1) k I ’

(1)

Withk = (%l+kz+kj+k$--1) and! := k/(2k,-1) E N, described in [ 131 6. The hypersurface configura- tion on the lhs can then be written in the form

5 It can be shown using the mirror transform of [ 15,131 that

these manifolds are indeed mirrors. 6As shown in [ 131, the assumption k/(2kl - 1) E N is merely

a matter of convenience and dropping this condition merely com-

plicates the discussion, as explicated with a concrete example. Re-

B. Hunt, R. Schimmrigk/ Physics Letters B 381 (1996) 427-436 429

M={zf+2:‘+p(22,23,24) =O}, (2)

where p( ~2, z3, zq) is a transverse polynomial. The relations ( l), which can be established via Landau- Ginzburg considerations utilizing fractional transfor- mations [ 15,131, are useful for the analysis of the Yukawa structure of heterotic/type II theories and im- mediately identify the type II mode of the heterotic dilaton.

In the following we will also be needing infor- mation about the the structure of the second coho- mology group7 . The Kithler sector of the theory re- ceives contributions from two different sources. First there is the ( 1,l )-form which comes from the restric- tion of the K%hler form of the ambient space. The remaining ( 1,l) -forms arise from the resolution of the singularities of the weighted spaces. Defining r = gcd( k2, k3, k4), one finds a fixed &-curve

Z2r : c = B(kz,w,) IIkI = (P(z2, z3,Zq) = o}, (3)

the resolution of which adds (2r - 1) further con- tributions to the K%hler sector. This curve lives in a weighted projective plane, whose resolution itself in- troduces a further number of NC ( 1,l) -forms, depend- ing on the relative divisibility properties of the weights of the curve as well as the type of its defining polyno- mial. Finally, there is the Zczk, _ 1) fixed point set. The precise structure of this set of singularities depends on the divisibility properties of (2kl - 1) relative to the rest of the weights. To be concrete we will present our discussion assuming * gcd( 2kl - 1,2ki) = 1, i = 2,3,4. In such a situation the singular set is

ZZkl-1 : IPI [ 2L] = 2& pts, (4)

the resolution of which leads to an additional 2Q kl - 1) ( 1,l )-forms. Thus we find a total of

/I(‘.‘) = 2r + NC + 2Q kl - 1) (5)

cently a heterotic/type II dual pair based on such a more general space with kr = 2, k2 = 4, k3 = 10 and Lt = 7 has been discussed in detail in [ 241.

‘The necessary ingredients for the following remarks can be found for example in [ 251. ’ In situations where this condition is not satisfied or the polyno-

mial is not of Fern-rat type in the first two variables the following formula has to be modified. This can easily be done following the discussion in 1251. For the example in footnote 6 the at, _, singular set is simply a Pt.

(l,l)-forms. The manifolds of type (2) have a fibration it4 -

Pi over the projective line whose generic fiber is an element in the K3 configuration

E p(2k,-l,kz,k&) [kl, (6)

which can be chosen to describe a quasismooth sur- face everywhere on the base P1 except at the 2e points Ai which solve ( 1 + Aze) = 0. Over these points the fibers degenerate. Important for the following will be detailed structure of these degenerations. One of the essential features of the class of manifolds ( 1) is that the complex structure of the fibers does not change as one moves in the fiber. They do change rather drasti- cally however when one hits one of the 28 base points Ai E Pi at which the fibers degenerate. At these points the coordinate ZI is completely unrestricted and the degenerate fibers are cones’ over the curve C em- bedded in the fiber K*; with vertex ( 1, Ai, 0, 0,O) E

p(2k,-1,2k~-l,kz.k3.~).

The discussion so far suffices to derive the Euler number of the spaces ( 1) purely in terms of the fiber structure, a result which we will have use for later on. If the fibration M - B of the Calabi-Yau man- ifold M over the base B were such that there are no singular fibers F its Euler number would simply be x(M) = ,y(B),y( F). As described above however singular fibers are generic for the spaces ( 1) and there- fore the necessary ingredients of this computation in- clude not only the Euler number of the base, x ( P1 ) = 2, the Euler number of generic fiber, x( K3) = 24, but also the number NS = 2L of singular base points as well as the Euler number of the degenerate fiber. The structure of the singular fibers depends crucially on whether kl is equal or larger than unity. If kl > I there

is the additional complication that each vertex of the cone over C is a &, -i-singular point on the Calabi- Yau manifold whose resolution introduces (kl - 1) spheres Pi. The Euler number of the degenerate fibers Fdeg therefore is given by 10

,I!(&) = x(c) +Qkl - 1) + 1. (7)

Q To be precise, they are cones on the nonresolved hypersurface. lo Here we understand x(C) to be Euler number of the resolved curve.

430 B. Hunt, R. Schimmrigk/Physics Letters B 381 (1996) 427436

The Euler number of the fibered threefold then takes the form

X(M) =2(1-!).24+2e(~(C)+2kt-1). (8)

The second crucial property of the manifolds ( 1) is that the monodromy transformation m on the coho- mology, induced by ((Y is the @’ root of unity)

& 3 m : (21, ...,Z4) - (w9z29Z3,z4), (9)

is nilpotent of degree C, i.e. me = 1. The structure of the fibrations ( 1) explicated thus

far allows us to draw on some general results of bira- tional geometry ’ ’ in order to get further insight into their structure. Namely, since the monodromy is nilpo- tent and the modulus is constant, it follows that these manifolds can in fact be described (birationally) as orbifolds of products I2 of the form Ce x K, where re : Cl - IPI is the projection of a particular e-fold cover of the base space of the fibration and K is the embedded K3-surface. In order to see this consider the action of the cyclic group Ze on the product

& : ce x K + Ce X K, (10)

defined by the projection re on the first factor and the monodromy action m (9) on the second factor. The action m leaves invariant the curve C and therefore the orbifold &\Ce x K will have a singular curve in each of the 2C singular fibers over the Ai. Instead of resolving these singularities we blow up the 3-fold Ce x K along the 2C fixed curves; the exceptional divisor over each is a ruled surface l3 over the curve, which we denote by ,?$. Now we take the quotient of the blown up Ce x K. The surface KA, has as quotient the weighted projective plane P(kZ,k3,h) while each Ei descends to the orbit space (being fixed under the action of &) , and is in the branch locus there. Thus on the resolved orbit space each singular fiber over a hi consists of two components, a plane IP(k2,k3,b) and a ruled surface

I1 For a single nilpotent fiber, by definition, the monodromy sat- isfies mS = 1, hence a ramified cover of the base, branched to order s at the base point of the singular fiber, has trivial mon- odromy. For a global fiber space with nilpotent monodromy, the same holds for a suitably chosen ramified cover of the base. r* On the ramified cover of the last footnote, the monodromy is

trivial, and the modulus is constant; this means we have a product. l3 A ruled surface is a fibration over a curve (here C) with fiber

PI.

Ei, the two intersecting in the curve C. Because C is just a hyperplane section of the original fiber KAY, it follows that the weighted projective plane can be blown down to a point. In this process the intersection curve C is blown down to a point (which is smooth for kt = 1) as well and the surface Ei becomes a cone over the curve C. This is precisely the structure we have previously found for the manifolds ( 1) and thus we have uncovered that the essential structure of the weighted hypersurfaces ( 1) is that of orbifolds of a global product involving K3 surfaces.

This may be described more explicitly as follows. Define

Cp : Ce x K c-t P(2k,-1,2k,-1,2k2,2k3,2k4) [2kl

1 .(~3(YO*Yl ..*7Y3)) H (1 +h(U)2e)i12eYi’2. (

(11)

and denote the coordinates in weighted 4-space by zi, i = 0, . . . . 4. Here the map is defined for re (a) 2e # - 1; we then take the closure in the projective space. Then clearly @~(p, K) is the hyperplane section zi = Fe (p) z2 of the hypersurface (2). The image is invari- ant under the action of Ze and for re (a) 2e = - 1 the image is the cone

Qa = {P(Z2,Z3, L1) = 0) C {ZI = Te((a)Z2}

c P(2&,-1,2k,-1,2k2,2k3,2k4). (12)

In this way the birational map described above is im- mediately performed and in particular one can deter- mine whether the vertex of the cone Qa for re ((T) 2e = - 1, namely the point ( 1, ne( a), 0, 0, 0), is a singu- lar point of the threefold. Looking at the equation, it is clear that the vertex is a quasi-smooth point of the threefold, and in particular, if ki = 1, it is a smooth point, while if ki > 1, it is a singular point of the ambient projective space and must be resolved.

We now see that the structure of the second coho- mology group of any of the spaces of type ( 1) is de- termined by a single K3 hypersurface and the action of the automorphism. We thus have reduced the prob- lem of deriving the heterotic gauge structure to the problem of deriving it from the K3. This can easily be achieved by considering the invariant part of the

B. Hunt, R. Schimmrigk/Physics Letters B 381 (1996) 427-436 431

Picard lattice with respect to the action which defines the fibrations, as we will discuss further below.

3. Hypersurface examples

Example I: We start with the K3 surface K = {y;” + yi + yz + y: = 0) E ~(1,6,14,21)[421; K has an auto-

morphism a: (yI,y2,y3vY4) H (ayl,Y:!,Y3,Y4) (a

a 42”d root of unity) 14, so our group Ze is Z42. Our curve C42 is a degree 42 cover of II? 1, branched at the 84 roots of -1 under the projection ~42 : C42 --+

PI 1. This curve (like any of the Cl) can be described explicitly as a weighted curve

c42 = {xi’ + xy4 + xt4 = 0) E Q2,1,1)Wl; (13)

the map ~42 is given explicitly by 7r4z(xo, xl, x2) ++ (XI, x2). The group Z42 acts on the curve C42 by x0 +-+ ~0, and the map ~42 is the map of C42 onto the quo- tient PI = ZQ\&. The action of 242 on the product then has 84 fixed divisors, namely the copies of K

lying over the points 41, . . . , qg4 E c42 which is the fixed point set of Z42 acting on C42. These fixed divi- sors are the degenerate fibers of the K3 fibration of the quotient &z\C42 x K over P’ (given by projection to the first factor). Instead of resolving the singularities of the quotient, we use the map ( 11) , which can be written as

+-+ (xl ($x2 (~)"*>y2.y3,y4). (14)

The image is just the well-known Calabi-Yau A4 E Itp, I ,I ,12,2&42) [ 841. To apply our formula it is then suf- ficient to look at the degenerate fibers of M as a K3 fibration. These are cones over the curve C = {y,’ + yz + yi = 0) in B(6,14,21)[ 421; it is easily checked that this curve is rational, and after resolution of the ambi- ent projective space, the Euler number is x(C) = 11. There is a &. a 23 and a ZI fixed point on the curve,

I4 Of the many possible actions of Z42 on K, this is the only one which preserves the curve on K defined by y1 = 0 aad therefore lifts to the resolution of singularities.

leading to 1,2 and 6 new curves, respectively, meeting as in the following picture:

From (7) we get X( F&g) = 12 and from (S), X(M) = -960.By (5) wehaveh’,‘(M) =2+9 = 11. These facts are of course well-known [ 181. Finally, to find the invariant part under the monodromy, we only have to determine the Picard lattice of K, as will be shown in Section 5. Our automorphism group 242 is the group denoted HK there, and this group leaves precisely the Picard lattice invariant. Now we apply the results of Kondo described there, which imply that K is the unique K3 surface with a Z-&2-automorphism, and that K is an elliptic surface (2 1) for which Kondo shows that the Picard lattice is & = U @ Es. Thus this is the lattice of the gauge group of the heterotic dual. In terms of the curve C this invariant lattice is described by the resolution diagram above, while in terms of the elliptic surface it is a union of a section and a singular fiber of type II* (these terms are de- fined in Section 5).

I

Thus we see that the heterotic dual should be de- termined by higgsing the first Es completely while re- taining the second Es. We also see that we should not fix the radii of the torus at some particular symmetric point but instead embed the full gauge bundle struc- ture into the Es. Doing precisely as instructed by the manifold we recover the heterotic model of [ 11.

Example II: Consider the manifold P( 1,1,2,4,4) [ 121 whose Hodge numbers were found [ 1 ] to match that of a particular heterotic model. The detailed under- standing of this space is of particular interest because it is known to be connected via a conifold transition to a codimension two Calabi-Yau manifold [ 131. We therefore wish to see whether we in fact can derive the heterotic theory from this Calabi-Yau manifold.

432 B. Hunt, R. Schimmrigk/Physics Letters B 381 (1996) 427-436

For this we have to determine the invariant sector of the Picard lattice of H2( K3) under the orbifolding. In order to do so we only need to observe that the orbifold singularities of the curve C = PC 1~2) [ 61 are three &-points whose resolution leads to a total of 3 ( 1,l )-forms which, together with the Kalrler form of the ambient space, determines the sublattice I(is3) c I’c3*19) = H2 (K3, Z). Taking into account the divisor coming from the curve C we find the resolution dia-

gram

+

with x(C) = 3. We see from this that the intersection matrix is precisely given by the Cartan matrix of the group SO( 8). We also see that in the heterotic dual we need to take the torus at the SU(3) point in the moduli space and break this SU( 3) by embedding the K3 gauge bundle structure groups appropriately. In this way we recover the heterotic construction of [ 11.

4. The general construction

In this section we abstract from the framework of weighted spaces and describe our construction for gen- eral K3 fibers which do not necessarily admit a de- scription as weighted hypersurfaces. For this we make the following assumptions. We are given a smooth K3 surface K with an automorphism group &, such that the fixed point set of the group is a curve on K, that is, there are no isolated fixed points. Then we use the curve Ce of Section 2, the e-fold cover of Pr branched at the N roots of unity on Pt, where e divides N. A somewhat involved computation then shows that the quotient Ce x K by Ze will be (birationally) Calabi- Yau if and only if

N = 2e. (15)

This is the value we found above, and it is true more generally. More precisely the statement is that when ( 15) holds there exists a Calabi-Yau space which is birational to the quotient Ze\Ce x K provided the sin- gularities of the quotient have Calabi-Yau resolutions, which is a purely local question. If this is the case then

the result will be a minimal Calabi-Yau manifold M with a K3 fibration and degenerate fibers over the Nti’ roots of unity.

Since the projection C’e - Pr has only fixed points over the @ roots of unity in PI, the fixed point locus of Ce x K will consist of the union of curves Cj in the fibers pj x K, where pj9 j = 1, . . . . N are the points of Ce mapping to the Nfh roots of unity. One can now pro- ceed to the quotient space Ze\Ce x K and resolve the singularities, or first blow up the curves Cj and then take the quotient. More precisely, one can continue to blow up loci in the fibers {pj} x K of the threefold Cj x K until the quotient is smooth, being a branched cover. Then there will be, on the quotient, exceptional surfaces in the degenerate fibers which can be blown down, and the result will be a minimal Calabi-Yau manifold M with a K3 fibration M - PI and degen- erate fibers over the N’i’ roots of unity.

A simple example from this point of view takes as starting point the quartic K3 IF“3 [4] 3 K = {xi yf = 0) in which case f? = 4 and the action of Z4 on K is given by xt I-+ ~~4x1. The fixed curve is C = {xi = 0) n K, a smooth genus 3 curve in the plane P2 = {xi = 0). Blowing up this curve on {pj} x K, we get the union of K and a ruled surface E - C. Taking the quotient, we get again for each 8* root of unity (N = 8) two components, M = Za\K and E, the latter of which constitute the branch locus of the projection onto the quotient. M is a projective plane and now has normal bundle 0~ ( - 1) (since K was blown up along a hyperplane section), and it is exceptional of the first kind and can be blown down to a smooth point of the threefold. This leaves as degenerate fiber the cone over the curve C the situation in the general case is similar, but more complicated.

Finally we mention that this construction, starting from just a smooth K3 with an automorphism, is ap- plicable to any such surface, and in particular, can be applied to elliptic K3 surfaces (see below) as well as to weighted complete intersection K3’s. This is espe- cially important, as the latter appear in the weighted conifold transitions described in [ 131, generalizing the splitting conifold construction of [ 271. Combin- ing the results of [ 131 with what we have learnt so far allows us to gain a complete understanding of the dual heterotic gauge structure of the conifold transi- tion. More precisely we need to collect the following ingredients: 1) The fibered Calabi-Yau threefold is

B. Hunt, R. Schimmrigk/Physics Leriers B 381 (1996) 427-436 433

completely determined by a single K3 surface. 2) The conifold transition connects a fibration with another fibration. For general conifold transitions this will not be the case, but as shown in [ 131 there exist conifold transitions for which this holds. It was furthermore shown in [ 131 that such transitions proceed via a de- generation of the fibers. 3) The singularity structure of the K3 surface determines the dual heterotic gauge group [ 121. Combining these facts we see that in coni- fold transitions between K3-fibered threefolds the res- olution graph of the K3 surface changes because of the vanishing and appearance of 2-cycles when the K3 fibers go through the degenerate configuration. Since it is this graph which determines the Dynkin diagram we thus gain an understanding of the heterotic dual of the transition.

5. Automorphisms of K3 surfaces

Our discussion in the previous sections shows that we need to understand the automorphisms of K3 sur- faces, in particular when group actions by some & exist. The first is that if 0~ denotes the non-vanishing holomorphic K, any automorphism via g*flK

an exact sequence

1 - ---+ Aut(K) -% & - (16)

where the cyclic group of k”’ roots of in and GK is the kernel, i.e., set (a of automorphisms preserving the form RK. This gives

of @ which is by results of the direct sum of of maximal

the Euler function. 231 that eigenvalues

Ze on @Q primitive roots unity. irreducible has maximal

ble namely Since 5 it that 5 Particularly are

automorphisms act on Picard This denoted HK, in the as (which the splits). a of we that an g HK, invari-

lattice g precisely the lattice. is from that main will in

of so is to more what possibly A result this

tion given Kondo. was in that unimodular C be divisor any the

in = Furthermore, 4(e) rk(TK) e precisely values S,

in cases exists unique to phism) surface given

For examples by list the lattice and complement are fol-

e SK

The situation for just the opposite of one for HK. The invariant sublattice is this TK, and the action on was described for abelian groups GK by Nikulin. The possible such GK can occur are the following:

(Z2)*, 0 I m Z4; Z2 Z8; Z3;

Since the largest cyclic group is follows if a surface admits a cyclic automorphism of

9, then is HK. There- fore, on our aims, it more useful to consider HK or GK. by mirror symmetry (which surfaces is is another K3 Km for which TK are exchanged. Consequently HK and GK switched also.

We apply these results to our construction. Let K a surface an automorphism of !?

HK, and let the Picard lattice. Then our construction above, we have

K3 fibration is the invariant of the lattice. Hence data (K, = HK, SK) deter-

to a heterotic string is isomorphic to other

words, to the invariant lattice, it is sufficient to the Picard lattice SK of Note that, to get K3 surface SK, sometimes

434 B. Hunt, R. Schimmrigk/Physics Letters B 381 (1996) 427-436

Table 1 In the last row we have listed the value of the .7-function, which is defined by 3 = g:/A. If J’ is constant, then the modulus of the elliptic curve is constant in the family.

Fiber IO I,,n > 0 II III IV IO’ I,‘,n>O II* III’ Iv*

v(g2) 0 0 11 1 22 2/>2/2 2 24 3 23 v(g3) 0 0 1 22 2 > 3 /3 /3 3 5 25 4 v(A) 0 n 2 3 4 6 n-l6 10 9 8

J # O,l,cu order n pole 0 1 0 1 /o/ # O,l,co order n pole 0 1 0

ficient to give a combination of singular fibers and an elliptic K3 surface with those singular fibers. Let us give a brief description of this class of K3 surfaces. An elliptic curve can always be realized as a cubic in IF)*. To get an elliptic surface, one lets this cubic curve in the plane vary. This is described by an affine equation of the type

Y2 = x3 - gz(t)n: - g3(t), (17)

where gi ( t) is a section of a line bundle LB2’ on some curve C (the base curve of the projection of the sur- face S - C). Here x and y are afline coordinates in a P2, and the entire surface is contained in a Pz- bundle over C. If S is a K3 surface, then necessarily C = Pt, and the sections gi(t) are just homogeneous polynomials of degrees 2i-deg( L), and for S to be K3 again we need deg( L) = 2. The fiber over a point t E Pt will be singular precisely when the discriminant of the Weierstral3 polynomial vanishes there, A(t) :=

g2w3 - 27g3w2 = 0. The type of singular fiber is completely determined by the degrees of vanishing of g2, gs and A at the point, according to Table 1.

Looking at the table, the following is clear. If we consider the dual graph of each fiber type (with the exception of II), then we get an extended Dynkin di- agram of one of the simple Lie algebras. The corre- spondence is given as follows:

IIIIIIV I, 1; Iv* III* II*

- A1 A2 &-I Dn+4 & EI Es

(see [ 281). In this way it is often possible to see what the lattice SK of such an elliptic surface is. More precisely, if there is a unique section, then the Picard lattice SK can be read off directly from the singular fibers. This is the situation with all the examples of Kondo.

We now describe the two examples of Kondo which we shall use later. These are all elliptic fibrations.

e = 66: There are two ways to construct this surface: Kondo describes it as an elliptic K3, with 12 fibers of type II at r = 0 and at the 11” roots of unity. The affine equation is

y2=x3+t&d,,), 1

(18)

and the automorphism is given by (x, y, t) H (c&x, c&y, &) . Alternatively, one can consider the (non-Gorenstein) weighted hypersurface

{X2 +y3 + z” + V@ = 0) E P(1,6,22,33)[661, (19)

which, upon resolution, yields a smooth K3 [29]. Here the automorphism is given by (n, y, z, w) H (x, y, z, ~~66 w) . From this second description we see that the fixed point set is the (total transform of the) CUl-W {X2 + y3 + Z” = 0) c P(6,22,33). hl the above description that curve is given by at most the fibers 7r-‘(O) and rTT-l (co) as 0 and 00 are the only fixed points oft, together with the zero section, the locus (in Pt x P2) given by setting x = 0 and y = 0. It should be noted that the fiber rTT-l (co) is smooth, hence the group acts on it, and does not fix it. Hence the fix point set is r-’ (0) U ao, where aa denotes the zero section. This will be used below.

l= 42: Here we again have a description as an elliptic surface,

I

Y2 =ns+Pn(,-LY:), (20) 1

and the automorphism is given by (x, y, t) H ( cyi2x, ai2y, (Y:$). It has a fiber of type II* at t = 0, and fibers of type II at all seventh roots of unity.

B. Hunt, R. Schimmrigk/ Physics Letters B 381 (1996) 427-436 435

This example may also be described as a weighted hypersurface:

{x2 + y3 + z7 + w4* = 0) c ~(1,6,14,21). (21)

Here the automorphism is given by (x, y, z, w) ++

(&y,Z,a42w). The important point following from these remarks

is that we can now pose the problem the other way around: given a gauge group of a heterotic string, we can find a Calabi-Yau with a K3 fibration such that the invariant lattice under the monodromy is the lattice of the given gauge group. More precisely, suppose we can find a K3 surface K, such that (i) it has SK given by the lattice of the given gauge group, and (ii) it has a non-trivial automorphism group HK. Then we can apply the construction above, and the result is a Calabi-Yau threefold, fibered in K3 surfaces, such that the invariants under the monodromy are exactly the lattice SK.

Now let us try to find some interesting lattices which could play the role of gauge groups for hererotic strings. Suppose, for example, we are look- ing for a type IIA string on a Calabi-Yau with gauge group SO( 8) and with ( IZ,, nh) = (8,272). First we note that the following combination of singular fibers would do the job: 11;, 9II; as mentioned above, this would give a Picard lattice on a K3 SK = D4 @ U. We now construct such an elliptic surface with an au- tomorphism of order 18, as follows. The Weierstral3 equation will be

9

Y2 =x3+t3n(t-rY&), I

(22)

and an automorphism is given by (x, y, t) ++ (a&x, c&y, a&t). From the general theory of el-

liptic surfaces, since g3 = t3nT(t - a;), which vanishes to order 3 at t = 0 and order 1 at t = a:, while A = gz - 27d = -27&~ vanishes to order 6 at r = 0 and to order 2 at t = (~6, we find that there are precisely the mentioned singular fibers, i.e., II;, 9II. We may do our construction above, using the curve Crs, and the result is a Calabi-Yau with 36 singular fibers. Furthermore, the number /I’,’ is easy to find - it is just one more than the corresponding value for the K3 surface (as this automorphism is in HK, the invariant lattice is just SK which has rank 6), that is

h’*’ = 7. We now calculate the Euler number of this fibration to find the other Hodge number. The singu- lar fibers are in this case also, cones over a reducible curve. On the elliptic surface, this curve is given by the fibers over 0 and the zero section. Note that this corresponds to the invariance of the Picard lattice SK, as SK is spanned by the classes: the fiber T-’ (0) and the pair (fiber,section), producing the lattice D4 @ U. It is the curve on the elliptic surface

and has Euler number x( C) = 7. There are 36 singular fibers, and our formula (8) for the Euler number gives x(X) = -528. Thus we find (h’*‘,h*~‘) = (7,271), precisely as needed.

We may also apply the above construction to the two Kondo examples with C = 66. In a sense, the situation of these examples is considerably easier than with the case C = 42, simply because there are not so many ra- tional curves. However, it still seems to be a difficult task to explicitly resolve the quotient singularities, so we will instead use the picture of weighted hypersur- face. For this we consider the k = 66 case, and note that the K3 surface is obtained from the resolution of the weighted Fermat hypersurface K E B(,,6,22,33) [66] by blowing down some curves. In fact, the canoni- cal bundle of the resolved surface is not trivial, it is concentrated on the proper transform C’ of the curve C = {xt = 0) n K. But this curve is on the resolved surface, call it K’, exceptional of the first kind and can be blown down. After blowing this curve down, fur- ther curves become exceptional of the first kind, and after blowing down everything we have, instead of the 5 original curves, now just 2. This surface, call it K”, is now a smooth K3 surface, and is in fact just the el- liptic surface with k = 66 as described by Kondo (that surface is automatically minimal).

We now consider the weighted threefold P(t,r,12,+s~[ 1321, again given by a Fermat equation. As in our examples above, this threefold again has a fibration, and after doing the usual resolution of singularities, the fibers are the non-minimal surface K’ above. Let X’ denote the corresponding threefold;

436 B. Hunt, R. Schimmrigk/Physics Lerters B 381 (1996) 427-436

then applying the adjuction formula to the fibers, we see that the canonical bundle of X’ is concentrated on the exceptional divisor over the curve C above, a ruled surface over C. Since in each fiber of the pro- jection X’ - IPI, the curve C can be blown down, it follows that the entire exceptional divisor on X’ may be blown down. Blowing down the further curves in each fiber, we get a smooth, minimal Calabi-Yau X” with a K3 fibration, which by (15) is Calabi-Yau, the general fibers of which are the surface K”, with 132 degenerate fibers at the 132”d roots of unity.

To calculate the Euler numbers of the degenerate fibers, it suffices to calculate the Milnor number of the singularity at the vertex. The singularity is y* = x3+(t-I) . l1 It is easy to see in this case that the Milnor number is 20, hence the degenerate fiber has Euler number x( F&g) = 4, and an application of the x-formula (8) yields

(51 C. Vafa and E. Witten, hep-th/9507050. [6] P. Aspinwall and J. Louis, hep-th/9510234;

I? Aspinwall, hep-th19511171. [7] G. Aldazabal, A. Font, L.E. lb&nez and E Quevedo, hep-

th/9510093. [8] P. Berglund, S. Katz and A. Klemm, hep-th/9506091. [9] T.M. Chiang, B.R. Greene, M. Gross and Y. Kanter, hep-

th/9511204. [ 101 A.C. Avram, P Candelas, D. Jan&C and M. Mandelberg,

hep-th/9511230. [ 11 I P. Berglund, S. Katz and A. Klemm, Type I1 Strings on

Calabi-Yau Manifolds and String Duality, NSF-ITP-95-161 preprint, to appear.

[ 121 E. Witten, Nucl. Phys. B 443 (1995) 85. [ 131 M. Lynker and R. Schimmrigk, hep-th/9511058. [ 141 P. Candelas, M. Lynker and R. Schimmrigk, Nucl. Phys. B

341 (1990) 383. [ 151 M. Lynker and R. Schimmrigk, Phys. Lett. B 249 (1990)

237.

x(X) = (2 - 132) .24 + 132.4 = -2592. (23)

[ 161 M. Lynker and R. Schimmrigk, Phys. Lett. B 215 (1988) 681; Nucl. Phys. B 339 (1990) 121; J. Fuchs, A. Klemm, C. Scheich and M.G. Schmidt, Phys. l&t. B 232 (1989) 317; Ann. Phys. 204 (1990) 1.

[ 17 ] A. Font, L.E. Ibanez and E Quevedo, Phys. Lett. B 224

Acknowledgement

The first author is indebted to Igor Dolgachev for discussions on the examples of Kondo and Mark Gross for communications. The second author is grateful to Per Berglund, Shyamoli Chaudhuri, Jens Erler, David Lowe and Andy Strominger for discussions. This work was supported in part by NSF grant PHY-94-07194.

[I81

[191

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