gorbounov v., ochanine s. - mirror symmetry formulae for the elliptic genus of complete...

Journal of Topology 1 (2008) 429445 c 2008 London Mathematical Societydoi:10.1112/jtopol/jtm014

Mirror symmetry formulae for the elliptic genus of complete intersections

Vassily Gorbounov and Serge Ochanine

Abstract

In this paper, we calculate the elliptic genus of certain complete intersections in products ofprojective spaces. We show that it is equal to the elliptic genus of the LandauGinzburg modelsthat are, according to Hori and Vafa, mirror partners of these complete intersections. Thisprovides additional evidence of the validity of their construction.

Introduction

Understanding the mathematics behind Quantum and in particular, Conformal Field Theorieshas been a challenge for more then 25 years. The usual ground for mathematical interpretationsof Quantum Field Theory predictions has been the Topological Quantum Field Theory, whichis a certain reduction of the genuine Quantum Field Theory. There have been signicantmathematical advances in this area. Mirror symmetry is among the major motivations behindthese advances. Vaguely stated, mirror symmetry is a duality between complex and symplecticgeometry. As was originally discovered by physicists [6], among its specic manifestations is astriking connection between the number of curves of a given genus in a symplectic manifoldand the periods of a holomorphic form of a dierent complex manifold. Ever since, the explicitconstruction of the mirror partner for a given manifold has become the central task for themathematicians. A vast amount of work has been done in this direction. In this paper, wefollow the line pioneered by Gepner [8] and developed by Vafa [25], who discovered that theConformal Field Theory dened by a manifold, the so-called sigma model, can be identical tothe Conformal Field Theory of a dierent type, the so-called LandauGinzburg model. Thispoint of view was further developed by Witten [28]. As far as the application of this idea tomirror symmetry is concerned, the major reference for the purposes of this paper is the work ofHori and Vafa [15]. They showed that the mirror partner of a large class of manifolds turns outto be a LandauGinzburg model of some kind, or its orbifold. The proof that these modelsform a mirror pair would consist of picking an invariant which is known to be identical for mirrorpartners and, by a calculation, showing that it is indeed the same for given hypothetical mirrorpartners. In [15] such an invariant is given by the periods of a holomorphic form on a manifoldand the so-called BPS masses in the LandauGinzburg model. Of course, the identities impliedby mirror symmetry in Topological Quantum Field Theory are a reduction of stronger identitiesin the original Quantum Field Theory. The rigorous mathematical structure behind QuantumField Theory is not known at the present time, so exploring the mathematical consequencesof mirror symmetry at this higher level is dicult. Some time ago, Malikov, Schechtman andVaintrob [19] introduced a construction of a mathematical approximation to the structure ofQuantum Field Theory dened by a manifold. It is the so-called Chiral de Rham complex.Mathematically, these ideas were further developed in [11, 10, 4, 5, 16]. The relevance ofthis construction to physics was explained only recently in [30, 17]. Despite being only an

Received 6 July 2007; published online 5 February 2008.

2000 Mathematics Subject Classication 14J32 (primary), 55N34, 14M10.

The rst author is partially supported by the NSF.

430 VASSILY GORBOUNOV AND SERGE OCHANINE

approximation to the genuine Quantum Field Theory, the Chiral de Rham complex carriesfeatures not available in the Topological Quantum Field Theory. One of them is the ellipticgenus. Introduced in mathematics in [21], it was almost immediately connected to quantumphysics [27, 26] and later shown to be identical for mirror partners, thus providing another testfor a pair of manifolds to be mirror partners. Moreover, as explained in [29], there is a physicscounterpart of the elliptic genus in a large class of Conformal Field Theories, in particular, inLandauGinzburg models and their orbifolds, and this physics elliptic genus coincides with theone dened in topology for the sigma model. Taking on Vafas and Wittens ideas, a numberof physicists came up with a new type of formulas for the elliptic genus of some classes ofmanifolds in terms of their mirror LandauGinzburg partner [18, 2, 3, 7]. For about ten years,no mathematical proof of these formulas was produced. The rst paper where such a proofwas given in the case of CalabiYau hypersurfaces was [9]. The scope of this paper was muchbroader and the result for the elliptic genus fell out as a simple consequence of a much deeperconnection found between the LandauGinzburg model and the Chiral de Rham complex of ahypersurface.

The purpose of the present paper is to prove, by more or less elementary means, thatthe LandauGinzburg mirror partners found in [15] for complete intersection in products ofprojective spaces have the physics elliptic genus identical to the topological elliptic genus. It isinteresting to note that our result provides an extra test for the constructions in [15], and assuch, renes the conditions for the existence of a mirror LandauGinzburg theory as a mirrorpartner for some complete intersections given in [15]. We are planning to return to the case ofcomplete intersections in more general toric varieties considered in [15] in a future work.

Although inspired by physics, with the exception of the introductory Section 1, this paper ismathematically self-contained. It starts with a brief introduction to Jacobi functions and thecorresponding elliptic genera. We then proceed to derive the main formula (Theorem 9) usingthe residue theorem for functions of several variables. The formula is derived for elliptic generaof any level, which includes the case of Hirzebruchs level N genus [13, 14]. In Section 11, theformula is specialized to the level 2 genus discussed in physics literature. A dierent proof ofthe level 2 case is given in Section 12.

1. LandauGinzburg orbifolds and their elliptic genus

In this section, we briey state the results about the elliptic genus of LG theories and theirorbifolds without going into details. In the physics literature, the elliptic genus is dened asa character of an action of an innite-dimensional Lie algebra, namely the so called N = 2algebra. For the purposes of this note, LandauGinzburg eld theories are described mathe-matically by a noncompact manifold and a function W on it, called a superpotential, which hasisolated singularities. These data are sucient for dening and calculating explicitly the ellipticgenus of some LG theories [29] that are relevant for this note. An important class of quantumeld theories related to LG is dened by orbifolds of LG with respect to a nite group action.The elliptic genus of the LG orbifolds relevant for us was calculated in a number of papers [2,3, 7, 18] (see also [9] for a more mathematical approach). Such a genus is either a Jacobi formor a modular function, depending on the data dening the LG model. In this note, we considerLG dened by (CN ,W ), where the superpotential W is a holomorphic and quasi-homogeneousfunction of z1 , z2 , . . . , zN and its orbifolds with respect to the action of some nite group ofsymmetries. The conditions on the W are the following: It should be possible to assign someweights ki Z to the variables zi for i N, and a degree of homogeneity d N to W , so that

W (ki zi) = dW (zi)

for all i and C. Let qi = ki/d.

ELLIPTIC GENUS OF COMPLETE INTERSECTIONS 431

Suppose that the potential is invariant under a nite abelian group of symmetries G. Denoteby Ri the function on G satisfying g(zi) = exp(2iRi(g))zi . The invariance of W means, ofcourse, that for all i and g G,

W (g(zi)) = W (zi).

The elliptic genus Ell(q, y) of such an LG orbifold dened by the data (W,G) following [2, 3],is given as follows:

Ell(q, y) =1|G|

g1 ,g2G

Ni=1

yRi (g1 )1((1 qi)z + Ri(g1) + Ri(g2) |)

1(qiz + Ri(g1) + Ri(g2) |) ,

where y = exp(2iz) and q = exp(2i).

Example 1. Consider the following LG data taken from [1]: In the space C9 , introducecoordinates Xa (a = 0, 1, 2, 3), Yb (b = 0, 1, 2), Zc (c = 0, 1), and dene the potential by theformula

W =1

r=0

(X3r + XrY2r + YrZ

2r ) + X

32 + X2Y

22 + X

33 .

In this case, d = 3 and qi = 1/3. It is easy to check that W is invariant under the followingaction of Z/12 with generator :

(Xa, Yb, Zc) = (4Xa, 2Yb, Zc).

Calculating the appropriate numbers Ri() which dene such an action, we obtain

Ra() = 1/3, Rb() = 1/6, Rc() = 1/12.Therefore, the above formula for the elliptic genus of the LG orbifold dened by the data(W,Z/12) becomes

Ell(q, y) =112

12i,j=0

(yj/3

1( 2z3 i+j3 |)1( z3 +

i+j3 |)

)4

(yj/6

1( 2z3 +i+j

6 |)1( z3 i+j6 |)

)3 (yj/12

1( 2z3 i+j12 |)1( z3 +

i+j12 |)

)2.

It is easy to check that this formula agrees with our formula for the elliptic genus of thecomplete intersection in CP3 CP2 CP1 given by equations

3a=0

X3a = 0,2

b=0

XbY2b = 0,

1c=0

YcZ2c = 0

(see Section 11).

In Sections 28, k will be an algebraically closed eld of characteristic p 0, G a nite groupof order n, and p n.

2. Regular representations

Consider the k-vector space V = Map(G, k) of all k-valued functions on G. Clearly, dimk V =n. The group G acts on V by the formula

(g f)(h) = f(hg) (g, h G, f V ).With this action, V is called the regular representation of G.


We now consider the case when G is abelian and write the group operation in G additively.Let G = Hom(G, k) be the character group of G. The following theorem is well known(cf. [23, 2.4]).

Theorem 1. (i) G V is a basis of V .(ii) The one-dimensional subspace V of V generated by G consists of those f V

which satisfy

f(u + g) (g)f(u) for all g G.(iii) The G-modules V are pairwise nonisomorphic.(iv) Every irreducible G-module is of degree 1 and is isomorphic to one of the V .

3. Generalized Jacobi functions

Let now E be an elliptic curve over k, let k(E) be the eld of rational functions on E, andlet G E be any nite subgroup of order n. As above, we assume that p n.

Let be the divisor

=gG

(g),

and let L(G) k(E) be the associated vector spaceL(G) = {f k(E) | div(f) } .

By the RiemannRoch theorem, dimk L(G) = n. The group G acts on E by translations andleaves unchanged. Therefore, L(G) is invariant under the induced G-action on k(E). Thus,L(G) is naturally an n-dimensional representation of G.Theorem 2. As a G-representation, L(G) is isomorphic to the regular representation

of G.

Proof. Let u0 E be any point such that nu0 = 0. Dene a G-linear map : L(G) V = Map(G, k)

by

(f)(g) = f(u0 + g) (f L(G), g G).This is well dened, since u0 G and therefore u0 + g is not a pole of f . Since L(G) and Vhave the same dimension n, we need only to prove that is injective.

Suppose that f = 0 and (f) = 0. Then, f(u0 + g) = 0 for all g G. Thus, f has at leastn zeroes. On the other hand, since f L(G), it has at most n poles. Since deg div(f) = 0, weconclude that f has a simple pole at each g G and a simple zero at each u0 + g; that is,

div(f) = gG

(g) +gG

(u0 + g).

Thus, the image of div(f) under Abels map

Div(E) Eis nu0 = 0. The contradiction shows that f = 0.

Corollary 1. For each G, there is a nonzero function f L(G) satisfyingf(u + g) (g)f(u)

for all g G. This function is determined uniquely up to a nonzero multiplicative constant.


Definition 1. We call f a (generalized) Jacobi function belonging to G.

Remark 1. The classical Jacobi functions correspond to the case where G is a cyclic groupof order 2 (char k = 2).

4. The divisor of a Jacobi function

We now describe explicitly the divisor of a Jacobi function.

Theorem 3. A nonzero function f L(G) is a Jacobi function if and only if div(f) isinvariant under translations by elements of G.

Proof. If f is a Jacobi function, the formula

f(u + g) (g)f(u)shows that the functions

u f(u) and u f(u + g)have the same divisor. Thus, div(f) is invariant by G.

Conversely, let f be a nonconstant function in L(G), and suppose that div(f) is invariantby translation. Then there is a constant (g) k such that

f(u + g) (g)f(u),for all u E. Since

(g1 + g2)f(u) = f(u + g1 + g2) = (g2)f(u + g1) = (g1)(g2)f(u),

we have (g1 + g2) = (g1)(g2). Thus, G, and f is a Jacobi function belonging to .

It is now easy to describe the divisors of the Jacobi functions. Let En be the group ofn-division points on E. Since n is the order of G, we have G En . For every coset En/G,let

=r

(r) Div(E).

Theorem 4. A nonconstant function f L(G) is a Jacobi function if and only ifdiv(f) = +

for some nonzero En/G.

Proof. Using Theorem 3, we need only to describe the nonzero principal divisors D satisfy-ing D and invariant under G. Since the polar part of D is nonzero and invariant underG, it must be equal to . Since D+ is invariant, and is of degree n, it is necessarily of theform

gG(r + g)

for some r G, that is, of the form .


5. Modulus and conjugation

Let f be a Jacobi function with character and divisor div(f) = + , En/G. Foreach r , the involution u r u takes div(f) to div(f). Therefore, f(u)f(r u) is anonzero constant, which we call the modulus of f and designate c(r). Thus, c : k. It iseasy to check that

c(r + g) = (g)c(r)

for all g G.Similarly, the involution u u takes + to + . This u f(u) is also a

Jacobi function with character 1 . In particular, when n = 2, En/G has order 2, and 1 = .It follows that f(u) = af(u), where a2 = 1. Thus, f is either even or odd. Since f has a poleof order 1 at O E, it has to be odd.

6. Jacobi functions on the Tate curve

Consider a local eld k complete with respect to a discrete valuation v, and let q k beany element satisfying v(q) < 1. It is well known (cf. [24, Appendix C, 14]) that E = k/qZcan be identied with the Tate curve

y2 + xy = x3 + a4x + a6 ,

where

a4 =m1

(5m3) qm

1 qm

and

a6 =m1

(5m

3 + 7m5

12

)qm

1 qm .

Let G E be a nite subgroup of order n. As usual, we assume that char k n. Let En/Gbe any nontrivial coset. We will construct an explicit Jacobi function for G, whose zeroes arethe points of .

If k, we will write for its image in E = k/qZ. Choose g1 = 1, g2 , . . . , gn k, sothat

G = {g1 , g2 , . . . , gn} .Then choose r1 , r2 , . . . , rn k, so that

= {r1 , r2 , . . . , rn}and

i

ri =i

gi .

This can be done as follows. First, choose r k, so that r E represents . Then clearly, = {rg1 , rg2 , . . . , rgn} .

Since rn = 1 in E, we have rn = qs for some s Z. Letr1 = qsr, ri = rgi (i 2).

Then i

ri = qsrn(

i

gi

)=i

gi .


Now, consider the basic Theta function

(u) = (1 u1)k1

(1 qku)(1 qku1)

=k1

(1 qku)k0

(1 qku1).

This is an analytic function on k that has simple zeroes at the points of qZ (cf. [22]) andsatises

(q1u) = u(u).Along with (u), we will consider its translates (u) ( k) dened by

(u) = (1u).

The function (u) has simple zeroes at the points of qZ and satises

(q1u) = 1u (u).Consider

f(u) =n

i=1

ri (u)gi (u)

.

Theorem 5. The function f is q-periodic and denes a Jacobi function for G having asthe coset of zeroes.

Proof. It is enough to prove the rst statement, since this would imply that f denes afunction on E with simple poles at the points of G and simple zeroes at the points of . Wehave

f(q1u) =n

i=1

ri (q1u)

gi (q1u)=

ni=1

(riu)ri (u)(giu)gi (u)

= f(u).

7. A special case

We now assume that q is not a root of 1, and choose an arbitrary nth root q0 of q. For n 1and a Z, dene

na (u) =

1, a(n)(1 qu)

0, a(n)

(1 qu1).

With the notation of the previous section, we have (u) = 10(u).

Theorem 6. The function

f(u) =n1(u)un0 (u)

denes a Jacobi function on E = k/qZ for the group G E consisting of the images of thenth roots of 1, and that has simple zeroes at the images of the nth roots of q.

Proof. Let 1 = 1, 2 , . . . , n be the nth roots of 1 in k. Thus,

G = {1 , 2 , . . . , n} , = {1 q0 , 2 q0 , . . . , n q0} .Notice that

(iq0) =(

i

)q.


Thus, we can take

r1 = q1q0 , ri = iq0 (i 2).According to the previous theorem, the function

ni=1

ri (u)i (u)

has all the required properties. It remains to identify the numerator and denominator explicitly.We will be using the identities

(1 it) =

(1 1i t) = 1 tn ,which are easily proved by noticing that the three polynomials in t have same degree n, thesame roots, and the same constant term 1.

We have i

i (u) =i

(1 iu1)k1

i

(1 qk1i u)k1

i

(1 qkiu1)

= (1 un )k1

(1 qnkun )k1

(1 qnkun )

= n0 (u).

Similarly,i

i q0 (u) =i

(1 iq0u1)k1

i

(1 qk1i q10 u)k1

i

(1 qkiq0u1)

= (1 qun )k1

(1 qnk1un )k1

(1 qnk+1un )

= n1(u).

Finally, i

ri (u) =i

i q0 (u)q1 q0 (u)q0 (u)

and

q0 (u) = (q10 u) = (q

1(qq10 u)) = qq10 uq1 q0 (u);that is,

n1(u)un0 (u)

is a constant multiple ofn

i=1

ri (u)i (u)

and therefore, has all the required properties.

Remark 2. In the case n = 2, the previous theorem gives

f(u) =1

u(1 u2)k1

(1 q2k1u2)(1 q2k1u2)(1 q2ku2)(1 q2ku2)

=1

u u1k1

(1 q2k1u2)(1 q2k1u2)(1 q2ku2)(1 q2ku2) .


The formal substitution u = ez/2 leads to

zf(ez/2) =z/2

sinh(z/2)

k1

(1 q2k1ez )(1 q2k1ez )(1 q2k ez )(1 q2k ez ) ,

which is a familiar expression for the generating function of the level 2 elliptic genus (seebelow).

8. Modulus and normalization on the Tate curve

Continuing with the situation of the preceding section, we rst compute the modulus for theJacobi function dened by;

f(u) =n1(u)un0 (u)

;

that is, we compute the constant value of f(u)f(ru1) for r . We start with r = q0 . Thefollowing formulas can be easily obtained from the denition of na :

n0 (u1) = unn0 (u), na (u1) = unna(u) (a 0(n))

and

n0 (q0u) = q1unn1 (u), n1(q0u) = n0 (u).It follows that

n0 (q0u1) = q1unn1(u), n1(q0u1) = unn0 (u),

and therefore

f(q0u1) =qun0 (u)q0n1(u)

= qn10 f(u)1 .

Thus,

c(q0) = qn10 .

Notice now that the formula for f does not depend on the choice of q0 . Therefore, we have thefollowing theorem.

Theorem 7. For the Jacobi function dened by

f(u) =n1(u)un0 (u)

,

and any nth root r of q in k, we have

c(r) = rn1 .

We now normalize f by requiring that

Resu=1

(f(u)

du

u

)=

1n.

This normalization is formally equivalent to the requirement that zf(ez/n ) = 1 + o(z). Since

(u 1)n1(u)un0 (u)

=u 1

u(1 un ) n1(u)

k1(1 qnkun )(1 qnkun )

=un1

1 + u + u2 + + un1 n1(u)

k1(1 qnkun )(1 qnkun ),


we see that the normalized Jacobi function is f(u)N , where

N =

k1(1 qnk )2

n1(1).

The modulus of the normalized function is

c(r) = rn1N 2

and satisesc(r)n = qn1N 2n .

In particular, if n = 2, we have

N =k1

(1 q2k

1 q2k1)2

and

c(r)2 = qk1

(1 q2k

1 q2k1)8

,

which is the familiar expression for the modular form for one of the three level 2 ellipticgenera.

9. Elliptic genus

From now on, k = C, E = C/L for some lattice L, G = L0/L and n = [L0 : L]. Let fbe a Jacobi function with character . Then f is an elliptic function with period lattice L.We normalize f , so that it has residue Resz=0(f) = 1. Then the Taylor expansion of zf(z)is a formal power series with constant term 1 and denes, via the Hirzebruch formalism, amultiplicative genus

: U C,which we refer to as the level n elliptic genus dened by f . The case n = 2 is best known. Inthis case, zf(z) is an even series in z and factors through a genus

: SO C.

10. Complete intersections

In this section, we will compute the elliptic genus of complete intersections satisfying anondegeneracy condition as a summation over some division points of E E . . .E.

Let M = (mij ) be a l t matrix over Z, and let P be the productP = CPN11 CPN21 . . . CPNt1 .

For 1 j t, let j be the pull-back over P of the canonical line bundle of CPNj1 . Then for1 i l, let i be the line bundle

i = mi 11 mi 22 . . . mi ttover P . We will write Hi for the stably almost complex manifold (hypersurface) dual to i , andlet X(M) be the transverse intersection

X(M) = H1 H2 . . . Hl.Writing Mz = (1z, 2z, . . . , lz), we dene l linear forms i : Ct C; that is,

iz = mi1z1 + mi2z2 + . . . + mitzt .


Let be the elliptic genus of level n dened by a Jacobi function for G E with character .The standard computation using Hirzebruchs formalism leads to the following statement.

Theorem 8. (X(M)) is the coecient of z11 z12 z1t in the Laurent expansion at

z = (0, 0, . . . , 0) of

F (z) = F (z1 , z2 , . . . , zt) =f(z1)N1 f(z2)N2 f(zt)Ntf(1z)f(2z) f(lz) .

In accordance with the notation [m] for the multiplication-by-m map on E, we will write

[M ] : Et Es, [i ] : Et E, [] : Et Efor the maps induced by left multiplication by M , by i , and by =

i .

For the rest of this section, we will assume that l = t and that detM = 0. Let Gt = G. . .G Et and H Et be the inverse image of Gt under [M ]. Clearly, H is a subgroup of Et containingGt (since the entries of M are integers), and it is easy to check that [H : Gt ] = (detM)2 .

For each j, 1 j t, choose a zero rj E of f , so thatdiv(f) =

g

(g) +g

(rj + g),

and let r = (r1 , r2 , . . . , rt) Et . Also, choose a s = (s1 , s2 , . . . , st) Et satisfying [M ]s = r(this uses detM = 0).

Theorem 9. If, for every j,

i mij Nj mod exp(G), then we have

(X(M)) =(1)t+1

(detM)c(r)

hH/Gt

([]h)f(s + h)N ,

where we use the abbreviations

c(r) = c(r1)c(r2) c(rt)and

f(s + h)N = f(s1 + h1)N1 f(s2 + h2)N2 f(st + ht)Nt ,and where the sum runs over representatives of the cosets in H/Gt .

Remark 3. The condition

i mij Nj mod exp(G) is equivalent to c1(X(M)) 0mod exp(G).

The proof of this theorem is based on the following, slightly modied version of the globalresidue theorem for functions of several complex variables, as described in [12, Chapter 5, pp.655656].

Theorem 10. Let V be a compact complex manifold of dimension t, and let

D1 ,D2 , . . . , D

(t ) be eective divisors having the property that the intersection of every t of them,Dn1 Dn2 . . . Dnt , is a nite set of points, whereas the intersection of every t+ 1 of themis empty. Let D = D1 + D2 + . . . + D , and let be a meromorphic t-form on V with polardivisor D. Then for every P Dn1 Dn2 . . . Dnt , the residue ResP is dened and

P

ResP = 0.


The residue ResP for P Dn1 Dn2 . . . Dnt can be dened as follows: choose localcoordinates (z1 , z2 , . . . , zt) near P and write near P as

=(z) dz1 dz2 . . . dzt

1(z)2(z) t(z) ,

where , 1 , 2 , . . . t are pairwise relatively prime holomorphic functions. Then

ResP =1

(2i)t

,

where is the suitably oriented real cycle

= {z | 1(z) = , 2(z) = , . . . , t(z) = } ,for a small .

It can be easily veried that if F (z) is a meromorphic function near O = (0, 0, . . . , 0) Ct , thenResO (F (z) dz1 dz2 . . . dzt) is the coecient of z11 z12 z1t in the Laurent expansion ofF (z) at O. We will also use the following formula, generalizing a fact well known for functionsof one variable.

Lemma 1. If is written near P as

=(z) dz1 dz2 . . . dzt

1(z)2(z) t(z) ,

and det(i/zj )(P ) = 0, then

ResP =(P )

det(i/zj )(P ).

We now turn to the proof of Theorem 10. In view of Theorem 9, and the above remarks, weneed to compute ResO , where

=f(z1)N1 f(z2)N2 f(zt)Nt dz1 dz2 . . . dzt

f(1z)f(2z) f(lz) .

We let V = Et and consider on V . According to Theorem 11, we haveP

ResP = 0,

where P runs through the points P Gt , corresponding to the poles of f(z) (rst kind), andthrough the points P s + H, which are the simultaneous zeroes of f([1 ]z), f([2 ]z), . . . ,f([l ]z) (second kind).

Notice that the condition i

mij Nj mod exp(G)

guarantees that F (z + g) = F (z) for all g Gt . It follows that the contribution of the pointsof the rst kind to the sum of residues is nt ResO .

Turning to the points of the second kind, say P = s + h, we rst notice that dierentiating

f(ri z) = c(ri)f(z)

with respect to z and taking the limit as z 0 givesf (ri) = c(ri).


Therefore,

f([i ]z)zj

(s + h) = f ([i ](s + h))mij = f (ri + [i ]h))mij = ([i ]h)c(ri)mij .

Thus,

det(f([i ]z)/zj )(s + h) = (1)tc(r)([]h) detM,and

Ress+h =(1)t([]h)f(s + h)N

(detM)c(r).

Also, notice that this residue remains unchanged when h is replaced by h + g with g Gt(because of the condition

j mij Nj mod exp(G)).

Theorem 10 is now an immediate consequence of the residue Theorem 11.

11. The level 2 case

We now specialize the above formula for the level 2 elliptic genus. Let H = {z C | (z) > 0}, L0 = ZZ , L = ZZ(2), G = {0, }, and let : G {1} be dened by() = 1. The divisor of the corresponding Jacobi function f is

div(f) = (0) () + (1/2) + (1/2 + ).By comparing the divisors and the 1/z2 terms in the Taylor expansions at 0, we easily concludethat

f(z)2 = (z|) (1/2 | ) = (z|) e1 .We will choose r = 1/2. Since f satises the dierential equation

(f )2 = f 4 2f 2 + (cf. [20]), we see that c(1/2)2 = f (1/2)2 = , and, with an appropriate choice of the squareroot, c(1/2) =

.

Consider rst, the case of a hypersurface X(m) CPN1 of degree m. Since we can takes = 1/2m, we have the following theorem.

Theorem 11. If N and m have same parity, then

(X(m)) =1

m

0a,b


Then, noticing that0 1 00 0 11 3 6

3 0 01 2 00 1 2

1 2 40 1 20 0 1

=1 0 00 1 00 0 12

,we easily conclude that

112

1 2 40 1 20 0 1

001

= 1/31/6

1/12

= vgenerates a cyclic subgroup of order 12 of H/G3 , so that one can take

{(a + b)v | 0 a, b < 12}

as representatives of the cosets of H/G3 . Also, notice that []v = 1, so

([](a + b)v) = (1)b .

We obtain the following theorem.

Theorem 12. For M as above,

(X(M)) =1

12 3/2

0a,b


u = z + s + h, we have

Resz=s+h

(1

f(mz)

)=

12i

dz

f(mz)

=1

2i

du

f(mu + r + mh)

=(mh)

2i

du

f(mu + r)

=(mh)2ic(r)

f(mu)du

= (mh)mc(r)

.

Theorem 13. We have

1f(mz)

= 1mc(1/2)

0a,b


The possible poles of F are 0, , and a,b with 0 a < 2m, 0 b < m. Let us compute theresidues of F at these points. First, we have

Res0(F ) = Res (F ) = 12mc(1/2)

0a


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arXiv:hep-th/0504078v3.

Vassily GorbounovDepartment of Mathematical SciencesKings CollegeUniversity of AberdeenAberdeen AB24 3UEUnited Kingdom

[email protected]

Serge OchanineDepartment of MathematicsUniversity of KentuckyLexington, KY 40506-0027USA

[email protected]

Landau--Ginzburg orbifolds and their elliptic genusRegular representationsGeneralized Jacobi functionsThe divisor of a Jacobi functionModulus and conjugationJacobi functions on the Tate curveA special caseModulus and normalization on the Tate curveElliptic genusComplete intersectionsThe level 2 caseA different approach for the level 2 caseReferences

gorbounov v., ochanine s. - mirror symmetry formulae for the elliptic genus of complete...

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