K3 Surfaces, Igusa Cusp Form and String Theory

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Toshiya Kawai

Abstract

It has recently b ecome apparent that the elliptic genera of K3 surfaces

(and their symmetric pro ducts) are intimately related to the Igusa cusp

form of weight ten. In this contribution, I survey this connection with an

emphasis on string theoretic viewp oints.

1. Intro duction

This is a survey of somewhat mysterious connections found recently

between K 3 surfaces and the Igusa cusp form of weight ten .

Gritsenko and Nikulin [GN1] following the metho d of Borcherds [Bor1]

proved a remarkable in nite pro duct formula of . The exp onents of the

factors app earing in their in nite pro duct coincide with the co ecients in

the expansion of the elliptic genus of K3 surfaces. The aim of this survey

is to try to lo ok at this coincidence more carefully.

Borcherds-typ e in nite pro ducts app ear naturally in some sort of mo d-

ular integrals asso ciated with one-lo op calculations of string theory as

p ointed out by Harvey and Mo ore [HM]. They used the metho d devel-

op ed by Dixon, Kaplunovsky and Louis (DKL) [DKL]. In sect.6, I present

a similar calculation [Kaw] for the case of our present interest. The ap-

proach taken is again that of DKL, which may now fairly be said to be

standard, but I will be more careful ab out the so-called \degenerate or-

bits" explaining the relevance of the Kronecker limit formulas.

The Igusa cusp form [Igu] made its app earance in the past in the

two-lo op calculations of critical b osonic string theory [Two]. Thus one

is naturally led to the exp ectation that K3 surfaces might be related to

b osonic string. In fact, Vafa and Witten [VW] p ointed out the connection

between K3 surfaces and b osonic string in the context of Gottsche's for-

mula [Go e]. They gave a physical derivation of the orbifold version [HH]

of Gottsche's formula. This metho d of Vafa and Witten has recently b een

extended by Dijkgraaf et al. [DMVV] for the computation of the gener-

ating function of the orbifold elliptic genera of symmetric pro ducts of K3

surfaces. In sect.7, I give some interesting interpretation of their result

2 T. Kawai

taking the relation between and the two-lo op vacuum amplitude of

b osonic string seriously.

Sects.2{5 are devoted to preliminary materials. In sect.2, we review

the de nitions of Siegel mo dular forms and Jacobi forms. In sect.3, we

summarize the fundamental prop erties of elliptic genera. In sect.4, we

investigate in detail the elliptic genus of K3 surfaces. In sect.5 we review

the in nite pro duct formula of Borcherds, Gritsenko and Nikulin.

Finally, in sect.8, we p oint out what remains to b e clari ed.

Notation.

R : the set of real numb ers.

C : the set of complex numb ers.

Z: the set of integers.

N : the set of p ositiveintegers.

N : the set of non-negativeintegers.

M(n; R ): the set of n n real matrices.

M(n; C ): the set of n n complex matrices.

M(n; Z): the set of n n integer matrices.

I : the n n identity matrix.

e[x]: exp (2 1 x).

2. Siegel mo dular forms and Jacobi forms

First I recall some fundamental materials from the theory of Siegel mo d-

ular forms [Fre] and that of Jacobi forms [EZ].

The Siegel upp er space of genus g 2 N is de ned by

H := f 2 M(g; C ) j = ; Im( ) > 0g : (2.1)

The symplectic group

0 I

Sp(2g; R) := f 2 M(2g; R) j J = J g; J = ; (2.2)

I 0

acts on H by

=(A +B)(C +D) ; (2.3)

where

A B

= 2 Sp(2g; R); A; B ; C ; D 2 M(g; R) : (2.4)

C D

K3 surfaces, Igusa cusp form and string theory 3

This action is not e ective, but Sp(2g; R)=fI g acts e ectively on H .

g g

An imp ortant subgroup of Sp(2g; R) is the Siegel mo dular group :=

Sp(2g; Z) which acts prop erly discontinuously on H .

For a function F on H ,we intro duce the weight-k slash op eration by

(F j ) ( ) := det (C +D) F( ) ; (2.5)

where k 2 Z and is as in (2.4).

De nition 2.1. Let be a nite index subgroup of (g 2 N ) and let % :

! C := C r f0g be a character. A holomorphic function F : H ! C

is cal ledamodular form of weight k (k 2 Z) with character % if F satis es

(F j ) ( ) = %( )F ( ); (8 2 ) ; (2.6)

(and for g =1 holomorphicity conditions at the cusps of .)

The space of mo dular forms is denoted by M (;%). In particular, we

g g

set M () := M (; id ). The normalized Eisenstein series E ( )( 2 H )

k 1

k k

are de ned by

E ( )=1 (n)q ; q = e[ ] ; (k 2 2N ) ; (2.7)

k k 1

n=1

where B are the Bernoulli numb ers and (n) = d . As is well-

k k

djn

C [ E ;E ]. Igusa [Igu] determined the structure of known, M ( )

4 6 1

k =0

the ring M ( ) as

k =0

C [E ; E ; ; ; ] ; (2.8) M ( )

4 6 10 12 35 2

k =0

where E and E are the genus two Eisenstein series of weight 4 and 6

4 6

resp ectively, while , and are resp ectively certain cusp forms of

10 12 35

weight 10, 12 and 35. The asterisk is to remind that the cusp form is

not algebraically indep endent with ( ) b eing expressed as a p olynomial

of the other generators.

Now we turn to the de nition of Jacobi forms. The theory of Jacobi

forms with integer weights and integer indices without characters is well

exp ounded in the monograph of Eichler and Zagier [EZ]. However, for

our purp ose, it is convenient to slightly extend the notion of Jacobi forms

so that Jacobi forms of half integral indices with characters can also be

treated. Here we follow the presentation given in [GN2].

4 T. Kawai

0 0 0 0 0 0 3

For any pair [(; );] and [( ; ); ] with (; ; ), ( ; ; ) 2 Z ,

let their pro duct be de ned by

0 0 0 0 0 0 0 0

[(; );][( ; ); ]=[(+; + ); + + ] : (2.9)

With this multiplication law the set H := f[(; );] j (; ; ) 2 Z g

b ecomes a group and is called the integer Heisenb erg group.

For A = 2 and (; ; ) 2 Z consider the matrix

0 10 1

1 0 0 a 0 b 0

B CB C

1 0 1 0 0

B CB C

[A; ; ; ]:= 2 : (2.10)

@ A@ A

0 0 1 c 0 d 0

0 0 0 1 0 0 0 1

The Jacobi group is the subgroup of de ned by

J 3

:= f[A; ; ; ] j A 2 and (; ; ) 2 Z g : (2.11)

Since

; f[A; 0; 0; 0] j A 2 g

1 1

(2.12)

J 3

H ; B f[I ;;;] j (; ; ) 2 Z g

Z 2

it follows that

n H : (2.13)

1 Z

Now one can intro duce a character of the integer Heisenb erg group

% :H !f1g by

H Z

+++ 3

% ([(; );]) = (1) ; (; ; ) 2 Z : (2.14)

Supp ose that a character % : !C is given. Then one can construct

1 1

l J

acharacter of the Jacobi group % % : !C for any integer l by

l l J

(% % )([A; ; ; ]) = % (A)% ([(; );]) ; [A; ; ; ] 2 : (2.15)

1 1 H

The Dedekind -function is de ned by

( ):=q (1 q ) ; q = e[ ] ; 2 H : (2.16)

n=1

( ) isaweight one mo dular form of with a character. This character

is denoted as : ! C . In this pap er we always assume that the

n l J

characters of the Jacobi group are of the form % : ! C where

n and l are integers.

K3 surfaces, Igusa cusp form and string theory 5

Z. A holomorphic function : De nition 2.2. Let k 2 Z and m 2

H C ! C is cal led a Jacobi form of weight k and index m with a

J J

character % : !C if the function :H !C de ned by

( ) := p (; z) ; = 2H ; p=e[]; (2.17)

satis es the same transformation law under the action of as for the

2 J J

elements of M ( ;% ) and if it has an expansion of the form

n r

(; z)= c(n; r ) q y ; q = e[ ] ; y = e[z ] ; (2.18)

n;r

4mnr 0

where n and r run respectively over the sets of rational numbers determined

by .

2 2

If we replace the condition 4mn r 0 by 4mn r > 0 in the

ab ove, is called a cusp Jacobi form, while if the condition 4mn r 0

is dropp ed is called a weak Jacobi form.

In the following we frequently use the notation

p = e[ ] ; q = e[ ] ; y = e[z ] : (2.19)

3. Elliptic genera

Lo osely sp eaking, elliptic genera [Lan1] [Wit] are the equivariant lo op-

space indices where equivariance is with resp ect to the natural circle action

on the lo op space. Physically, they are the extensions of the Witten index

in the sense that they are asso ciated with two dimensional sup ersymmetric

quantum eld theories while the ordinary Witten index is asso ciated with

a sup ersymmetric quantum mechanics.

In fact, there are several versions of elliptic genera, however what we

shall be concerned with corresp onds to the following setting. For a com-

pact Kahler manifold M , let E ! M b e a xed holomorphic vector bundle

and let T ! M be the holomorphic tangent bundle. Using the splitting

principle, the total Chern classes of E and T can be expressed as

l d

Y Y

c(E )= (1 + ) ; c(T )= (1 + ) ; (3.1)

n i

n=1

i=1

where l := rank E and d := dim M . The elliptic genus is a function on

H C de ned by

l d

Y Y

Z [M ](; z):= P (; z + ) ; (3.2)

E n

P (; )

n=1

i=1

6 T. Kawai

where

1 1

n1 1 n

12 2

y P (; z)= 1q (1 yq )(1 y q ) : (3.3)

n=1

Notice that we have P (; z) = # (; z)= ( ) by using one of the Jacobi

theta functions.

For any holomorphic vector bundle F ! M of rank m, we intro duce

the generating functions for the antisymmetric as well as symmetric p owers

of F by

m 1

^ X X

s s s s

F = t (^ F ) ; S F = t (S F ) : (3.4)

s=0 s=0

Then the elliptic genus can be expanded as

12 2

1) q y Z [M ](; z)=(

1 1

O ^ O ^

ch E E

n1 1 n

yq y q

n=1 n=1

(3.5)

1 1

O O

n n

S T S T td(M )

q q

n=1 n=1

h i

12 2

1) q y (E )+O(q) ; =(

where

s s

(E ):= (y ) (^ E ) : (3.6)

s=0

In the second equality of (3.5) wehave used the Riemann-Ro ch-Hirzebruch

theorem

n n

(E ):= (1) dim H (M; E )= ch(E ) td(M ) : (3.7)

n=0

In particular we have

m+n m;n m

(T )= (1) h y ; (3.8)

m;n=0

K3 surfaces, Igusa cusp form and string theory 7

m;n n m

where h := dim H (M; ^ T ) are the Ho dge numb ers of M . In the

sequel we will write (M ) for (T ). (M ) is essentially the -genus

y y y y

of Hirzebruch.

The elliptic genus (3.2) enjoys nice functional prop erties if appropriate

top ological conditions on E and T are met. The following summarizes

what has b een shown in [KYY][KM].

Theorem 3.1. The el liptic genus Z [M ]:H C !C is a weak Jacobi

E 1

l d l J

form of weight 0 and index l=2 with character " % : ! C if

c (E )=0 and ch (E )= ch (T).

1 2 2

Corollary 3.2. If M is a Calabi-Yau manifold, then the el liptic genus

Z [M ] : H C ! C is a weak Jacobi form of weight 0 and index d=2

T 1

d J

with character id % : !C .

Corollary 3.3. If M is an even dimensional Calabi-Yau manifold, then

the el liptic genus Z [M ]:H C !C is a weak Jacobi form of weight 0

T 1

and index d=2 without character.

In harmony with this theorem and corollaries, there is a uniform and

concrete pro cedure [KYY] [KM] to calculate the elliptic genera of the

Landau-Ginzburg orbifolds corresp onding to certain realizations of Calabi-

Yau manifolds and bundles on them.

In the rest of this pap er we shall exclusively be concerned with the

situation of Corollary 3.3. Thus the theory of Jacobi forms covered in

[EZ] is enough.

Instead of Z [M ](; z)we will simply write Z [M ](; z)oreven Z (; z)

when the manifold M is clear from the context.

4. The case of K3 surfaces

K3 surfaces are the rst interesting cases of even dimensional Kahler

manifolds of vanishing rst Chern classes for which the elliptic genus is

non-trivial. (The elliptic genus of ab elian surfaces vanishes.) So let M

be a K3 surface. Then from the knowledge of the Ho dge numb ers of K3

surfaces we obtain that

(M ) = 2+20y +2y : (4.1)

Corollary 3.3 tells us that the elliptic genus Z (; z)isaweak Jacobi form

of weight 0 and index 1 with the expansion

Z (; z)=2y +20+2y +O(q): (4.2)

8 T. Kawai

The rst task in this section is to nd explicit expressions of Z (; z).

There are two cusp Jacobi forms of index 1

E ()E (; z) E ( ) E (; z)

6 4;1 4 6;1

(; z)= ;

10;1

144

(4.3)

E () E (; z) E ( ) E (; z)

4 4;1 6 6;1

(; z)= ;

12;1

144

which are of weight 10 and 12, resp ectively [EZ] and are the rst co e-

cients in the Fourier-Jacobi expansions of the Igusa cusp forms and

, resp ectively. Here E (; z) (k 2 2N ; k 4) is the Eisenstein-Jacobi

12 k;1

series of weight k and index 1 [EZ]. Accordingly there are twoweak Jacobi

forms of index one

(; z) (; z)

10;1 12;1

e e

(; z)= ; (; z)= ; (4.4)

2;1 0;1

( ) ( )

whose weights are resp ectively 2 and 0. Here we have intro duced the

discriminant

3 2

E () E ()

4 6

( ):= = ( ) : (4.5)

1728

Now comparing the expansions of Z and we nd that

0;1

Z (; z)=2 (; z) : (4.6)

0;1

This is one way of expressing the elliptic genus of K3 surfaces.

To nd another expression, we notice the following identity proved in

[EZ]:

(; z) (; z)

12;1 0;1

= =12}(; z) ; (4.7)

(; z)

10;1

(; z)

2;1

where the Weierstra }-function is de ned as

8 9

> >

< =

1 1 1 1

+ : (4.8) }(; z):=

2 2 2

> >

z (z ! ) !

(2 1)

: ;

! 2Z+Z

! 6=0

Since we use several prop erties of the }-function in this pap er we summa-

rize them b elow.

K3 surfaces, Igusa cusp form and string theory 9

The }-function satis es the nonlinear di erential equation

1 1

2 3

fD }(z )g =4}(z) E }(z )+ E

y 4 6

12 216

(4.9)

=4(}(z)e )(}(z ) e )(}(z ) e ) ;

1 2 3

. The ro ots e , where we have intro duced the Euler derivative D := y

1 y

e , e are related with the half p erio ds

2 3

1 +1

! = ; ! = ; ! = ; (4.10)

1 2 3

2 2 2

by e = }(! ) ( =1;2;3) and satisfy

e + e + e =0;

1 2 3

E ; e e +e e +e e =

4 1 2 2 3 3 1

(4.11)

e e e = E :

1 2 3 6

864

Furthermore we have useful identities

2 2 2 2

E ; (e e ) =3(e +e +e )=

4 1 2 3

(;; )2S

2 3 3 3

(e e ) e = 3(e + e + e )= E ; (4.12)

1 2 3 6

(;; )2S

2 2 4 4 4

E ; (e e ) e = e + e + e =

4 1 2 3

1152

(;; )2S

where S := f(1; 2; 3); (2; 3; 1); (3; 1; 2)g.

If we de ne the theta functions with characteristics by

p p

# (; z):= exp 1 (n + a) +2 1(n + a)(z + b) ;

n2Z

(4.13)

then the Jacobi theta functions are expressed as

# (; z)=# (; z) ; (; z) ; # (; z)=#

1 2

(4.14)

0 0

# (; z)=# (; z) ; # (; z)=# (; z) :

3 4

0 2

10 T. Kawai

The ro ots can b e written in terms of theta constants as

4 4

e = # (0) + # (0) ;

1 3 4

4 4

# (0) # (0) e = ; (4.15)

2 4 2

4 4

# (0) + # (0) : e =

2 3 3

We also recall that

0 3

# (; 0) = 2( ) ; (4.16)

where the prime represents the derivative with resp ect to the second vari-

able.

Wenow intro duce the function

p p

# (; z) # (; z)

1 1

=2 ; (4.17) 1 1 K (; z):=

( ) # (; 0)

which plays an imp ortant role later in this pap er. This function is related

to the weak Jacobi form by

2;1

(; z)=K(; z) : (4.18)

2;1

One of the highlights of the theory of Jacobi forms is the following

structure theorem:

Theorem 4.1. (Eichler-Zagier [EZ], Feingold-Frenkel [FF]). The bigraded

ring of al l the weak Jacobi forms of even weight is a polynomial algebra

e e

over C [ E ;E ] generated by and .

4 6 0;1 1;2

Corollary 4.2. If we assign weights 4, 6 and 2 respectively to E , E and

4 6

}, any weak Jacobi form of weight 2k (k 2 N ) and index m (m 2 N )

0 0

can be expressed as

p (E ;E ;})K ;

2k +2m 4 6

where p (E ;E ;}) is a weight 2k +2m homogeneous polynomial in

2k +2m 4 6

E , E and }.

4 6

Now it trivially follows from (4.6), (4.7) and (4.18) that

Z (; z)=24}(; z) K (; z) ; (4.19)

which is of course in accordance with Corollary 4.2. The expression (4.19)

has an interesting interpretation as we shall see.

K3 surfaces, Igusa cusp form and string theory 11

Since the }-function is related to the Jacobi theta functions through

the identities [EMOT]

# (z )

K (z ) ; ( =1;2;3) ; }(z )=e +

# (0)

(4.20)

1 # (z )

= K (z ) ;

3 # (0)

the elliptic genus can also b e expressed as

( )

# (z )

Z (; z)=24 + e K (z ) ; ( =1;2;3) ; (4.21)

# (0)

# (z )

Z (; z)=8 : (4.22)

# (0)

These formulas are related to the Kummer surface which is realized by

resolving the singularities of the orbifold C =Z .

As it easily follows from (4.3), (4.7) and (4.18) we have the formulas

E = E } E K ;

4;1 4 6

(4.23)

2 2

E = E } E K ;

6;1 6 4

which are again in accordance with Corollary 4.2. There is another way

of expressing the Eisenstein-Jacobi series E and E which reads

4;1 6;1

2 6

# (z ) # (0) ; E =

+1 +1 4;1

(4.24)

2 6

}(z + ! )# (z ) # (0) : E =6

+1 +1 6;1

It is not dicult to check that the two expressions indeed coincide by use

of formulas

8 2

# (0) = 16(e e ) ; (; ; ) 2S;

(4.25)

(e e )(e e )

}(z+! )=e + ; (; ; ) 2S;

}(z ) e

The formula given in [KYY] corresp onds to the choice =2.

12 T. Kawai

as well as (4.20) and (4.12). The second expressions (4.24) app ear in

certain compacti cations of heterotic string.

The theta functions asso ciated with the integrable representations at

(1)

level m (m 2 N ) of the ane Lie algebra A are de ned by

r r

4m 2

(; z):= y ; 2 Z mo d 2mZ : (4.26) q

r2Z

rmo d 2m

Any (weak) Jacobi form (; z) of index m can be expressed in terms of

these theta functions as [EZ]

(; z)= h ( ) (; 2z) ; (4.27)

(mod2m)

for some set of functions h ( ).

The elliptic genus of K3 surfaces can b e expanded as

2 n r

Z (; z)= c(4n r )q y ; (4.28)

n2N ;r2Z

where we used the fact that the co ecients dep end only on the combina-

tion 4n r . By de ning

X X

N N

4 4

h ( ):= c(N )q ; h ( ):= c(N )q ; (4.29)

0 1

N0 mo d 4 N1 mo d 4

the elliptic genus can thus be written as

Z (; z)=h () (; 2z)+h () (; 2z) : (4.30)

0 0;1 1 1;1

The q -expansions of the functions h ( ) and h ( ) are given by

0 1

2 3 4

h ( ) = 20 + 216 q + 1616 q + 8032 q + 33048 q + ;

(4.31)

2 3 4

q h ()=2128 q 1026 q 5504 q 23550 q :

Note in particular that

c(0) = 20 ; c(1) = 2 ; c(N )=0 if N <1: (4.32)

There are explicit expressions of h and h in terms of theta constants:

0 1

4 5

10 (0) (0) 2 (0)

0;1 1;1 1;1

; h =

(4.33)

5 4

2 (0) 10 (0) (0)

0;1 0;1 1;1

h = :

K3 surfaces, Igusa cusp form and string theory 13

This can b e demonstrated by applying the formulas

# (z ) = (0) (2z ) (0) (2z ) ;

1 1;1 0;1 0;1 1;1

# (z ) = (0) (2z )+ (0) (2z ) ;

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

(4.34)

# (z ) = (0) (2z )+ (0) (2z ) ;

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

# (z ) = (0) (2z ) (0) (2z ) :

4 0;1 0;1 1;1 1;1

5. Pro duct formulas of Borcherds, Gritsenko and Nikulin

Let V be a 2 + n-dimensional real vector space and let ( ; ): V V !R

be a non-degenerate bilinear form of signature (2;n) on V . The orthog-

onal group O (V ) consists of four connected comp onents. The index two

subgroup consisting of the elements of spinor norm 1 in O (V ) is denoted

as O (V ). The connected comp onent that contains the identity is denoted

+ + +

O (V )=fI g. Consider an orthogonal by SO (V ) so that SO (V )

2+n

decomp osition

V = V V ; (5.1)

such that ( ; )j is p ositive de nite and ( ; )j is negative de nite. Let

V V

K be the subgroup of SO (V ) which resp ects this decomp osition. K is

a maximal compact subgroup of SO (V ). Thus inequivalent decomp osi-

tions (5.1) are parametrized by SO (V )=K which is a hermitian symmetric

space of typ e IV in the classi cation of E. Cartan. Let D be one of the

two connected comp onents of

f[w ] 2 P(V C ) j (w; w)= 0 and (w; w ) > 0g ; (5.2)

where [w ] is a line corresp onding to w 2 V C r f0g. There exists an

D. To see this, let w =Rew and w =Imw isomorphism SO (V )=K

1 2

for [w ] 2D. Then the conditions in (5.2) are equivalent to

(w ;w )= (w ;w ) > 0 and (w ;w )= 0: (5.3)

1 1 2 2 1 2

Thus [w ] 2 D determines a real two dimensional (p ositive) vector space

[w ]

spanned by the (oriented) normalized orthogonal basis fw^ ; w^ g where V

1 2

w^ := w = w ; (i =1;2) : (5.4)

i i i

(w; w )

(w ;w )

i i

To b e sp eci c, assume in the following that V is spanned by vevectors

e , e , f , f and with their only non-vanishing inner pro ducts given by

1 2 1 2

: (5.5) (e ;e )=(e ;e )=1=(f ;f )= (f ;f ); (;)=

1 2 2 1 1 2 2 1 2

14 T. Kawai

Then consider the lattice

(1) (2)

L := H H Z; (5.6)

(1) (2)

where H = Ze + Ze and H = Zf + Zf , together with its sublattice

1 2 1 2

(2)

:=H Z: (5.7)

Obviously V = L R has signature (2; 3). The cone fx 2 R j (x; x) >

0g consists of two connected comp onents. We x one of them (the future

light-cone) and denote it by C (). Then the tube domain

H := R + 1C () C ; (5.8)

gives a realization of D with the isomorphism H !D given by

(; )

2H ![w]2D; w =e e + : (5.9)

1 2

In the present case, we have a further isomorphism H !H given by

= 2H ! =f + f +2z 2H: (5.10)

2 1 2

In what follows we will freely use these isomorphisms. Thus, for instance

we have

(w; w) = 2(Im( ); Im( )) = 4 det Im( ) > 0 : (5.11)

The symplectic group Sp(4; R ) acts on H while the group O (V ) acts

SO (V ) on the domain D . We have an isomorphism Sp(4; R )=fI g

compatible with the isomorphism H D . If we de ne O (L)=O(L)\

+ + +

O (V) and SO (L)=O(L)\SO (V ) where O (L)isthe automorphism

SO (L) = group of L, then there exists an isomorphism =fI g

2 4

(L)=fI g. (An explicit demonstration of this for the dual lattice L O

can b e found in [GN1].) Hence one can regard Siegel mo dular forms on H

with resp ect to (a nite index subgroup of ) as automorphic forms on

the typ e IV domain D with resp ect to (a nite index subgroup of ) O (L).

For the pro duct of all the genus two theta functions with even charac-

teristics denoted as ( ), Gritsenko and Nikulin [GN1] proved an in nite

pro duct formula

1 1 1

k l r c(4klr )=2

2 2 2

( ) = p q y (1 p q y ) ; (5.12)

(k;l;r)>0

K3 surfaces, Igusa cusp form and string theory 15

where p = e[ ], q = e[ ] and y = e[z ] as b efore and c(N ) are the co e-

cients app earing in (4.28). The pro duct in (5.12) is taken over all triplets

of integers (k; l; r) satisfying the conditions (k; l; r) > 0 where

(k; l; r) > 0 () either k>0; l0;

or k 0; l>0;

or k = l =0; r<0:

Since the relation ( ) = ( ) holds with a suitable choice of the

10 5

multiplicative constant, we have an in nite pro duct representation of the

Igusa cusp form of weight ten:

k l r c(4klr )

( ) = pq y (1 p q y ) : (5.13)

(k;l;r)>0

In order to prove this in nite pro duct representation Gritsenko and

Nikulin used the lifting pro cedure intro duced by Borcherds [Bor1]. The

Borcherds-lifting makes use of the Hecke op erators for the Jacobi group

as essential ingredients.

The Hecke op erators V (l 2 N ) on a function : H C ! C is de ned

` 1

[EZ] by

(j V )(; z):=

k;m `

`mcz `z a + b

k k1

(c + d) e ` ; ; (5.14)

c + d c + d c + d

2 nM

where M := fA 2 M(2; Z) j det A = `g. By cho osing the standard set of

representatives we may take [Ap o]

a b

a; d 2 N , ad = ` and b =0;1;::: ;d 1. : n M =

1 `

0 d

(5.15)

Therefore we have more explicitly

a + b

k1 k

(j V )(; z)=` d ;az : (5.16)

k;m `

ad=` b=0

Using these Hecke op erators, a straightforward calculation leads to the

identity

Y X

k l r c(4klr ) `

(1 p q y ) = exp p (Z j V )(; z) : (5.17)

0;1 `

k>0 `=1

(k;l;r)>0

16 T. Kawai

Consequently we obtain

( ) = p (; z) exp p (Z j V )(; z) ; (5.18)

10 0;1 `

`=1

where we have set

l r c(r )

(; z):=qy (1 q y ) ; (5.19)

(0;l ;r )>0

which is a Jacobi form of weight c(0)=2=10and index 1 as is clear from

an alternative expression

(; z)=()K(; z) : (5.20)

The RHS of (5.18) should b e compared with the equation given on p.192

in [Bor1]. The factor (; z) (or its square ro ot for ) corresp onds to the

\denominator" of the ane vector system in [Bor1].

We will see in the following that the decomp osition into factors (5.18)

p ossesses some interesting interpretations.

6. Mo dular integral and Igusa cusp form

[w ] [w]

For a given [w ] 2D, let V b e the orthogonal complementof V in V

[w ] [w ] [w ]

so that V = V V . For each 2 L, let denote the pro jection

[w ] [w ]

of onto V . From the obvious relation = (; w^ )^w it follows

i i

i=1

that

2j(; w )j

[w ] [w]

; = : (6.1)

+ +

(w; w)

We divide the lattice L into two sublattices

(1) (2)

L := H H 2Z and L := L + ; (6.2)

0 1 0

so that L = L [ L .

0 1

For a xed = 2H we will consider the integral

X X

d 1

h () +h () R (; ; ; [w ]) ; (6.3) I ( ) =

0 1

2 Im()

2L 2L

0 1

P nH

1 1

K3 surfaces, Igusa cusp form and string theory 17

where d = d Re() d Im(), P := =fI g and

1 1 2

1 1

[w ] [w] [w] [w]

R (; ; ; [w ]) :=e ; ;

+ +

2 2

(6.4)

[w ] [w]

=e (; ) exp 2 Im() ; ;

+ +

It should b e understo o d that we are here implicitly using the isomorphisms

given by (5.9) and (5.10). As a matter of fact, the integral (6.3) is divergent

and we need a regularization for it to b e meaningful.

Similar integrals have app eared in several physical contexts of string

theory. Dixon, Kaplunovsky and Louis [DKL] were the rst to explicitly

evaluate this typ e of integral. Their metho d was extended in the work of

Harvey and Mo ore [HM] where the connections to in nite pro ducts a la

Borcherds [Bor1] were p ointed out. A further extension can be found in

[Bor2].

The particular case of (6.3) was treated in [Kaw]. The upshot of the

calculation is

Theorem 6.1. The integral (6.3) with an appropriate regularization is

equal to

10 2

log Y j ( )j ; (6.5)

where Y := det Im( ).

Below we sketch the outline of the pro of. First parametrize 2 L in

terms of ve integers m , m , n , n and r as

1 2 1 2

= m e + m e + n f + n f r : (6.6)

1 1 2 2 1 1 2 2

Then, we nd that

R (; ; ; [w ]) = e[(m m n n )]

1 2 1 2

Im()

exp m + n + n + m ( z )+rz : (6.7)

2 2 1 1

Substituting this into (6.3) and p erforming the Poisson resummations over

18 T. Kawai

m and n ,we arriveat the result

2 2

X X X

d Y

I ( ) = h () +h ()

0 1

2Im() Im()

b22Z b22Z+1

G2M(2;Z)

P nH

1 1

e[r =4]e[ det G] exp(F [G])

Y d

2Im() Im()

P nH

1 1

X X

c(N )e[(N + r )=4]e[ det G] exp(F [G]) ;

N;r2Z

G2M(2;Z)

(6.8)

where F [G] assumes a complicated expression which I refrain from writing

down. Notice that we have extended the de nition of c(N ) by assigning

c(N )=0 if N 1 (mo d 4) or N 2 (mo d 4). The crucial p oint is that

the right multiplication of any element of P on G 2 M(2; Z) can be

absorb ed to the action of the same element of P on . Actually this is

not manifest in (6.8) but can be seen by using the mo dular prop erties of

h and h . Thus rather than summing over M(2; Z) and integrating over

0 1

the fundamental region P nH , one may,by restricting oneself to the sum

1 1

over the representatives in the orbit decomp osition of M(2; Z) with resp ect

the right action of P ,integrate over more a ordable regions larger than

the fundamental region.

Explicitly the orbit decomp osition is given by

M(2; Z)= O P [O P [O P ; (6.9)

0 1 1 1 2 1

where

0 0

O = ;

0 0

0 m

O = m; n 2 Z; (m; n) 6=(0;0) ;

(6.10)

0 n

[ [

1 0

O = M = [ Q(M = ) ; Q = ;

2 ` 1 ` 1

0 1

`2N `2N

with

a b

M = = a; d 2 N , ad = ` and b =0;1;::: ;a 1. :

` 1

0 d (6.11)

K3 surfaces, Igusa cusp form and string theory 19

Accordingly one may consider the decomp osition

I ( ) = I ( ) + I ( ) + I ( ) : (6.12)

0 1 2

Before evaluating each contribution in this, some intuitive remarks are

in order. We started from (6.3) and reached (6.8) by the Poisson resum-

mations. Physically this means switching from the Hamiltonian picture to

the Lagrangian or the path integral picture of a sigma mo del with a single

Wilson line whose worldsheet is the elliptic curve E := C =(Z + Z) and

the target space is again the elliptic curve E := C =(Z + Z). The target

elliptic curve E is further parametrized by the complexi ed Kaheler pa-

rameter . Thus the factor e[ det G] app earing in (6.8) can b e regarded

as the usual one counting the instanton numb er.

The contribution I ( ) corresp onds to the non-zero degree maps whose

images wrap prop erly the target elliptic curve p ossibly many times. One

may intuitively exp ect that if the area of the image is very large, i.e.

the image wraps the target elliptic curve large number of times, its con-

tribution is exp onentially suppressed and I ( ) is not divergent. I ( )

2 0

is asso ciated with the map whose image shrinks to a p oint in E . This

either should not give rise to a divergence. On the other hand, I ( ) cor-

resp onds to the maps whose images degenerate to non-trivial 1-cycles in

E . These cycles can be arbitrarily long and wrap the basis of H (E ; Z)

in nitely many number of times without any costs since they have zero

areas. Consequently they can be p otential sources of divergence.

Now we turn to the actual calculations. Since O P = O , we have

0 1 0

to integrate over the fundamental region to calculate I ( ). However this

is not dicult and we obtain that

I ( ) = (c(0)+2c(1)) : (6.13)

6Im()

As for I ( ), we note that P acts e ectively on each element of O .

2 1 2

Thus we can extend the integral region from the fundamental region to

the whole upp er plane H . The calculation is more or less straightforward

and the result reads

10 k l r c(4klr )

I ( ) = log Y (1 p q y ) : (6.14)

k>0

(k;l;r)>0

The expression inside the absolute value is precisely the one related to the

Hecke op erators in the previous section. This should not b e to o surprising

since determining a holomorphic map of degree ` from E to E is equiva-

lent to determining a sublattice of index ` in the target lattice Z + Z C

20 T. Kawai

as the image of the worldsheet lattice Z + Z. However the set of all the

sublattices of index ` in Z + Z is isomorphic to n M and the Hecke

1 `

op erator V asso ciates with the lattice Z + Z the sublattices of index ` in

Z + Z , with multiplicity one.

Nowwe turn to the subtle case of I ( ). Notice rst that each element

1 k

of O is xed by the right action of f j k 2 Zg P . Thus

1 1

the integral region can be extended only to the strip of width one in

H . Furthermore, as mentioned in the ab ove, I ( ) must be regularized

1 1

somehow. As a regularized expression of I ( ), we adopt, for a complex

number s with Re(s) > 1, the integral

Z Z

1=2 1

Y d Im()

I ( ;s)= dRe()

1+s

2Im() Im()

1=2 0

X X

0 m

c(N )e[(N + r )=4] exp F

0 n

m;n

N;r2Z

X X

Y dIm()

0 m

= c(r ) exp F ;

0 n

1+s

2Im() Im()

r =0;1 m;n

(6.15)

where means taking the sum over all pair of integers (m; n) except

m;n

for (m; n)=(0;0) and

2 1r Y

0 m

jm + n j + Im(z (m + n)) : F =

0 n

Im()Im() Im( )

(6.16)

To rewrite this expression we need to recall the de nitions of some well-

known functions in numb er theory. Weintro duce two kinds of them known

as the Epstein zeta functions or alternatively as the non-holomorphic

Eisenstein series. For 2 H and s 2 C with Re(s) > 1, the rst zeta

function is de ned by

Im( )

E (; s)= : (6.17)

jm + n j

m;n

Similarly, for 2 H and s 2 C with Re(s) > 1, the second zeta function

is de ned for real numb ers and which are not b oth integers by

Im( )

e[m + n ] E ( ; ; ; s)= : (6.18)

jm + n j m;n

K3 surfaces, Igusa cusp form and string theory 21

Now exchanging the sums and the integral in the last expression of (6.15),

which is p ossible by virtue of our regularization, and recalling the de ni-

s1 t

dt t e , (Re(s) > 1) we obtain tion of the Gamma function (s)=

that

(s) Im( )

I ( ;s)= c(0)E (; s)+ 2c(1)E ( ; ; ; s) ;

2 Y

(6.19)

with

Im(z ) Im(z )

= ; = : (6.20)

Im( ) Im( )

In order to evaluate this expression near s = 1 we recall the Kronecker

limit formulas [Sie][Lan2].

First limit formula. E (; s) can be analytical ly continued into a func-

tion regular for Re(s) > except for a simple pole at s = 1. It has the

expansion

+2 log 2 log Im( ) j ( )j + O (s 1) ; E (; s)=

(6.21)

1 1

where = lim 1+ + + log n is Euler's constant.

2 n

n!1

Second limit formula. E ( ; ; ; s) can be analytical ly continued into

and a function regular for Re(s) >

# (; )

: (6.22) e[ =2] E ( ; ; ; 1) = log

( )

Thus analytically continuing I ( ;s) and using (s)= 1 (s1) +

1 E

O ((s 1) ), we obtain the expansion

1 1

I ( ;s)= c(0) + log(4 ) log Y logj ( )j

2(s 1) 2 2

# (; z) Im(z )

+ c(1) log +2 + O (s 1) : (6.23)

( ) Im( )

Using this the \degree zero" part amounts to

reg

10 18 2

I ( ) + I ( ) = log Y e[ ] ( ) # (; z)

0 1

(6.24)

= log Y p (; z) ;

22 T. Kawai

where we have de ned

1 c(0)

reg

I ( ) := lim I ( ;s) + log 4 : (6.25)

1 E

s!1

2 s 1

reg

Finally,if we de ne the regularization of (6.3) by I ( ) := I ( ) +

reg

I ( ) + I ( ), it follows that

reg 10 k l r c(4klr )

I ( ) = log Y pq y (1 p q y )

(6.26)

(k;l;r)>0

10 2

= log Y j ( )j :

This is the desired result.

7. Symmetric pro ducts and Igusa cusp form

(m) m

Supp ose that M is a smo oth pro jective surface. Let M := S M denote

th [m]

the m symmetric pro duct of M 's and let M := Hilb (M ) denote the

Hilb ert scheme of zero dimensional subschemes of length m. The Hilb ert

[m] (m)

scheme M is a smo oth resolution of M . Gottsche [Go e] proved that

[m]

the generating function of the Euler characteristics of M 's is given by

1 1

X Y

m [m]

: (7.1) p (M )=

k (M )

(1 p )

m=0

k=1

This, in particular, means that if M is a K3 surface,

m [m]

p (M )= : (7.2)

( )

m=0

This formula of Gottsche has aroused much interest [HH][VW][Nak].

In particular, Vafa and Witten [VW] gave a physical derivation of the

orbifold formula [HH] which reads for a K3 surface M as

m (m)

p (M )= : (7.3)

( )

m=0

(m)

In this formula (M ) should b e understo o d as the orbifold Euler char-

(m)

acteristic of the symmetric pro duct M where the orbifold pro cedure is

with resp ect to the symmetric group S . Vafa and Witten, based on this

formula, p ointed out a mysterious connection between K3 surfaces and

b osonic string. Subsequently their observation played an imp ortant role

K3 surfaces, Igusa cusp form and string theory 23

in understanding the dualitybetween heterotic string compacti ed on T

and typ e I IA string compacti ed on a K3 surface. See for instance [BSV]

[YZ].

In the sequel M is assumed to b e a K3 surface. The pro cedure of Vafa

and Witten for the orbifold Euler characteristic of the symmetric pro ducts

was extended for the orbifold elliptic genera of the symmetric pro ducts by

Dijkgraaf, Mo ore, E. Verlinde and H. Verlinde [DMVV] who proved that

X Y

m (m)

p Z [M ](; z)= ; (7.4)

k l r c(4klr )

(1 p q y )

m=0

k>0

(k;l;r)>0

(m)

where again Z [M ] should be understo o d as the orbifold elliptic genus

(m)

of M . With the use of (5.17) this can also b e written as

1 1

X X

` m (m)

p (; z) ; (7.5) p Z [M ](; z) = exp

m=0

`=1

where we have set

:= Z [M ] V ; (` 2 N ) : (7.6)

` `

0;1

Consequently, we see that

(m)

Z [M ]= s ( ;::: ; ); (m =0;1;2;:::); (7.7)

m 1 m

where s are the Schur p olynomials. For instance,

(1)

Z [M ](; z)=s ( )= =Z(; z) ;

1 1 1

(2)

+ Z [M ](; z)=s ( ; )=

1 2 2 1 2

(7.8)

+1 1

;z + Z ;z ; = Z(; z) + Z (2; 2z)+Z

2 2 2

where Z (; z)=Z[M](; z).

[m]

The Hilb ert scheme Z [M ] of a K3 surface M is an 2m dimensional

Calabi-Yau manifold [Bea]. Thus it is tempting to think that one can

(m) [m]

replace Z [M ] by Z [M ] in the ab ove equations. This is true at least

at the level of -genera . To see this, we take the limit q ! 0 in (7.4) or

On the contrary to the misleading impression given in [KYY] and [KM], the infor-

mation of the -genus is not enough to determine the elliptic genus completely if the

index of the elliptic genus as a weak Jacobi form, i.e. the dimension of the manifold or

the rank of the bundle is suciently large. I am grateful to V. Gritsenko for discussion

on this p oint.

24 T. Kawai

(7.5) to obtain that

1 1

X Y

m (m)

(p=y ) (M )=

k 20 k 2 k 1 2

(1 p ) (1 p y ) (1 p y )

m=0

k=1

(7.9)

1 (M )

: = exp (p=y )

a 1 p

a=1

(m) [m]

That this result with the replacementof (M )by (M ) holds was

y y

proved in [GS][Che].

String theorists are very familiar to the Igusa cusp form since it

app ears in the two-lo op vacuum amplitude of b osonic string [Two]. Now

I wish to provide an interpretation of (7.4) from this viewp oint. Eq.(7.4)

can be more suggestively rewritten as

(m) 2

1 Z [M ](; z)K (; z)

= p ; (7.10)

( ) ( )

m=0

where we used (5.18), (5.20) and (7.5). The LHS is the chiral part of the

two-lo op vacuum amplitude of b osonic string. Since b eing expanded in

powers of p, the RHS should represent the contributions of the one-lo op

two-p oint functions of the physical states at various mass levels according

to the factorization principle when one of the two handles of the genus

two Riemann surface is very thin and long.

As is well-known the physical states of b osonic string are classi ed as

25;1

follows. Let k 2 R (m 2 N ) be the momenta satisfying the mass

m 0

shell conditions

(k ;k )

m m

+ m =1; (7.11)

1 25;1

and let X (z ) (z 2 P ) denote chiral free b osons taking values in R .

The physical states corresp onding to the momenta k has a (mass) of

x 25 ;1

2m 2 . Let e ;::: ;e b e the basis of R whose Gramm matrix is given

1 26

25;1

by := diag(+1;::: ;+1; 1). The comp onents of vectors in R with

resp ect to this basis are represented by Greek sup erscripts and lowered by

the metric as usual. The vertex op erator emitting the tachyon is given

1(k ;X (z ))

: : (7.12) : e

Here we are working in the convention of op en string theory.

K3 surfaces, Igusa cusp form and string theory 25

The vertex op erators corresp onding to the emissions of massless and mas-

sive states are

1(k ;X (z ))

: U (v ;X)(z )e : ; =1;2;::: ;p (m) ; (7.13)

m 24

where q=( )= p (n)q and

n=0

m n

X X X Y

j j

U (v ;X)(z ):= 1 @ X v (z ) :

;::: ;

n=1 ;::: ;

m 1;::: ;m 1 j =1

1 n

m ++m =m

(7.14)

The p olarization tensors v are determined so that the vertex op era-

;::: ;

tors (7.13) have conformal weights 1. They are totally symmetric and are

transverse to the momentum k , i.e. (k ) v =0.

m m

; ::: ;

Thus the factorization principle suggests

(m) 2

Z [M ](; z)K (; z) =

(2 1)

p (m)

D E

p p

1(k ;X (z )) 1(k ;X (0))

m m

: U (v ;X)(z )e :: U (v ;X)(0)e : :

m m

(7.15)

For the evaluation of this expression, rst notice that the function

1), where K (; z) is as de ned in (4.17), is the K (; z):=K(; z)=(2

prime form [Mum] b ehaving K (; z) z as z ! 0. Therefore we have

D E

p p

1(k ;X (z )) 1(k ;X (0)) (k ;k ) 2m2

m m m m

^ ^

: e :: e : = K (; z) = K (; z) :

(7.16)

Secondly, note that due to the transversality of the p olarization tensors

v to the momentum k there are no contractions between U (v ;X)

m m

1(k ;X )

and e . Thus the only remaining problem is to takeinto account

the contraction between U (v ;X)(z ) and U (v ;X)(0). This is rather

m m

subtle since we are dealing with chiral b osons, but we adopt the following

rule

h@X (z)@X (0)i! }^(; z) ; (7.17)

where }^(; z) := (2 1) }(; z) and we remind that the }-function

satis es

1 1

E ( )= + : (7.18) }(; z)=D log K (; z)+

2 y

12 (log y )

26 T. Kawai

As an example, consider the massless case (m = 1). There are p (1) =

24 p olarization (contravariant) vectors v satisfying (v ;v ) = . The

contribution from the two-p oint functions of massless states should b e

24 26

E D

X X

p p

1(k ;X (z )) 1(k ;X (0))

1 1

: v @X (z):e : v @X (z):e

=1 ; =1

(k ;k )

1 1

=24^}(; z)K (; z) =24^}(; z) :

(7.19)

Hence we have

(1) 2 2

Z [M ](; z)= 24} ^(; z)K (; z) =24}(; z) K (; z) :

(2 1)

(7.20)

However this is precisely the expression (4.19) we encountered b efore!

As for massive states we cannot b e as explicit as the massless case since

we have to determine the p olarization tensors case by case. However we

can make some general statement. The contraction b etween U (v ;X)(z )

and U (v ;X)(0) yields the expression containing the following terms

(D }) ; (7.21)

where a are nonnegative integers such that (i +2)a = 2m. Using

i i

the prop erties of the }-function (4.9) and

D E ; (7.22) } =6}

y 4

D } =12}D }; (7.23)

y y

as well as taking into account the fact that ia is even, it is easy

to deduce that (7.21) is a weight 2m homogeneous p olynomial of E , E

4 6

(m)

and }. This is the desired result since the elliptic genera Z [M ] (and

[m]

Z [M ]) are weak Jacobi forms of weight 0 and index m and hence they

must be of the form p (E ;E ;})K according to Corollary 4.2.

2m 4 6

Ihave calculated several cases explicitly to nd the results:

(m)

m Z [M ](; z)

1 24 }K

2 4

2 (324 } + E ) K

64 10

3 6

3 (3200 } + E } + E ) K

4 6

3 27

1329 45 199

4 2 8

4 (25650 } + E } + E } + E ) K

4 6 4

4 4 384

5 3 2 10

5 (176256 } + 3720 E } + 186 E } +17E }+ E E ) K

4 6 4 4 6 72

K3 surfaces, Igusa cusp form and string theory 27

For these lower values of m's the information of -genera was sucient

to determine the elliptic genera and hence we can without doubt replace

(m) [m] m 2m

M by M in the table. Notice that the co ecientof} K coincides

m 2m

with p (m). This must b e so since } K tends to 1 as z ! 0 while the

rest of terms vanish in this limit.

(m)

To summarize, the el liptic genus Z [M ] is equal to the sum of the

(chiral) two-point functions of al l the physical states having the mass

squared 2m 2 on an el liptic curve, divided by the (chiral) tachyon two-

point function on the same el liptic curve.

Finally we recall that in addition to the relation b etween b osonic string

and K3 surfaces Vafa and Witten [VW] p ointed out the one between su-

p erstring and ab elian surfaces. The vacuum two-lo op as well as one-lo op

amplitude of sup erstring vanishes and there is no analog of . This is in

parallel with the fact that the elliptic genus of ab elian surfaces vanishes

identically.

8. Discussion

Perhaps one of the still op en problems regarding the phenomena we ob-

served in this pap er is to clarify what kind of algebras are b ehind them

and what kind of roles these algebras are playing. Gritsenko and Nikulin

asso ciated with or more precisely , a generalized Kac-Mo o dy sup er-

10 5

algebra whose real simple ro ots are characterized by the Cartan matrix

0 1

2 2 2

@ A

2 2 2

; (8.1)

2 2 2

As a submatrix, this Cartan matrix contains (in two p ossible ways) that

(1)

of A

2 2

: (8.2)

2 2

(1)

This ane Lie algebra A has a very clear physical origin. If M is a

[m]

K3 surface M is a hyp erkahler manifold [Bea] of complex dimension

[m]

2m. The elliptic genus Z [M ] is asso ciated with the sigma mo del with

[m]

M as the target space. This sigma mo del is governed by the N = 4

sup erconformal algebra. This N =4 sup erconformal algebra has a Vira-

(1)

soro central charge 6m and contains A at level m as a subalgebra. This

[m]

is the physical origin of why the elliptic genus Z [M ] as a weak Jacobi

(1)

form of index m can b e expanded in terms of the A theta functions as in 1

28 T. Kawai

(4.27). Thus the generalized Kac-Mo o dy sup eralgebra contains a tower of

(1)

integrable representations of A at arbitrary levels. This is reminiscent

of the situation in [FF].

On the other hand, if we take the viewp oint in the last section seri-

ously, N = 4 sup erconformal invariance must somehow b e realized at each

mass level of b osonic string theory. This exp ectation clearly needs further

investigation.

As is well-known, the tachyon vertex op erators of b osonic string are

utilized in the Frenkel-Kac-Segal constructions for the basic representa-

tions of (simply-laced) ane Lie algebras. It has long b een anticipated by

many that the massless as well as massivevertex op erators might also play

roles in realizing some larger algebraic structures. Although the situation

discussed in this pap er may not b e the right one for this anticipation since,

for instance, wewere dealing with the uncompacti ed b osonic string, it is

still interesting to observe that with a genus two Riemann surface is as-

so ciated the big algebraic structure of the Gritsenko-Nikulin algebra and

if we pinch one of the handle of the Riemann surface and use the fac-

torization principle we see that vertex op erators of all mass levels app ear

naturally.

Acknow ledgments. I should like to express my gratitude to the Taniguchi

foundation and the organizers of this symp osium. I esp ecially thank K.

Saito for discussion at various o ccasions. I am also grateful to V. Gritsenko

for clarifying corresp ondence.

References

[Ap o] T.M. Ap ostol, Modular Functions and Dirichlet Series in Number

Theory, Graduate Texts in Mathematics 41, Springer-Verlag, 1976.

[Bea] A. Beauville, Varietes kahleriennes dont la premiere classe de Chern

est nul le, J. Di . Geom. 18 (1983), 755{782.

[Bor1] R.E. Borcherds, Automorphic forms on O (R) and in nite

s+2;2

products,Invent. Math. 120 (1995), 161{213.

[Bor2] R.E. Borcherds, Automorphic forms with singularities on Grass-

mannians, alg-geom/9609022.

[BSV] M. Bershadsky, V. Sadov and C. Vafa, D-Branes and Topological

Field Theories, Nucl. Phys. B463 (1996) 420{434.

[Che] J. Cheah, On the cohomology of Hilbert schemes of points, J. Alg.

Geom. 5 (1996), 479{511.

K3 surfaces and Igusa cusp form 29

[DKL] L.J. Dixon, V. Kaplunovsky and J. Louis, Moduli dependence of

string loop corrections to gauge coupling constants, Nucl. Phys. B355

(1991) 649{688.

[DMVV] R. Dijkgraaf, G. Mo ore, E. Verlinde and H. Verlinde, El liptic

Genera of Symmetric Products and Second Quantized Strings, Com-

mun. Math. Phys. 185 (1997) 197{209.

[EMOT] A. Erdeli, W. Magnus, F. Ob erhettinger and F.G. Tricomi,

Higher Transcendental Functions,Vol. 2, McGraw-Hill, 1953.

[EZ] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in

Mathematics 55, Birkhauser, 1985.

[FF] A.J. Feingold and I.B. Frenkel, A hyperbolic Kac-Moody algebra and

the theory of Siegel modular forms of genus 2, Math. Ann. 263 (1983)

87{114.

[Fre] E. Freitag, Siegelsche Modulformen, Grundlehren der mathematis-

chen Wissenschaften, Bd. 254, Springer-Verlag, 1983; Singular Modu-

lar Forms and Theta Relations, Lecture Notes in Mathematics 1487,

Springer-Verlag, 1991.

[GN1] V.A. Gritsenko and V.V. Nikulin, Siegel automorphic form correc-

tions of some Lorentzian Kac{Moody Lie algebras, Amer. J. Math. 119

(1997), 181{224, alg-geom/9504006; The Igusa modular forms and \the

simplest" Lorentzian Kac{Moody algebras, alg-geom/9603010.

[GN2] V.A. Gritsenko and V.V. Nikulin, Automorphic Forms and

Lorentzian Kac{Moody Algebras, part I and part I I, alg-geom/9610022

and alg-geom/9611028.

[Go e] L. Gottsche, The Betti numbers of the Hilbert Schemes of Points on

a Smooth Projective Surface, Math. Ann. 286 (1990) 193{207; Hilbert

Schemes of Zero-dimensional Subschemes of Smooth Varieties, Lecture

Notes in Mathematics 1572, Springer-Verlag, 1994.

[GS] L. Gottsche and W. So ergel, Perverse sheaves and the cohomology of

Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993),

235{245.

[HH] F. Hirzebruch and T. Ho er, On the Euler number of an Orbifold,

Math. Ann. 286 (1990) 255{266.

30 T. Kawai

[HM] J.A. Harvey and G. Mo ore, Algebras, BPS States, and Strings, Nucl.

Phys. B463 (1996) 315{368.

[Igu] J. Igusa, On Siegel Modular Forms of Genus Two, Amer. J. Math. 84

(1962) 175{200; On Siegel Modular Forms of Genus Two (II), Amer.

J. Math. 86 (1964) 392{412.

[Kaw] T. Kawai, N =2 heterotic string threshold correction, K 3 surface

and generalized Kac-Moody superalgebra,Phys. Lett. B372 (1996) 59{

64.

[KM] T. Kawai and K. Mohri, Geometry of (0,2) Landau-Ginzburg Orb-

ifolds, Nucl. Phys. B425 (1994) 191{216.

[KYY] T. Kawai, Y. Yamada and S.-K. Yang, El liptic Genera and N=2

Superconformal Field Theory, Nucl. Phys. B414 (1994) 191{212.

[Lan1] P.S. Landweb er (Ed.), El liptic Curves and Modular Forms in Alge-

braic Topology, Lecture Notes in Mathematics, 1326, Springer-Verlag,

1988.

[Lan2] S. Lang, El liptic Functions, Addison-Wesley, 1973.

[Mum] D. Mumford, Tata Lectures on Theta II, Birkhauser, 1984.

[Nak] H. Naka jima, Heisenberg algebra and Hilbert schemes of points on

projective surfaces, alg-geom/9507012.

[Sie] C.L. Siegel, Advanced Analytic Number Theory, Studies in Mathe-

matics 9, Tata Institute of Fundamental Research, 1980.

[Two] A.A. Belavin and V.G. Knizhnik, Complex geometry and the the-

ory of quantum strings,Sov. Phys. JETP 64 (1986) 214{228, Zh. Eksp.

Teor. Fiz. 191 (1986) 364{390; Algebraic geometry and the geometry

of quantum strings,Phys. Lett. 168B 201{206.

A. Belavin, V. Knizhnik, A. Morozov and A. Perelomov, Two and three

loop amplitudes in the bosonic string theory, JETP Lett. 43 (1986) 411{

414, Phys. Lett. B177 (1986) 324{328.

G. Mo ore, Modular forms and two loop string physics, Phys. Lett.

B176 (1986) 369{379.

A. Kato, Y. Matsuo and S. Odake, Modular invariance and two loop

bosonic string vacuum amplitude,Phys. Lett. B179 (1986) 241{246.

[VW] C. Vafa and E. Witten, A strong coupling test of S-duality, Nucl.

Phys. 431 (1994) 3{77.

K3 surfaces and Igusa cusp form 31

[Wit] E. Witten, The index of the Dirac Operator in Loop Space, pp. 161{

181 in [Lan2].

[YZ] S-T. Yau and E. Zaslow, BPS States, String Duality, and Nodal

Curves on K3, Nucl. Phys. B471 (1996) 503{512.

Toshiya Kawai

Research Institute for Mathematical Sciences

Kyoto University

Kyoto 606-01, Japan

e-mail: [email protected]. jp