The Candelas-de la Ossa-Green-Parkes Formula @

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ELSEVIER

IUCLEAR PHYSIC c

Nuclear Physics B (Proc. Suppl.) 67 (1998) 106--114

P R O C E E D I N G S S U P P L E M E N T S

The Candelas-de la Ossa-Green-Parkes Formula

Bong H. Lian ~, Kefeng Liu b and Shing-Tung Yau c

a D e p a r t m e n t of M a t h e m a t i c s , Brandeis, University, Waltham, MA 02154 b D e p a r t m e n t of Mathematics, Stanford University, Stanford, CA 94305 c D e p a r t m e n t of M a t h e m a t i c s , H a r v a r d University, Cambridge, MA 02138

In this note we discuss a recent proof of the formula for the worldsheet instanton prepotential predicted by Mirror Symmetry for the quintics in pa. One of the key ingredients in the proof is the equivariant cohomology groups on the so-called linear sigma model moduli spaces. We introduce the notion of admissible data on the equivariant cohomology groups of the linear sigma model. An admissible data may be thought of as a sequence of equivariant classes satisfying certain algebraic conditions. It arises naturally from Kontsevich's stable map compactification of moduli spaces of maps from curves into projective manifolds. The structures of admissible data help reduce many counting problems to checking certain combinatorial structure of a compactification. The mirror transformation of Candelas et al turns out to be a transformation between two admissible data associated respectively to the linear and the non-linear sigma models. As an application, we prove the formula for the worldsheet instanton prepotential in terms of hypergeometric series. At the end we also interprete an infinite dimensional transformation group, called the mirror group, acting on admissible data, as a certain duality group of the linear sigma model.

1. I N T R O D U C T I O N

For the r e m a r k a b l e history of the Mirror Con- jecture, we refer the reader to articles in [14].

In 1990, Candelas, de la Ossa, Green, and Parkes conjectured a formula for counting the n u m b e r nd of rational curves in every degree d on a general quintic in p a . Their c o m p u t a t i o n is inspired by an earlier construction of a mirror manifold by Greene-Plesser. It has been conjectured earlier, by Clemens, t h a t the n u m b e r of rational curves in every degree is finite. T h e conjectured formula agrees with a classical result in degree 1, an earlier c o m p u t a t i o n by S. K a t z in degree 2, and has been verified in degree 3 by Ellingsrud- Stromme. In 1994 following some ideas of Gro- m o v and W i t t e n , R u a n - T i a n introduced the notion of a symplectic G r o m o v - W i t t e n (GW) invariants. I n d e p e n d e n t l y Kontsevich proposed an algebraic geometric notion of G r o m o v - W i t t e n (GW) invariants. Significant generalizations of his definition have been given by Kontsevich- Manin, L i - T i a n and Behrend-Fentachi. For an excellent introduction to stable maps, see the pa-

per of Fulton-Pandharipande [16]. A recent p a p e r of Li-Tian shows t h a t the symplectic version and the algebraic geometric version of the G W t h e o r y are essentially the same in the projective catetory.

Beautiful applications of ideas from q u a n t u m cohomology have recently been done by Caporaso- Harris [10], C r a u d e r - M i r a n d a [11], DiFrancesco- Itzykson [12] and others, solving m a n y impor- t a n t enumerative problems. Significant connec- tion between q u a n t u m cohomology and the geometry of FYobenius manifolds a p p e a r s in the work of Dubrovin [13] and Manin [32].

Closer to mirror s y m m e t r y , a t h e o r e m of Manin says t h a t the degree k G W invariants for p 1 (the so-called multiple-cover contribution) is given by k -3. This was conjectured by Candelas et M, and was justified by Aspinwall-Morrison using a different compactification. See also the recent paper of Voisin [35]. According to the definition of Kontsevich, the n u m b e r

g d = ~ nd/k k-3 kid

is the degree of the Euler class Csd+l(Ud) where 0920-5632/98/$19.00 Elsevier Science B.V.

PII S0920-5632(98)00126-1

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B.H. Lian et al./Nuclear Physics B (Proc. Suppl.) 67 (1998) 106-114 107

Ud is the obstruction bundle over the degree d (genus zero with no marked point) stable map moduli space AT40,0(d, p4), induced by the line bundle 49(5) --+ p4. We shall call Kd the degree d Kontsevich-Manin number.

Using the torus action on p 4 and the Atiyah- B o t t localization formula, Kontsevich computes the nmnbers Kd for d = 1, 2, 3, 4, and verifies t h a t they agree with the conjectured formula. In a series of recent papers, Givental and Kim introduce a number of important ideas, among which is an equivariant version of Kontsevich's approach•

Givental then proposes to use fixed point formu- las on stable moduli to compute equivariant Eu- ler classes, along the lines proposed by Kontse- vich. More recently, Graber-Pandharipande have also applied fixed point m e t h o d to study Gromov- W i t t e n invariants of p n generalizing some results of Kontsevich.

In two recent papers [18] [19], Givental proposes a proof of the conjectured formula. Since his papers, there has been many seminars devoted to a t t e m p t s to understand the papers• The papers contain many beautiful ideas which have led to i m p o r t a n t new insights into the conjectured formula. However the ideas have not been car- ried through in details. (See for example [32]).

Inspired by Givental's achievement, we have recently given a complete proof of the conjectured formula [29]. Our goal in this note is to give sketch of our proof. For details we refer the read- ers to our forth-coming paper [29].

We now formulate one of our main theorems in this paper. Let M be a projective manifold and fl ~ H 2 ( M , Z ) . Let dQg,k(fl, M) be the stable map moduli space of degree fl, arithmetic genus g, with k marked points [26]. T h r o u g h o u t this paper, we shall only deal with the case with g = 0.

Let

K d = o(d,p4)

F(T) = 5T3

--d-+

E KaedT

d>l

where Ud --+ dQ0,0(d,P 4) is the bundle whose fiber at ( f , C ) is given by the section space

H°(C,f*(5H)). Consider the fourth order hypergeometric differential operator:

( d ) 4 t d d

L := - 5e (577 + 1 ) . . . ( 5 7 7 + a).

By the Frobenius method, it is easy to show t h a t

• d>O I ' I m = l ( H -b m ) 5 d

i = 0,1,2,3,

form a basis of solutions to the differential equa- t i o n L . f = 0 . Let

f l Jr(T) = 5 , f l f2 f3 )

T=~oo' 7t~oo fo So "

T h e o r e m 1.1 (The Mirror Conjecture) F ( T ) =

re(T).

We remark t h a t the definitions of F, -7" in [8] dif- fer from those given above by a quadratic polynomial in T. T h e functions F(T), Jr(T) are known respectively as a type IIA and a type IIB pre- potential functions. T h e transformation on the functions fi given by the normalization

f i H - - A f0

and the change of variables f l

t~--~T= - -

fo

are the mirror transformation of Candelas et al.

By their construction, the functions f0, .., f3 are periods of a family of Calabi-Yau threefolds. By the theorem of Bogomolov-Tian-Todorov, these periods in fact determine the complex structure of the threefold.

A similar Mirror Conjecture formula holds true for a three dimensional Calabi-Yau complete intersection in a toric Fano manifold [30]. This will turn out to agree with the beautiful construction of the mirror manifolds of Batyrev [3], Batyrev-Borisov [4], as well as the m a n y mirror s y m m e t r y computations of Morrison [33], Libgober-Teitelboim [28], Batyrev-van Straten [5], Candelas-Font-Katz-Morrison [9], Hosono- Klemm-Theisen-Yau [22] and Hosono-Lian-Yau [23].

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108 B.H. Lian et al./Nuclear Physics B (Proc. Suppl.) 67 (1998) 106-114

T h e application of the a p p r o a c h outlined in this p a p e r turns out to be rather broad. It applies to manifolds with torus action and m a n y of their submanifolds. We have also obtained the multiple cover formula for p 1 , and a formula for the Euler classes of the obstruction bundles induced by t h e canonical bundle on p 2 in this way. In this note, to m a k e t h e ideas clear we restrict our- selves to the simplest case, genus 0 curves in some submanifolds of P n . In a future p a p e r [30], we extend our discussions to toric varieties, homogeneous manifolds, their submanifolds, and for higher genus moduli spaces. We hope to even- tually u n d e r s t a n d from this point of view the far reaching results for higher genus of Bershadsky- Cecotti-Ooguri-Vafa [7], the beautiful c o m p u t a - tions of Getzler [17] for elliptic G W invariants and of B a t y r e v - C i o c a n F o n t a n i n e - K i m - v a n S t r a t e n [6]

o1~ G r a s s m a n n i a n s .

2. A D M I S S I B L E D A T A

One of the key ingredients in our approach is the linear sigma model, first introduced by Wit- ten [36], and later used to s t u d y mirror s y m m e t r y by Morrison-Plesser [21] resulting in new insights into t h e origin of hypergeometric series. In this paper, we consider t h e S 1 × T-equivariant cohomology of t h e linear sigma model.

T h r o u g h o u t this paper, we fix a positive integer n. Let p, ~, a , A0, .., An be formal variables. We denote A = (A0, .., An). We will first introduce notations and m e n t i o n a few facts which will be used t h r o u g h o u t this paper.

Let T be an r-dimensional real torus with a complex linear representation on C g+~. Let rio, -., fin be t h e weights of this action. We consider the induced action of T on p N , and the T-equivariant cohomology with coefficients in Q, which we shall denote by

HT(--).

Now

HT(pt)

is given by t h e s y m m e t r i c algebra on the dual of the Lie algebra of T. We can now regard the j3~

to be elements of

H~(pt).

By a choice of basis of

H~(pt), HT(pt)

becomes a polynomial algebra with r generators of degree 2. T h r o u g h o u t this paper, we shall follow t h e convention t h a t such generators have degree 1.

It is known t h a t the equivariant cohomology of

p N is given by [24]

HT(pN) = HT(pt)[~]/ ( Ni~=o(~--/3i)) •

Here ~, which we shall call the equivariant hyperplane class, is a fixed lifting of the hyperplane class of p y . Each one-dimensional weight space in C N+I becomes a fixed point p~ in p N .

We shall identify the rings H~(pi) and H~(pt) = H(BT).*

T h e r e are N + 1 canonical restriction m a p s ~p~ :

HT(P N) -- HT(pt),*

given by ~ H fli, i = 0, .., N. There is also a push-forward m a p

HT(P N) -- HT(pt)*

given by integration along the fiber. By the localization formula, it is given by

w ~- Res¢. N*

0d

I-i~=0 (¢ - fli)

We now specialize the above to two different situations which will be used frequently in this paper. First consider the s t a n d a r d action of T = (S1) n+l on C n+l, and let (A0,..,An) denote the weights. In this case, there are obviously n + 1 isolated fixed points given by the coordinate lines in C n+l. We shall denote the equivariant hyperplane class by p, the canonical restriction m a p s simply by Lp~

:w(p,

A) H w(Ai, A), and the push-forward m a p by

p f : HT(P n) H HT(pt).

We shall often use the evaluation m a p Aj ~ 0 on the ring H T ( P n ) , and shall call this the nonequivariant limit. Thus in this limit, p becomes the hyperplane class H c H * ( P n ) , and the push- forward m a p becomes the 'degree m a p ' H ( P n) =

Q[H]/(H n+l)

--* Q given by H k ~-* 5k,n.

We now consider the second situation. For each d = 0, 1, 2, .., consider the following complex linear action of the group S 1 × T on C (n+l)(d+l). We let the group act on the ( i j ) - t h coordinate line in C (n+l)(d+l) by the weights As +

ja,

i = 0, ..,n, j = 0,..,d. Thus there are ( n + l ) ( d + l ) isolated fixed points Pij on the projective space

p(n+l)d+n,

given by those coordinate lines. In this case, we shall denote the equivariant hyperplane class by a, the canonicM restriction m a p s b y e * _Pij :w(t~,a,A) ~ - * w ( A i + m a , a , A ) and the

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B.H. Lian et al./Nuclear Physics B (Proc. Suppl.) 67 (1998) 106-114 109

push-forward m a p by Pfd. Throughout this paper we shall denote

~'~ : : HS1 ×T(pt) ~ Q[c~, ),], 7~ -1 := quotient f i e l d o f ~ .

T h e n the push-forward m a p is given by

pfd:

7¢[~]/& -~ 7¢,

f(~, s, ~) f ( n , s , , ~ ) ~-~ Res~ n a

I]~=0 1-lm=0(~ - ,xj - m s ) where Jd is the ideal

n d

J~ = ( I I I ] ( ~ - ~J - m < )

j = 0 rn=0

Here we have abused the notation n, using it to represent a class in ~[n]/Jd for every d. But it should present no confusion in the context it arises.

Observe t h a t we have a chain of natural inclu- sions of ideals in 7Z[n]:

J d C J d - 1 , d = 1,2,..

which gives rise to a chain of linear maps Hs* × T ( P (~+l)d+'~) ~ n[n]/Jd --+ n [ n l / J d - 1 --+ 7~[n]/Jo ~ g T ( P ~ ) [ s ] .

We denote the last isomorphism by ~* which is given by L* p,~ : n ~-+ p.

Let Nd be the space of nonzero (n + 1)-tuple of degree d polynomials in two variables w0, wl, modulo scalar. T h e r e is a canonical way to identify Nd with p(n+l)d+n. Namely, a point z C

p(n+l)d+n

corresponds to the polynomial tuple

[ E r zorwow ~ , r d - r . . , E r znrwow ~ ] ~ Nd. r d - r This identification will be used throughout this paper.

I t is easy to see t h a t the natural T-action on (n + 1)-tuples together with a generic Sl-action on [w0,wl] e p 1 , induces a S 1 x T-action on Nd which coincides with the S ~ x T-action on

p(n+l)d+n

described earlier.

D e f i n i t i o n 2.1 We call the sequence of projective spaces {Nd} the linear sigma model for P'~.

Clearly the ground ring H s l × T ( p t ) is canoni- cally a subring of Hs~ x T ( N d ) . We can localize the latter by tensoring with the quotient field 7Z -1.

We shall denote the resulting ring simply by 7 Z - I H S l x T ( N d ) = ~ - l [ n ] / J d.

2.1. A d m i s s i b i l i t y

Fix a class ft E H ~ ( P ~) with the p r o p e r t y t h a t its restriction at the i-th fixed point has

~p~(fl) ¢ 0 for i = 0, .., n. If ft only depends on p, we often denote its restriction simply by f~(Ai).

Given w E 7~, let & be the class obtained from w by replacing s by -c~.

D e f i n i t i o n 2.2 (Notations) We denote by Qn the set of sequences of cohomology classes

Q: ed(<s,~) ~ n-IHs~×T(Nd),

d = 1,2,..., and Qo = fL

D e f i n i t i o n 2.3 We call a sequence of cohomology classes

Q: Qd(.,s,~) e H;~×r(Nd),

d = 1 , 2 , . . , and Qo = f~

an f~-admissible data if for all d, and r = O, ..,d, i = O, .., n,

* ~ * I,*

(*) bpi,o( ) %,~(Qd)= pi,,,(G)

G.,~_,.(Qd-r)"

More explicitly condition (*) can be w r i t t e n as

~(~i, ~)Q~(~ + r~, ~, ~) =

Q r ( ~ i + r s , s , ~ ) Q ~ _ r ( ~ - (d - r > , - s , ~).

We observe t h a t the set of admissible d a t a is a monoid, ie. it is closed under the product QdQ~, and has the unit given by Qd = 1 for all d.

Hence the p r o d u c t of an ft-admissible with an fZ'-admissible d a t a is an f t ~ ' - admissible data.

In a general geometric setting, this nmltiplica- tive p r o p e r t y corresponds to taking intersection of two suitable projective manifolds. T h e equivariant class Ft E H~-(P n) will play the role of the equivariant T h o m class of the normal bundle of such a projective manifold. Throughout this pa- per, we shall deal only with one fixed class f~ at

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110 B.H. Lian et aL /Nuclear Physics B (Proc. Suppl.) 67 (1998) 106-114

a time. When no confusion arises, we shall say admissible rather than ~-admissible.

Example 1. Let 1 be a positive integer. P u t

ld

P : Pd(n,a,)~) = H ( l a - m s ) .

m ~ O

It is straightforward to check t h a t P is an ~- admissible d a t a with f~(a) = la. We leave this as an exercise to the reader.

Example 2. Let M ° := Azi0,0((1,d),P 1 × p n ) be the moduli space of holomorphic m a p s p 1 __.

p 1 x P = of bidegree (1, d). Since each such m a p can represented by id × [f0, .., f=] where fi are degree d polynomials in two variables w0, Wl (which are homogeneous coordinates of p 1 ) , there is an obvious m a p ~ : M~ --~ Nd which sends a m a p in M ° to an (n + 1)-tuple [f0, .., f,~] e Nd. This m a p is clearly equivariant with respect to S 1 × T.

Since this m a p is an embedding and since Nd is c o m p a c t , this shows t h a t Nd is a compactification of M °. We can also compactify M ° by embedding it into the moduli space of stable m a p s Md := AJl0,0((1, d), p 1 × p = ) .

D e f i n i t i o n 2.4 (Notations) We call the sequence of the stable moduli spaces Md the non- linear sigma model for p n .

W i t h a bit of work, it can be shown t h a t the m a p ~ has an equivariant regular extension to : Md ~ Nd, by collapsing certain components of t h e curve C at each point (f, C) c Md.

T h e idea of using a collapsing m a p to relate two moduli p r o b l e m s is not new. T h e collapsing m a p ~ was known to G. T i a n in 1995, and a similar idea also a p p e a r e d in a p a p e r of J. Li in Donaldson t h e o r y [27] in which a collapsing m a p was used to relate the definitions of Donaldson invariants on two different compactifications of the moduli space of vector bundles. T h e m a p ~ was also used by Givental in [18].

Now each positive power IH of the hyperplane bundle H --* P ~ induces an equivariant bundle Wd --* Md similar to the bundle Ud A740,0(d,P n) we have seen earlier. Namely, the fiber of Wd at (f, C) E Md will be t h e section space H ° ( C , (7r2 o f ) * ( l H ) ) . Here rr2 : p 1 x p n __+

P ~ is the projection onto t h e second factor. Let

~d be the equivariant Euler class of Wd. We can now push-forward these classes for d = 1, 2, .. via the equivariant ~ and obtain a sequence equivariant cohomology classes ~!(Xd) E H~I xT(Nd).

For d = 0, we define ~!(Xo) to be the class la.

T h e o r e m 2.5 The sequence ~ ! ( ~ d ) above is an

~-admissible data with ~ = lg.

In E x a m p l e 1, the classes Pd in the first admissible d a t a obviously resembles the coefficients in the hypergeometric series in the Introduction.

W h e r e a s in E x a m p l e 2, the classes ~!(Xd) in the second admissible d a t a arise n a t u r a l l y from equivariant Euler classes of the bundle Vd. Recall t h a t the Mirror Conjecture Formula is a formula obtained by transforming certain hypergeometric functions to a generating function of the Euler classes of Ud (ie. the instanton prepotential F ) . This t r a n s f o r m a t i o n is the mirror t r a n s f o r m a t i o n of Candelas et al described in the Introduction.

This inspires the following strategy:

1. Relate the admissible d a t a Pd to hypergeometric series.

2. Relate the admissible d a t a ~v! ()~d) to the in- s t a n t o n prepotential F.

3. Construct an a p p r o p r i a t e mirror transfor- m a t i o n from the Pd to t h e ~! (Xd)-

D e f i n i t i o n 2.6 Two admissible data P , Q are said to be linked if the restrictions L* (Pd) and _Pi,o

~pl.o(Qd) agree at a = (hi - A j ) / d for all distinct i , j E {0,..,n} and all d - - 0 , 1,2,...

T h e o r e m 2.7 The admissible data P : Pd = I ] ~ = 0 ( l a - m ~ ) and Q: Qd = (~!()(.d) are linked.

T h e above two t h e o r e m s are proved by care- fully studying the localization of Xd to the fixed points in M4, and c o m p a r e with t h e localization of ~o!(Xd) in Nd. In fact the c o m p u t a t i o n of

~p,,o(Qd) at (~ = (A~ - A j ) / d by using localization gives us the expression:

ld

Lo(Qd) = H - -

m ~ O

In fact this is one of our motivations for in- troducing the (hypergeometric) admissible d a t a

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B.H. Lian et aL /Nuclear Physics B (Proc. Suppl.) 67 (1998) 106-114 111

v'~ ld t l

P~ = 11~=0~ ~ - m s ) .

This e x a m p l e illustrates t h e typical w a y in our a p p r o a c h t o solve an e n u m e r a t i v e problem.

Namely, to s t u d y equivariant classes such as Xa we p u s h t h e m into Nd t o o b t a i n an admissible d a t a Q. T h e n we c o m p u t e their values by localization m e t h o d at a = (Ai - Aj)/d. We t h e n c o n s t r u c t an explicit admissible d a t a P which is linked t o Q. A n d finally we t r y t o relate P a n d Q in some explicit m a n n e r . See [29] for m a n y other examples.

T h e o r e m 2 . 8 (Uniqueness)

Suppose P, Q are any two linked admissible data.

If deg~,,o(Pd - Qd) <_ ( n + l ) d - 2 for all i, then P = Q .

We now discuss t h e relationship with hyperge- o m e t r i c series. Given a n y Q E Q a , we define t h e following associated hypergeometric series:

HG[P](t) : =

<>o [I~=01-Im=~(p- A~ - m s ) '

which is a H ~ ( P ~) c o h o m o l o g y valued series.

Here we use t h e convention t h a t t h e d e n o m i n a - t o r in d = 0 t e r m in t h e s u m m a t i o n is 1. N o t e t h a t for P : Pd = 1 - I ~ = 0 ( l n - m s ) as given in E x a m p l e I above, it is easy to see t h a t in the limit

-~ 0, we have

~ = o ( l H m s )

HG[P](t) = e -Ht/" E e dt

d ?no~)n+l

d > o I ] m = ~ ( H -

where H on t h e right h a n d side is t h e nonequivariant h y p e r p l a n e class of p n . T h e coefficients of ( - H / h ) i for i = 0, 1, .., n - 1, are exactly solutions to a h y p e r g e o m e t r i c differential equa- t i o n (for l = n + 1), hence t h e n a m e hypergeometric series, discussed in t h e I n t r o d u c t i o n . 2.2. M i r r o r t r a n s f o r m a t i o n s

Let G be t h e g r o u p of lower t r i a n g u l a r unipo- t e n t m a t r i c e s f = [f~]r,s=oa,2,.. with entries in 7~ -1 [a] which are h o m o g e n e o u s functions of degree 0. We shall always assume that the entries are regular along n = A~, s = (A~ - Aj)/d for all d and alIi 7~ j.

We now i n t r o d u c e t h r e e invertible o p e r a t i o n s Q ~-~ Q / , Q ~-* I Q , a n d Q H Q ° on t h e set Q~

defined as follows:

d - 1 n

+ Er=0 f~,rQr(~, s, ~) [L=0

d - 1 )k n

+ ~ = 0 f d , ~ Q ~ ( ~ - (d - r ) s , s , ) I l k = 0

Hd

- r - l ~

m=0 t ~ - A k - m ~ ) Q ~ : Q ~ ( n , s , A ) = Q 4 ( a - d s , - s , A )

for f E G. O n e can check t h a t these o p e r a t i o n s are well-defined on Q a , ie. t h e right h a n d sides are i n d e p e n d e n t of t h e choice of representative of Qd E T~-IHs~×T(Ng). It is obvious t h a t t h e op- erations Q ~-~ Q/, Q ~-~ f Q preserve t h e p r o p e r t y t h a t Q0 = ~t. Since ~t is i n d e p e n d e n t of s , b y as- s u m p t i o n , it follows t h a t t h e o p e r a t i o n Q H Q ° also preserves t h e p r o p e r t y t h a t Q0 = ft. It is easy to verify t h a t for f, g ~ G, we have

(Q~)~ = Q~f

~(~Q) = ~IQ (Q~)~ = Q.

D e f i n i t i o n 2 . 9 Let G ~ be the group of transforma- tions on Q~ generated by all three types of trans- formations (2.1). An element of ~' is called a mirror transformation if it sends an admissible data to an admissible data. We denote by .hi ~ the group of mirror transformations. Two admis- sible data lying in the same orbit of .M ~ are called mirror transforms of one another.

Example. Let f be a u n i p o t e n t m a t r i x of t h e form:

i

^o ^o ^{. . .}

:1

a I 1 0 0 . . .

a2 al 1 0 0 .-.

3. a2 al 1 0 0 . •

with ai E Q. T h e n it can be shown t h a t Q ~ 0 = ( ( ( ( Q ~ V ) ~ V

defines a mirror t r a n s f o r m a t i o n . O n e can also express this t r a n s f o r m a t i o n in t e r m s of t h e associated h y p e r g e o m e t r i c series HG. Namely,

HG[Q](t) = g(et) • HG[Q](t)

(2.1)

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112 11.11. Lian et al. /Nuclear Physics B (Proc. Suppl.) 67 (1998) 106-114

where g(q) -- 1 + alq + a2q 2 + ' . . . (cf. t h e mirror t r a n s f o r m a t i o n in t h e Introduction). For more details, see [29].

T h e o r e m 2 . 1 0 Admissible data in the same orbit of the mirror group are linked. In particular if P is linked to Q, then any mirror transform of P is linked to Q.

3. T H E M I R R O R C O N J E C T U R E

T h r o u g h o u t this section, we assume t h a t l = n + 1 = 5, a n d we fix Ft(~) = la.

Consider t h e h y p e r g e o m e t r i c differential equa- tion

( d ,~ ~ ) - let(l d + 1) . - . ( l ~ + n ) ) f = 0. d

It is trivial to show t h a t (cf. Introduction) t h a t there exists unique solutions of the forms u(t) = 1 + O(e t) and v(t) = u ( t ) t + O(et). Recall t h a t T ( t ) - v(t)

u(t - t + o ( e ' )

is the mirror m a p of Candelas et al. As in the In- troduction, we have four solutions f0,--, f3 with

u = f o , v = f l .

T h e o r e m 3.1 The linked admissible data P : Pd = l-[m=o(la-- m a ) and Q : Qd = ~!(Xd) ld are mirror transforms of one another. Their re- spective hypergeometric series are related by

e k

HG[Q](T(t) ) - HG[P](t).

u

where g is some power series in e t.

One constructs a mirror t r a n s f o r m a t i o n by simply i m m i t a t i n g t h e mirror t r a n s f o r m a t i o n of Can- delas et al described in the Introduction. T h e asserted equality can t h e n be shown by applying the Uniqueness T h e o r e m .

T h e o r e m 3.2 (Special Geometry) In the

nonequivariant limit A --~ O, we have HG[Q](T) = 5H(1 + T H + F'sH--~- ~

+ ( T F ' - 2 F ) 5H----~) F ( T ) .-'- 5T._._.~*6 + ~-~d>_O K d e d T H G [ P ] ( t ( T ) ) = 5Hu(t)(1 + T H + 7 H2

+ ( T T - )

J : ( T ) . _ 5rf~ f~ ~_~

" - - 2 ~ ' f o f o - - f o I"

It is easy to c o m p u t e the a s y m p t o t i c behaviour of Y, namely

5T 3

~ ( T ) : - - C + O ( e T ) '

which is the same as the a s y m p t o t i c behaviour of F.

T h e preceding theorems imply t h a t F ' = 9 r' .

This shows t h a t F - F = const. B u t since F, ~- have the same a s y m p t o t i c behaviour, the const.

must be zero. Hence F = 9 ~.

3.1. C o n c l u d i n g r e m a r k s

Much of the machinery we have introduced here can be generalized to a large class of counting problems. For example any convex bundle (see [29] for definition) V -~ P'~ induces a sequence of equivariant bundles Vd --* 2Qo,o(d, p n ) on the stable moduli spaces, in the same way as 1H does.

T h e push-forwards F! (X y ) of the equivariant Eu- ler classes of Vd t u r n out to form an ~-admissible d a t a with gt being the equivariant Euler class of V. T h e general theory of admissible d a t a allows us to s t u d y the classes ~! (X y ) on the linear sigma model. Similarly any concave bundle (see [29] for definition), such as - l H , also induces an admissible d a t a by considering the corresponding obstruction bundles on A40,0(d, P n ) . One obtains the admissible d a t a for the multiple cover formula of p1 and the local mirror s y m m e t r y formula for the canonical bundle on p 2 . See [29] for details.

As we have seen, the set of linked admissible d a t a has an infinite dimensional transfor- m a t i o n group - the mirror group. T w o special linked admissible data: P arising from hypergeometric functions and the other from the classes

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B.H. Lian et al. /Nuclear Physics B (Proc. Suppl.) 67 (1998) 106-114 113

Q : Qd = ~!(Xd) on the non-linear sigma model are related by a particular mirror transformation of Candelas et al. Since the mirror group is so big, there are many other admissible data which are linked to P. and can be obtained simply by acting on P by the mirror group. From the phys- ical point of view, P arises from type IIB string theory while Q arises from type IIA string t h ~ ory, and mirror s y m m e t r y is a duality between the two. This relationship manifests itself on the linear sigma model as a duality transformation.

This suggests t h a t other admissible data linked to P may arise from some other string theories which are dual to type IIA and IIB, via more general mirror transformations. From the point of view of moduli theory, P is associated to the linear sigma model compatification Nd of the moduli space M ° we discussed in the Introduction.

Whereas Q is associated to the non-linear sigma model Md, which is the stable map compaetifi- cation of M °. This suggests that other admissible d a t a linked to P may correspond to other compactifications of M °. If true, we will have an association between string theories, linked admissible data, and compactifications of moduli space of maps, all in the same picture, whereby there is a duality in each kind which one sees in the linear sigma model. It would be interesting to m:derstand this duality more precisely.

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The Candelas-de la Ossa-Green-Parkes Formula @

@

The Candelas-de la Ossa-Green-Parkes Formula

--d-+

T=~oo' 7t~oo fo So "

re(T).

fo

HT(--).

HT(pt)

H~(pt).

H~(pt), HT(pt)

HT(pN) = HT(pt)[~]/ ( Ni~=o(~--/3i)) •

We shall identify the rings H~(pi) and H~(pt) = H*(BT).

HT(P N) --* HT(pt),

HT(P N) --* HT(pt)

w ~-* Res¢. N

:w(p,

p f : HT(P n) H HT(pt).

Q[H]/(H n+l)

ja,

p(n+l)d+n,

pfd:

p(n+l)d+n

p(n+l)d+n

Q: ed(<s,~) ~ n-IHs~×T(Nd),

Q: Qd(.,s,~) e H;~×r(Nd),

G.,~_,.(Qd-r)"

Hd

i

:1

u = f o , v = f l .

We shall identify the rings H~(pi) and H~(pt) = H(BT).*

HT(P N) -- HT(pt),*

HT(P N) -- HT(pt)*

w ~- Res¢. N*