Mirror symmetry (see [Y]) discovered in string theory few years ago gives a mysterious correspondence between symplectic and complex manifolds. More precisely, we consider only Calabi-Yau manifolds, i.e. compact complex manifolds with $c_1=0$ and with a Kähler metric satisfying Einstein equation. The first numerical observation was that for almost all families of Calabi-Yau manifolds there exists a dual family with the reflected Hodge diamond: $h^{p,q}(V)=h^{p,n-q}(W)$, $n=\dim V=\dim W$. Nowadays, MS means that generating functions for numbers of rational and higher genus curves satisfying all possible incidence functions for numbers of rational and higher genus curves satisfying all possible incidence conditions on one such a manifold can be expressed through periods of Hodge structures of mirror manifolds, and vice versa. It was checked in many examples for curves of small degrees. Recently [K2] we developed a method for computing numbers of rational curves of all degrees in many cases.
Here we are not concentrated on explicit checkings, but trying to describe mathematical structures hidden in MS. Our guess leads to the following picture: there should be a kind of twistor correspondence between lagrangian subvarieties on one side and complexes of holomorphic vector bundles on the other side.
1. Gromov-Witten invariants.
We expect that for any compact symplectic manifolds $(V,\omega)$ so
called Gromov-Witten classes are defined. These invariants depend on a
homology class $\beta\in H_2(V;{\bf Z})$ and a pair $(g,n)$ of
non-negative integers satisfying the inequality $2-2g-n<0$,
$I_{g,n;\beta}\in H_{\rm even}(\overline{\scr M}_{g,n}\times V^n;{\bf
Q})$. $\overline{\scr M}_{g,n}$ denotes the coarse moduli space of
stable $n$-punctured complex curves of genus $g$.
Intuitively, the geometrical meaning of Gromov-Witten classes can be described as follows. Denote by $\chi_{g,n\beta}$ the space of equivalence classes of $(C,x_1,\cdots, x_n,\phi)$ where $C$ is a smooth complex curve of genus $g$, x_i$ are distinct points on $C$, and $\phi\colon C\to V$ is a pseudo-holomorphic map (i.e., a solution of a generic perturbation of the Cauchy-Riemann equation) with the image of the fundamental class of $C$ equal to $\beta$. There is a natural map from $\chi_{g,n;\beta}$ to ${\scr M}_{g,n}\times V^n$ associating with $(C,x_*,\phi)$ the equivalence class of $(C,x_*)$ and the sequence of points $(\phi(x_1),\cdots,\phi(x_n))$. We define $I_{x,n;\beta}$ as the image of the fundamental class of a natural compactification $\overline{\scr X}_{g,n;\beta$.
The system of Gromov-Witten classes should satisfy certain properties (axioms from [KM]), the most important of which describes intersections of $I_{g,n;\beta}$ with boundary divisors of $\overline{\rm scr M}_{g,n}$. A consequence of these properties is the associativity equation from the next section.
There are still transversality problems in rigorous definitions of Gromov-Witten classes. There is a solution valid for genus zero curves on Calabi-Yau and Fano manifolds (Y. Ruan, G. Tian [RT]) and a general approach for all symplectic varieties (see [K2]). Also, there are now purely algebro-geometric definitions of GW classes in many cases.
2. Hodge structures arousing from symplectic manifolds.
For a compact symplectic manifold $(V,\omega)$ denote by $H\coloneq
\oplus H^k(V;{\bf C})$ the total cohomology space of $V$ considered as
a super-vector space and also as a complex flat super-manifold. Define
a function $\Phi$ (pre-potential) on $H$ by the formula
$$\Phi(\gamma)=\sum_{\beta\in H_2(V,{\bf
Z})}e^{-\int_\beta\omega}\sum^\infty_{n=3}\frac
1{n!}\int_{I_{0,n;\beta}1_{\overline{\scr
M}_{0,n}}\otimes\gamma\otimes\cdots\otimes\gamma.$$
Here $\gamma$ denotes a non-homogeneous cohomology class on $V$, and
$1_{\overline{\scr M}_{0,n}}$ is the identity in the cohomology ring
of $\overline{\scr M}_{0,n}$. This series is expected to be absolutely
convergent in some open domain in $H$, if the cohomology class
$[\omega]\in H^2(V,{\bf R})$ is sufficiently large.
Function $\Phi$ satisfies a system of non-linear differential equations of the third order (due ro R. Dijkgraaf, E. Verlinde, H. Verlinde, and E. Witten, see [W]). Let us choose a base $x_\alpha$ of the space $H$. Denote by $(g_{\alpha\beta})$ the matrix of the Poincaré pairing, $g_{\alpha\beta}\coloneq \int_Vx_\alpha\wedge x_\beta$, and by $(g^{\alpha\beta})$ the inverse matrix. For all $\alpha,\beta,\delta$, we have (modulo appropriate sign corrections for odd-degree classes): $$\sum_{\epsilon,\epsilon'}{\partial^3\Phi\over \partial x_\alpha \partial x_\beta\partial x_\epsilon}g^{\epsilon\epsilon'}{\partial^3\Phi\over \partial x_\gamma\partial x_\delta\partial x_{\epsilon'}}=\sum_{\epsilon,\epsilon'}{\partial^3\Phi\over \partial x_\alpha\partial x_\gamma\partial x_\epsilon}g^{\epsilon\epsilon'}{\partial^3\Phi\over \partial x_\beta\partial x_\delta\partial x_{\epsilon'}}$$
This equation can be reformulated as the condition of associativity of the algebra given by structure contsants $A^\gamma_{\alpha\beta}\coloneq \sum_{\gamma'}g^{\gamma\gamma'}\partial_{\alpha\beta\gamma'}\Phi$. In invariant terms it means that $\Phi$ defines a commutative associative multiplication on the tangent bundle to $H$ (quantum cohomology ring). The equation above was studied by B. Dubrovin [D]. He discovered that it is a completely integrable system in many cases (but not for CY maifolds). For example, for $V\cong {\bf C}P^2$ the associativity equation is equivalent to the Painlevé VI equation
Let us introduce a connection $\nabla$ on the tangent bundle to $H$ by the formula $\nabla=\nabla_0+A$, where $\nabla_0$ is the standard zero connection. The associativity equation implies flatness of $\nabla$.
Suppose that $c_1(V)=0$, $H^{2,0}(V)=)$, and $V$ carries at least one integrable complex structure compatible with $\omega$. For any such complex structure we have a Hodge decomposition $H=\oplus H^{p,q}$. We expect that all cycles of degrees independent on $g,\beta$. It follows that the restriction of $\nabla$ to the submanifold $H^2=H^{1,1}$ of $H$ maps $H^k$ to $H^{p+1,q+1}\otimes \Omega^1(H^{1,1})$.
We introduce filtrations $\oplus_{p\leq p_0}H^{p,q}$ on trivial bundles over $H^{1,1}$ with fibers equal to $\oplus_{{p-q}\text{ is fixed}}H^{p,q}$. Hence we have flat connections and filtrations satisfying the Griffiths transversality conditions. One can prove by using formal arguments with Hodge-Tate groups that the equivalence classes of these complex variations of Hodge structures do not change under deformations of the complex strucure on $V$.
Mirror Conjecture. These variations of Hodge
structures are algebro-geometric.
In almost all examples these variations of Hodge structures are
equivalent to total cohomology groups of a mirror family of complex
manifolds with $c_1=0$.
Recently physicists (see [BCOV]) discovered a remarkable way to predict numbers of higher genus curves on Calabi-Yau manifolds using finitely many terms of perturbative expansions for a quantum field theory on the mirror manifold obtained by a quantization of the Kodaira-Spencer deformation theory.
Fukaya's triangulated category.
A recent work by Kenji Fukaya [F] leads to a construction of a
triangulated category $F(V)$ for any symplectic manifold $(F,\omega)$
with $c_1=0$ and sufficiently large $[\omega]$. Recall that
triangulated categories form an arena of the modern homological
algebra. The basic example of a triangulated category is the derived
category of an abelian category.
K. Fuakaya assigned to $V$ not a triangulated category, but a more refined structure $\tilde F(V)$, called an $A_\infty$-category. Objects of this category are closed lagrangian submanifolds of $V$. Morphisms are defined only when lagrangian submanifolds are in general position, $V$. Morphisms are defined only when lagrangian submanifolds are in general position, ${\rm Hom}_{\tilde F}({\scr L},{\scr L}')\coloneq {\bf C}^{{\scr L}\cap {\scr L}'}$. One can introduce a structure of a complex on ${\rm Hom}_{\tilde F}({\scr L},{\scr L}')$ with ${\bf Z}$-grading arising from the Maslov index and the differential $d_{\scr L,{\scr L'}$ being a modified Floer differential. The matrix coefficient of $d_{{\scr L},{\scr L}'}$, associated with two intersection points $p_1,p_2\in{\scr L\cap {\scr L}'$ is the number of pseudo-holomorphic discs $\phi\colon D\to V$ with $$\phi(-1)=p_1,\phi(1)=p_2,\phi({\rm upper\ boundary}\subset{\scr L},\phi({\rm lower\ boundary}\subset{\scr L}'$$ counted with the weight $\exp(-{\rm area\ of}\ D$). Higher multiplications (= Massey operations), needed for the structure of an $A_\infty$-category, are defined analogously using pseudo-holomorphic discs in $V$ with the boundary in a union of lagrangian subvarieties.
One can enlarge $F(V)$ considering local systems on lagrangian submanifolds. The enlraged category $F(V)$ is endowed with the functorial isomorphism $R{\rm Hom}(X,Y)^*\cong R{\rm Hom}(Y^,X[n])$, wher $2n$ is the dimension of $V$. The same kind of duality holds for $n$-dimensional Calabi-Yau manifolds (Serre duality).
Homological Mirror Conjecture. A suitably enlarged
category $F(V)$ is equivalent to the derived category of coherent
sheavs on the mirror complex manifold $W$.
There are several reasons to believe this conjecture: (1) lagrangian
submanifolds and holomorphic bundles form natural super-symmetric
boundary conditions for open string theories. (2) on the tangent
bundle to the "super-moduli space of triangulated categories" there
exists a natural associative commutative multiplication (cup-product
on the Hochschild cohomology), this is the quantum cohomology ring for
Fukaya's category and the cup-product in $\oplus H^p(W,\wedge^qT)$ for
${\scr D}^b({\rm Coh}(W))$ in the case of complex variety $W$, (3)
there exists an algebraic construction of cohomology classes on ${\scr
M}_{g,n}$ (see [K1]) which can be extended, presumabley, to categories
with CY duality.
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