Alexander Givental

Equivariant Gromov-Witten invariants

The objective of this paper is to describe some construction and applications of the equivariant counterpart to the Gromov-Witten (GW) theory, i.e. intersection theory on spaces of (pseudo-) holomorphic curves in (almost-) Kahler manifolds. Given a Killing action of a compact Lie group G on a compact Kahler manifold X, the equivariant GW-theory provides, as we will show in Section 3, the equivariant cohomology space H G(X) with a Frobenius structure (see [2]). We discuss applications of the equivariant theory to the computation ([7],[11]) of quantum cohomology algebras of flag manifolds (Section 5), to the simultaneous diagonalization of the quantum cup-product operators (Sections 7,8), to the S-equivariant Floer homology theory on the loop space LX (see Section 6 and [10],[9]) and to a “quantum” version of the Serre duality theorem (Section 12). In Sections 9 — 11 we combine the general theory developed in Sections 1 — 6 with the fixed point localization technique [3] in order to prove the mirror conjecture (in the form suggested in [10]) for projective complete intersections. By the mirror conjecture one usually means some intriguing relations (discovered by physicists) between symplectic and complex geometry on a compact Kahler Calabi–Yau nfold and respectively complex and symplectic geometry on another Calabi-Yau n-fold called the mirror partner of the former one. The remakable application [16] of the mirror conjecture to enumeration of rational curves on Calabi–Yau 3-folds (1991, see the theorem below) raised a number of new mathematical problems — challenging maturity tests for modern methods of symplectic topology.

Introduction to Symplectic Field Theory

We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new invariants of contact manifolds and their Legendrian submanifolds. Moreover, we hope that the applications of SFT go far beyond this framework.1

A Mirror Theorem for Toric Complete Intersections

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions. Revision 03.03.97: we correct an error in Introduction.

QUANTUM COHOMOLOGY OF FLAG MANIFOLDS AND TODA LATTICES

We discuss relations of Vafas quantum cohomology with Floers homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.

Semisimple Frobenius structures at higher genus

In the context of equivariant Gromov-Witten theory of tori actions with isolated fixed points, we compute genus g > 1 Gromov-Witten potentials and their generalizations with gravitational descendents. Both formulas, with and without descendents, are stated in a form applicable to any semisimple Frobenius structure and therefore can be considered as definitions in the axiomatic context of Frobenius manifolds. In (nonequivariant) Gromov-Witten theory, they become conjectures expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product.

Homological Geometry and Mirror Symmetry

A homogeneous polynomial equation in five variables determines a quintic 3-fp;d in ℂP4. Hodge numbers of a nonsingular quintic are know to be: h p, p = 1, p = 0, 1, 2, 3 (Kahler form and its powers), h3, 0 = h0,3 = 1 (a quintic happens to bear a holomorphic volume form), h2,1 = h1, 2 = 101 = 126 - 25 (it is the dimension of the space of all quintics modulo projective transformations, and h2,1 is responsible here for infinitesimal variations of the complex structure) and all the other h p,q = 0.

Symplectic geometry of Frobenius structures

The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in two-dimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GL N of hidden svmmetries.

Homological geometry I. Projective hypersurfaces

Introduction Consider a generic quintic hypersurfaceX inCP . It is an example of Calabi– Yau 3-folds. It follows from Riemann–Roch formula, that rational curves on a generic Calabi–Yau 3-fold should be situated in a discrete fashion. Therefore a natural question of enumerative algebraic geometry arises: find the number nd of rational curves in X of degree d for each d = 1, 2, 3, .... In 1991 Candelas, de la Ossa, Green and Parkes [1] ‘predicted’ all the numbers nd simultaneously: they conjectured that the generating function

Simple singularities and integrable hierarchies

The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobenius manifold. In [12], it is proved that the total descendent potential corresponding to K. Saito’s Frobenius structure on the parameter space of the miniversal deformation of the A n−1-singularity satisfies the modulo-n reduction of the KP-hierarchy. In this paper, we identify the hierarchy which is satisfied by the total descendent potential of a simple singularity of ADE type. Our description of the hierarchy is parallel to the vertex operator construction of Kac-Wakimoto [17] except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac-Wakimoto theory are studied on a case-by-case basis and remain, generally speaking, unknown.

A symplectic fixed point theorem for toric manifolds

In this paper, by a toric manifold we mean a non-singular symplectic quotient M = ℂ n //T k of the standard symplectic space by a linear torus action. Such a toric manifold is in fact a complex Kahler manifold of dimension n - k. We denote p(M) and c(M) the cohomology class of the Kahler symplectic form and the first Chern class of M respectively. They both are effective, that is, Poincare-dual to some holomorphic hypersurfaces. We call a homology class in H 2(M, ℤ) effective if it has non-negative intersection indices with fundamental cycles of all compact holomorphic hypersurfaces in M, and denote ℰ the set of all non-zero effective homology classes. Our main result is the following