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Equivariant Gromov - Witten Invariants

Alexander B. Givental

TL;DR

This work develops an equivariant Gromov–Witten framework for ample Kahler manifolds, establishing a robust Frobenius-structure and WDVV theory in the presence of group actions. It derives explicit, computable formulas for quantum products via fixed-point localization, and demonstrates how vector-bundle twists extend GW theory to complete intersections. For projective complete intersections with $\sum l_j<n$, it yields closed-form hypergeometric solutions and mirror-symmetric differential equations; in the borderline Calabi–Yau case $\sum l_j=n+1$, it verifies mirror predictions through Picard–Fuchs equations and period integrals, connecting GW data to mirror geometry. The results unify Frobenius-manifold structures, D-module interpretations, and loop-space perspectives, providing concrete tools to compute quantum cohomology algebras and to confirm mirror symmetry in broad classes of projective complete intersections.

Abstract

We develop general theory of equivariant quantum cohomology for ample Kahler manifolds and prove the mirror conjecture for projective complete intersections.

Equivariant Gromov - Witten Invariants

TL;DR

This work develops an equivariant Gromov–Witten framework for ample Kahler manifolds, establishing a robust Frobenius-structure and WDVV theory in the presence of group actions. It derives explicit, computable formulas for quantum products via fixed-point localization, and demonstrates how vector-bundle twists extend GW theory to complete intersections. For projective complete intersections with , it yields closed-form hypergeometric solutions and mirror-symmetric differential equations; in the borderline Calabi–Yau case , it verifies mirror predictions through Picard–Fuchs equations and period integrals, connecting GW data to mirror geometry. The results unify Frobenius-manifold structures, D-module interpretations, and loop-space perspectives, providing concrete tools to compute quantum cohomology algebras and to confirm mirror symmetry in broad classes of projective complete intersections.

Abstract

We develop general theory of equivariant quantum cohomology for ample Kahler manifolds and prove the mirror conjecture for projective complete intersections.

Paper Structure

This paper contains 11 sections, 5 theorems, 139 equations.

Table of Contents

  1. Moduli spaces of stable maps
  2. Equivariant correlators
  3. The WDVV equation
  4. Ample vector bundles
  5. Quantum cohomology
  6. Floer theory and $D$-modules
  7. Frobenius structures
  8. Fixed point localization
  9. Projective complete intersections
  10. Complete intersections with $l_1+...+l_r=n$
  11. Calabi-Yau projective complete intersections

Key Result

Theorem 1.1

If $X$ is ample then all non-empty moduli spaces $X_{n,d}$ of genus $0$ stable maps are compact nonsingular complex orbifolds of "right" dimension $\langle c_1(T_x),d\rangle +\operatorname{dim}_{\Bbb C}X+n-3$.

Theorems & Definitions (5)

  • Theorem 1.1: KnBM
  • Theorem 3.1
  • Theorem 6.1
  • Corollary 6.2
  • Proposition 7.1