Gromov-Witten Invariants via Algebraic Geometry

TL;DR

This paper explains how to accurately compute Gromov-Witten invariants on Calabi-Yau manifolds by incorporating deformation and excess intersection theories, addressing discrepancies caused by degenerate instantons.

Contribution

It introduces methods to account for continuous families of instantons and the role of degenerate instantons in Gromov-Witten invariant calculations.

Findings

Incorporates deformation theory to handle continuous instanton families.

Uses excess intersection theory for accurate counts.

Highlights the importance of degenerate instantons.

Abstract

Calculations of the number of curves on a Calabi-Yau manifold via an instanton expansion do not always agree with what one would expect naively. It is explained how to account for continuous families of instantons via deformation theory and excess intersection theory. The essential role played by degenerate instantons is also explained. This paper is a slightly expanded version of the author's talk at the June 1995 Trieste Conference on S-Duality and Mirror Symmetry.