Gromov-Witten classes, quantum cohomology, and enumerative geometry

TL;DR

This paper explores the mathematical foundations of topological quantum field theory, focusing on Gromov-Witten classes and their applications to enumerative geometry, including counting rational curves on specific algebraic varieties.

Contribution

It provides an axiomatic framework for Gromov-Witten classes, analyzes their properties for Fano varieties, and applies these concepts to enumerative problems in algebraic geometry.

Findings

Tree level theories are determined by their correlation functions.

Applications include counting rational curves on del Pezzo surfaces.

Discussion of properties of Gromov-Witten classes for Fano varieties.

Abstract

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov-Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Applications to counting rational curves on del Pezzo surfaces and projective spaces are given.