The structure of 2D semi-simple field theories

TL;DR

This paper classifies all 2D semi-simple cohomological field theories using Frobenius algebras, demonstrating how they influence Gromov-Witten invariants and confirming Givental's higher-genus reconstruction conjecture.

Contribution

It provides a complete classification of 2D semi-simple cohomological field theories and confirms a key conjecture on reconstructing Gromov-Witten invariants.

Findings

Classification of cohomological 2D field theories based on Frobenius algebras

Expression of effects on Gromov-Witten potential via Givental's Fock space

Reconstruction of Gromov-Witten invariants from quantum cup-product and Chern class

Abstract

I classify all cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of kappa-classes and by an extension datum to the Deligne-Mumford boundary. Their effect on the Gromov-Witten potential is described by Givental's Fock space formulae. This leads to the reconstruction of Gromov-Witten invariants from the quantum cup-product at a single semi-simple point and from the first Chern class, confirming Givental's higher-genus reconstruction conjecture. The proof uses the Mumford conjecture proved by Madsen and Weiss.