Cohomological field theory calculations

TL;DR

This paper introduces cohomological field theories (CohFTs), discusses their classification, and surveys explicit calculations of CohFTs related to Witten's classes, Verlinde bundles, and Gromov-Witten invariants, highlighting open questions.

Contribution

It provides an accessible overview of CohFT classification and explicit calculations, connecting foundational theory with recent computational advances.

Findings

Classification of semisimple CohFTs via Givental group

Explicit calculations of CohFTs from Witten's classes and Verlinde bundles

Identification of open problems in the field

Abstract

Cohomological field theories (CohFTs) were defined in the mid 1990s by Kontsevich and Manin to capture the formal properties of the virtual fundamental class in Gromov-Witten theory. A beautiful classification result for semisimple CohFTs (via the action of the Givental group) was proven by Teleman in 2012. The Givental-Teleman classification can be used to explicitly calculate the full CohFT in many interesting cases not approachable by earlier methods. My goal here is to present an introduction to these ideas together with a survey of the calculations of the CohFTs obtained from Witten's classes on the moduli spaces of r-spin curves, Chern characters of the Verlinde bundles on the moduli of curves, and Gromov-Witten classes of the Hilbert schemes of points of the plane. The subject is full of basic open questions.