Vertex algebras of CohFT-type

TL;DR

This paper demonstrates that vertex algebras of CohFT-type produce semisimple cohomological field theories with tautological Chern classes, connecting algebraic structures to geometric invariants on moduli spaces.

Contribution

It establishes that bundles from CohFT-type vertex algebras define semisimple cohomological field theories and provides a formula for their Chern characters in terms of fusion rules.

Findings

Bundles define semisimple cohomological field theories

Chern classes are tautological

Explicit expression for total Chern character

Abstract

Representations of certain vertex algebras, here called of CohFT-type, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DGT2]. We show that such bundles define semisimple cohomological field theories. As an application, we give an expression for their total Chern character in terms of the fusion rules, following the approach and computation in [MOPPZ] for bundles given by integrable modules over affine Lie algebras. It follows that the Chern classes are tautological. Examples and open problems are discussed.