Classes on the moduli space of Riemann surfaces through a noncommutative Batalin-Vilkovisky formalism

TL;DR

This paper develops a noncommutative Batalin-Vilkovisky formalism to construct cohomology classes on the moduli space of Riemann surfaces, linking algebraic data to geometric invariants via Feynman diagrams.

Contribution

It introduces a novel approach using noncommutative Batalin-Vilkovisky formalism to generate nontrivial cohomology classes on moduli spaces from Frobenius algebras.

Findings

Constructed cohomology classes from Frobenius algebra data.

Expressed evaluations in terms of Feynman diagram expansions.

Demonstrated nontrivial classes through explicit examples.

Abstract

Using the machinery of the Batalin-Vilkovisky formalism, we construct cohomology classes on compactifications of the moduli space of Riemann surfaces from the data of a contractible differential graded Frobenius algebra. We describe how evaluating these cohomology classes upon a well-known construction producing homology classes in the moduli space can be expressed in terms of the Feynman diagram expansion of some functional integral. By computing these integrals for specific examples, we are able to demonstrate that this construction produces families of nontrivial classes.