Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals

TL;DR

This paper introduces a novel class of supersymmetric matrix models linked to solutions of the noncommutative Batalin-Vilkovisky quantum master equation, revealing connections to moduli space homology and tautological classes.

Contribution

It develops a new framework connecting noncommutative BV geometry with matrix integrals and moduli space cohomology, extending the Kontsevich model to supersymmetric cases.

Findings

Asymptotic expansion of matrix integrals yields homology classes in moduli space.

Constructs cohomology classes from queer matrix superalgebra with odd differentiation.

Supersymmetric extension of the Kontsevich model for 2D gravity.

Abstract

We associate the new type of supersymmetric matrix models with any solution to the quantum master equation of the noncommutative Batalin-Vilkovisky geometry. The asymptotic expansion of the matrix integrals gives homology classes in the Kontsevich compactification of the moduli spaces, which we associated with the solutions to the quantum master equation in our previous paper. We associate with the queer matrix superalgebra equipped with an odd differentiation, whose square is nonzero, the family of cohomology classes of the compactification. This family is the generating function for the products of the tautological classes. The simplest example of the matrix integrals in the case of dimension zero is a supersymmetric extenstion of the Kontsevich model of 2-dimensional gravity.