Matrix De Rham complex and quantum A-infinity algebras

TL;DR

This paper links non-commutative BV-formalism with super-invariant matrix integration, representing quantum A-infinity-algebras via de Rham differentials on matrix spaces related to Bernstein-Leites algebras and gl(N|N).

Contribution

It establishes a novel connection between non-commutative BV-equations and matrix integrals, providing a geometric interpretation of quantum A-infinity-algebras.

Findings

Representation of BV-equation via de Rham differential on matrix spaces.

Identification of matrix integral Lagrangians as equivariantly closed forms.

Connection between non-commutative BV-formalism and super-invariant matrix integration.

Abstract

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A-infinity-algebras, introduced in "Modular operads and Batalin-Vilkovisky geometry" IMRN, Vol. 2007, doi: 10.1093/imrn/rnm075, is represented via de Rham differential acting on the matrix spaces related with Bernstein-Leites simple associative algebras with odd trace q(N), and with gl(N|N). I also show that the Lagrangians of the matrix integrals from "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals", Comptes Rendus Mathematique, vol 348 (2010), pp. 359-362, arXiv:0912.5484, are equivariantly closed differential forms.