Topological Open Membranes

TL;DR

This paper investigates topological open membranes within a BV formalism, exploring how bulk deformations influence boundary operator algebras, leading to structures like homotopy Lie algebras and Courant algebroids, with implications for deformation quantization.

Contribution

It introduces a framework connecting topological open membranes to homotopy Lie algebras and Courant algebroids, proposing a method to address Courant algebroid quantization.

Findings

Boundary operator algebra forms a homotopy Lie algebra

Models relate to quasi-Lie bialgebras and Courant algebroids

Proposes a tool for quantizing Courant algebroids

Abstract

We study topological open membranes of BF type in a manifest BV formalism. Our main interest is the effect of the bulk deformations on the algebra of boundary operators. This forms a homotopy Lie algebra, which can be understood in terms of a closed string field theory. The simplest models are associated to quasi-Lie bialgebras and are of Chern-Simons type. More generally, the induced structure is a Courant algebroid, or ``quasi-Lie bialgebroid'', with boundary conditions related to Dirac bundles. A canonical example is the topological open membrane coupling to a closed 3-form, modeling the deformation of strings by a C-field. The Courant algebroid for this model describes a modification of deformation quantization. We propose our models as a tool to find a formal solution to the quantization problem of Courant algebroids.