Topological Open Membranes

TL;DR

This work develops a BV-formalism framework for topological open membranes in three dimensions, showing that bulk deformations induce a boundary $L_ olinebreak_infty$-algebra that encodes the boundary dynamics as a homotopy Lie structure. In simple cases the boundary algebra reduces to quasi-Lie bialgebras, while in general it forms a Courant algebroid or quasi-Lie bialgebroid; the master equation organizes these structures through derived brackets and canonical transformations. The authors relate the membrane quantization to a formal deformation quantization of Courant algebroids, via a path integral map from Hochschild cohomology to boundary $G_ olinebreak_infty$ structures, and connect to CFTs such as CS/WZW and to Dirac-gerbe geometry. The results unify several mathematical frameworks—quasi-Lie bialgebras, Courant algebroids, Dirac structures, and gerbes—with the physics of M-theory branes, offering a formal route to quantize Courant algebroids and to study C-field deformations of open string boundaries.

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.