BV Quantization of Topological Open Membranes

TL;DR

This paper develops a BV sigma-model description of topological open membranes with a WZ coupling to a 3-form, framing bulk-boundary interactions as deformations of the boundary $L_\infty$ (closed string field) structure. By computing propagators and tree-level correlators, the authors demonstrate that a bulk 3-form induces a trilinear bracket on boundary polyvector fields, deforming the usual Schouten–Nijenhuis structure and realizing a semi-classical quasi-Lie bialgebroid. The results establish a concrete link between Courant algebroid deformations, boundary algebra, and a potential universal quantization to quasi-Hopf algebroids, setting the stage for higher-order and nonperturbative quantization (to be explored in future work). The work also clarifies the role of point-splitting regularization in extracting topological, deformation-theoretic brackets from bulk couplings. Overall, it provides a first explicit bridge from topological membranes to quasi-Lie bialgebroid quantization via boundary string field theory.

Abstract

We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids.

Figures (1)

• Figure 1: The point splitting regularization, where $x$ and the contour over $y$ are taken on the boundary, and the bulk integral is performed over a region a distance $\epsilon$ away from the boundary (shaded region).