Deformation of BF theories, Topological Open Membrane and A Generalization of The Star Deformation

TL;DR

The paper analyzes the deformation of the $n$-dimensional abelian BF theory using the antifield BRST (BV) formalism to classify all consistent interactions, uncovering a new class of topological gauge theories. By formulating the BV superfield action and imposing the master equation, it derives deformations $S = S_0 + g S_1$ with $(S,S) = 0$ and $(S_1,S_1) = 0$, revealing an underlying $L_\infty$-algebra structure governing the deformed gauge symmetry. The work explores explicit lower-dimensional examples (2D Poisson sigma model and a 3D deformation with six structure maps) and extends to the quantum BV formalism, including gauge fixing and boundary observables that lead to a generalized star-product on boundaries and an $L_\infty$-structure of correlators. It further connects deformations of worldvolume BF theory to the physics of topological open membranes, proposing higher-dimensional formality and potential links to mirror symmetry via A/B-model perspectives.

Abstract

We consider a deformation of the BF theory in any dimension by means of the antifield BRST formalism. Possible consistent interaction terms for the action and the gauge symmetries are analyzed and we find a new class of topological gauge theories. Deformations of the world volume BF theory are considered as possible deformations of the topological open membrane. Therefore if we consider these theories on open membranes, we obtain noncommutative structures of the boundaries of open membranes, and we propose a generalization of the path integral representation of the star deformation.