Topological Open P-Branes

TL;DR

This work develops a BV-quantized framework for topological open $p$-branes, showing that a flat $C$-field induces a deformation of boundary polyvector algebras into higher homotopy structures while the bulk carries a $(p+1)$-algebra, with a bulk/boundary correspondence reflecting the generalized Deligne conjecture. It renders a deformation-quantization interpretation of $(p-1)$-brane observables and connects these deformations to Hochschild cohomology and the Hochschild–Kostant–Rosenberg-type structures in the $p$-algebra setting. The paper provides concrete model realizations for strings and membranes, including explicit BV actions, descent observables, and gauge-fixing schemes, and demonstrates how turning on background fields like the $C$-field yields tangible boundary theories in bivector or bivector-background settings. Finally, it maps these constructions to extended moduli spaces and extended B-model frameworks, arguing for a generalized homological mirror symmetry that encompasses $p$-algebras and open $p$-branes, with potential applications to $A_\infty$-deformations of $D^{b}\mathrm{Coh}(X)$ and a broader categorical mirror symmetry landscape.

Abstract

By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3-form C-field leads to deformations of the algebras of multi-vectors on the Dirichlet-brane world-volume as 2-algebras. This would shed some new light on geometry of M-theory 5-brane and associated decoupled theories. We show that, in general, topological open p-brane theory has a structure of (p+1)-algebra in the bulk, while a structure of p-algebra in the boundary. The bulk/boundary correspondences are exactly as the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of p-algebras. It also imply that the algebras of quantum observables of (p-1)-brane are ``close to'' the algebras of its classical observables as p-algebras. We interpret above as deformation quantization of (p-1)-brane, generalizing the p=1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to topological strings and conjecture that the homological mirror symmetry has further generalizations to the categories of p-algebras.