Topological Open P-Branes

TL;DR

This paper explores the deformation quantization of topological open p-branes using BV quantization, revealing algebraic structures and bulk-boundary correspondences that generalize known mathematical conjectures and have implications for M-theory and string theory.

Contribution

It introduces a framework connecting BV quantization of open p-branes with p-algebra structures and generalizes the Deligne conjecture to higher algebraic contexts.

Findings

Topological open p-branes have (p+1)-algebra structures in the bulk and p-algebra structures on the boundary.

Bulk-boundary correspondences align with the generalized Deligne conjecture for p-algebras.

The algebra of quantum observables approximates classical observables as p-algebras, indicating deformation quantization.

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.