A Classical Bulk-Boundary Correspondence

TL;DR

This paper establishes a classical bulk-boundary correspondence using $bP_0$-factorization algebras, linking Poisson BV theories on manifolds to their associated bulk-boundary systems, generalizing ideas behind deformation quantization.

Contribution

It introduces a novel framework connecting Poisson BV theories with bulk-boundary systems via $bP_0$-factorization algebras, extending the understanding of classical field theories.

Findings

Established a bulk-boundary correspondence for Poisson BV theories

Generalized the deformation quantization insight of Kontsevich

Utilized operadic homotopy theory of $bP_0$-algebras

Abstract

In this article, we use the language of $\mathbb{P}_0$-factorization algebras to articulate a classical bulk-boundary correspondence between 1) the observables of a Poisson Batalin-Vilkovisky (BV) theory on a manifold $N$ and 2) the observables of the associated universal bulk-boundary system on $N\times \mathbb{R}_{\geq 0}$. The archetypal such example is the Poisson BV theory on $\mathbb{R}$ encoding the algebra of functions on a Poisson manifold, whose associated bulk-boundary system on the upper half-plane is the Poisson sigma model. In this way, we obtain a generalization and partial justification of the basic insight that led Kontsevich to his deformation quantization of Poisson manifolds. The proof of these results relies significantly on the operadic homotopy theory of $\mathbb{P}_0$-algebras.