Linear Batalin-Vilkovisky quantization as a functor of $\infty$-categories

TL;DR

This paper develops a homotopical framework for linear BV quantization as a symmetric monoidal functor of ∞-categories, extending classical concepts to derived algebraic geometry and higher algebra contexts.

Contribution

It introduces a novel ∞-categorical construction of linear BV quantization with properties extending to derived geometry and higher Morita categories.

Findings

Constructs a symmetric monoidal functor for BV quantization

Extends BV quantization to derived algebraic geometry

Provides algebraic construction of E_n-enveloping algebras

Abstract

We study linear Batalin-Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of $\infty$-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of $E_n$-algebras to produce a "higher BV quantization" functor, and when restricted to formal moduli problems, it behaves like a determinant. Along the way we also use our machinery to give an algebraic construction of $E_n$-enveloping algebras for shifted Lie algebras.