Quantum BV $\mathcal{L}_{\infty}$-algebras I: Derived geometric foundations

TL;DR

This paper introduces quantum BV $$-algebras, exploring their fundamental properties and homotopy structures relevant to Chern-Simons theory, laying the groundwork for future applications.

Contribution

It defines quantum BV $$-algebras, studies their core properties, and introduces homotopy BV data and orientations in the context of derived geometry.

Findings

Introduction of quantum BV $$-algebras

Analysis of homotopy Lie structures in Chern-Simons theory

Foundational properties for future applications

Abstract

We introduce the concept of a quantum BV $\mathcal{L}_{\infty}$-algebra and study fundamental properties. In particular, we investigate homotopy Lie theoretic structures that naturally arise in the context of Chern-Simons theory. Of note, are the notions of homotopy BV data and of a BV orientation. The sequel of this paper will involve the direct application of these constructions to the setting of Chern-Simons theory.