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

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.