One-dimensional Chern–Simons theory and the $\hat{A}$ genus

Abstract

We construct a Chern–Simons gauge theory for dg Lie and $L$–infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin–Vilkovisky formalism and Costello’s renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a $1$–manifold into a cotangent bundle $T^{\ast }X$, as such a Chern–Simons theory. Our main result is that the effective action of this theory is naturally identified with the $Â$ class of $X$. From the perspective of derived geometry, our quantization constructs a projective volume form on the derived loop space $\mathcal{ℒ}X$ that can be identified with the $Â$ class.