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

Understand

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*X, as such a Chern-Simons theory.
• Our main result is that the partition function of this theory is naturally identified with the A-genus of X.
• From the perspective of derived geometry, our quantization construct a volume form on the derived loop space which can be identified with the A-class.