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*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.

61 pages, figures, final version

References in corpus (1)

• The tangent complex and Hochschild cohomology of E_n-rings

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• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Hopf algebras arising from dg manifolds