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

Owen Gwilliam and Ryan Grady

Algebraic & Geometric Topology 14 (2014) 2299–2377

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 TX, as such a Chern–Simons theory. Our main result is that the effective action of this theory is naturally identified with the  class of X. From the perspective of derived geometry, our quantization constructs a projective volume form on the derived loop space X that can be identified with the  class.

$\hat{A}$ genus, BV formalism, Chern–Simons theory, topological quantum mechanics
Mathematical Subject Classification 2010
Primary: 57R56
Secondary: 18G55, 58J20
Received: 15 November 2012
Revised: 27 November 2013
Accepted: 28 November 2013
Published: 28 August 2014
Owen Gwilliam
Department of Mathematics
University of California, Berkeley
970 Evans Hall
Berkeley, CA 94720
Ryan Grady
Department of Mathematics and Statistics
Boston University
111 Cummington Mall
Boston, MA 02215