#### Volume 14, issue 4 (2014)

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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
##### Abstract
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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 $\mathsc{ℒ}X$ that can be identified with the $Â$ class.

##### Keywords
$\hat{A}$ genus, BV formalism, Chern–Simons theory, topological quantum mechanics
##### Mathematical Subject Classification 2010
Primary: 57R56
Secondary: 18G55, 58J20