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One-dimensional
Chern–Simons theory and the $\hat{A}$ genus
Owen Gwilliam and Ryan Grady |
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Algebraic & Geometric Topology 14 (2014) 2299–2377
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Abstract |
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We construct a Chern–Simons gauge theory for dg Lie and –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 –manifold into a cotangent bundle , as such a Chern–Simons theory. Our main result is that the effective action of this theory is naturally identified with the class of . From the perspective of derived geometry, our quantization constructs a projective volume form on the derived loop space that can be identified with the class. |
Keywords
$\hat{A}$ genus, BV formalism, Chern–Simons theory,
topological quantum mechanics
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Mathematical Subject Classification 2010
Primary: 57R56
Secondary: 18G55, 58J20
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References |
Publication
Received: 15 November 2012
Revised: 27 November 2013
Accepted: 28 November 2013
Published: 28 August 2014
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Authors |
| Owen Gwilliam | |
| Department of Mathematics University of California, Berkeley 970 Evans Hall Berkeley, CA 94720 USA |
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| Ryan Grady | |
| Department of Mathematics and
Statistics Boston University 111 Cummington Mall Boston, MA 02215 USA |
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