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Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

Gukov, Sergei (2005) Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial. Communications in Mathematical Physics, 255 (3). pp. 577-627. ISSN 0010-3616.

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We study three-dimensional Chern-Simons theory with complex gauge group SL(2,ℂ), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,ℂ) partition function and the colored Jones polynomial.

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URLURL TypeDescription Paper
Gukov, Sergei0000-0002-9486-1762
Additional Information:© 2005 Springer-Verlag. Received: 15 July 2003. Accepted: 5 October 2004. Published online: 2 March 2005. It is a pleasure to thank D. Bar-Natan, R. Dijkgraaf, N. Dunfield, S. Garoufalidis, R. Gopakumar, G. Horowitz, D. Long, M. Mariño, S. Minwalla, H. Ooguri, F. Rodriguez-Villegas, L. Rozansky, C. Vafa, E. Witten, S.-T. Yau, and especially K. Krasnov, G. Moore, A. Strominger, and D. Thurston for valuable and stimulating discussions. This research was conducted during the period S.G. served as a Clay Mathematics Institute Long-Term Prize Fellow. This work is also supported in part by RFBR grant 01-01-00549 and RFBR grant for Young Scientists 02-01-06322. I would also like to thank the University of California at Santa Barbara, Stanford University, California Institute of Technology, and Rutgers University for kind hospitality while this work was in progress. Communicated by G.W. Gibbons
Funding AgencyGrant Number
Russian Foundation for Basic Research01-01-00549
Russian Foundation for Basic Research02-01-06322
Clay Mathematics InstituteUNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20160509-071118060
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ID Code:66723
Deposited By: Ruth Sustaita
Deposited On:09 May 2016 17:25
Last Modified:09 Mar 2020 13:18

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