Dimofte, Tudor and Gukov, Sergei (2011) Quantum Field Theory and the Volume Conjecture. In: Interactions between hyperbolic geometry, quantum topology, and number theory. Contemporary Mathematics (541). American Mathematical Society , Providence, RI, pp. 41-67. ISBN 978-0-8218-4960-6 http://resolver.caltech.edu/CaltechAUTHORS:20110707-110110717
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The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, nontrivial manner. The goal of the present lectures is to review the original statement of the volume conjecture and its recent extensions and generalizations, and to show how, in the most general context, the conjecture can be understood in terms of topological quantum field theory. In particular, we consider: a) generalization of the volume conjecture to families of incomplete hyperbolic metrics; b) generalization that involves not only the leading (volume) term, but the entire asymptotic expansion in 1/N; c) generalization to quantum group invariants for groups of higher rank; and d) generalization to arbitrary links in arbitrary three-manifolds.
|Item Type:||Book Section|
|Additional Information:||© 2011 American Mathematical Society. Submitted on 25 Mar 2010 (v1), last revised 26 Mar 2010 (this version, v2). The work of SG is supported in part by DOE Grant DE-FG03-92-ER40701, in part by NSF Grant PHY-0757647, and in part by the Alfred P. Sloan Foundation. Opinions and conclusions expressed here are those of the authors and do not necessarily reflect the views of funding agencies. We would like to thank Edward Witten, Don Zagier, and Jonatan Lenells for enlightening discussions on subjects considered in these notes. We would also like to thank the organizers of the workshop Interactions Between Hyperbolic Geometry, Quantum Topology, and Knot Theory and Columbia University for their generous support, accommodations, and collaborative working environment.|
|Classification Code:||2010 MSC Primary: 58J28, 81T45, 57M27|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||11 Jul 2011 16:44|
|Last Modified:||08 Mar 2013 22:08|
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