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. No.541. American Mathematical Society , Providence, RI, pp. 4167. ISBN 9780821849606 . http://resolver.caltech.edu/CaltechAUTHORS:20110707110110717

PDF
 Published Version
See Usage Policy. 1486Kb 
Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20110707110110717
Abstract
The volume conjecture states that for a hyperbolic knot K in the threesphere 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 threemanifolds.
Item Type:  Book Section  

Related URLs: 
 
Additional Information:  © 2011 American Mathematical Society. Submitted on 25 Mar 2010 (v1), last revised 26 Mar 2010 (this version, v2). T he work of SG is supported in part by DOE Grant DEFG0392ER40701, in part by NSF Grant PHY0757647, 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.  
Group:  Caltech Theory  
Funders: 
 
Classification Code:  2010 MSC Primary: 58J28, 81T45, 57M27  
Record Number:  CaltechAUTHORS:20110707110110717  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20110707110110717  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  24336  
Collection:  CaltechAUTHORS  
Deposited By:  Ruth Sustaita  
Deposited On:  11 Jul 2011 16:44  
Last Modified:  03 May 2016 04:43 
Repository Staff Only: item control page