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Quantum field theory and the volume conjecture. (English) Zbl 1236.57001
Champagnerkar, Abhijit (ed.) et al., Interactions between hyperbolic geometry, quantum topology and number theory. Proceedings of a workshop, June 3–13, 2009 and a conference, June 15–19, 2009, Columbia University, New York, NY, USA. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4960-6/pbk). Contemporary Mathematics 541, 41-67 (2011).
Summary: 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 $$S^3\setminus K$$. 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; 6) 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.
For the entire collection see [Zbl 1214.00022].

##### MSC:
 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57M50 General geometric structures on low-dimensional manifolds 81T45 Topological field theories in quantum mechanics 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)