Quantum Field Theory and the Volume Conjecture
arXiv:1003.4808
Abstract
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.
32 pages, 6 figures; acknowledgements updated
References in corpus (7)
- Ricci flow with surgery on three-manifolds
- Analytic Continuation Of Chern-Simons Theory
- Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
- On the characteristic and deformation varieties of a knot
- SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
- The colored Jones polynomials and the Alexander polynomial of the figure-eight knot
- Some limits of the colored Jones polynomials of the figure-eight knot