Quantum Field Theory and the Volume Conjecture

TL;DR

This paper reviews the volume conjecture linking hyperbolic volume and colored Jones polynomials, exploring its extensions through topological quantum field theory, including generalizations to incomplete metrics, asymptotic expansions, higher rank quantum groups, and arbitrary links.

Contribution

It provides a comprehensive overview of the volume conjecture and introduces recent extensions and generalizations within the framework of topological quantum field theory.

Findings

Extension to incomplete hyperbolic metrics

Asymptotic expansion in 1/N beyond leading order

Generalization to higher rank quantum groups and arbitrary links

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.