Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial

TL;DR

This paper explores the deep connections between three-dimensional Chern-Simons theory with complex gauge group, hyperbolic 3-manifold invariants, and knot polynomials, revealing new relations and generalizations of existing conjectures.

Contribution

It establishes a novel link between the A-polynomial, quantum invariants, and hyperbolic geometry in SL(2,C) Chern-Simons theory, extending known conjectures.

Findings

A-polynomial describes classical and quantum properties of the theory.

New relations between A-polynomial, colored Jones polynomial, and hyperbolic invariants.

Generalizations of the volume conjecture and Melvin-Morton-Rozansky conjecture.

Abstract

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.