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

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

2003-06-17

arxiv.org/abs/hep-th/0306165v1

Commun.Math.Phys. 255 (2005) 577-627

Physics

High Energy Physics

High Energy Physics - Theory

67 pages, 13 figures, harvmac

Scientific paper

10.1007/s00220-005-1312-y

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.

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