Article,
Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
Communications in Mathematical Physics, 255 (3): 577--627 (Jun 17, 2003)
DOI: 10.1007/s00220-005-1312-y
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.
