ThreeDimensional Quantum Gravity, ChernSimons Theory, and the APolynomial
Abstract
We study threedimensional ChernSimons theory with complex gauge group SL(2,C), which has many interesting connections with threedimensional quantum gravity and geometry of hyperbolic 3manifolds. We show that, in the presence of a single knotted Wilson loop in an infinitedimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the Apolynomial of a knot. Using this approach, we find some new and rather surprising relations between the Apolynomial, the colored Jones polynomial, and other invariants of hyperbolic 3manifolds. These relations generalize the volume conjecture and the MelvinMortonRozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 April 2005
 DOI:
 10.1007/s002200051312y
 arXiv:
 arXiv:hepth/0306165
 Bibcode:
 2005CMaPh.255..577G
 Keywords:

 High Energy Physics  Theory;
 General Relativity and Quantum Cosmology;
 Mathematics  Geometric Topology;
 Mathematics  Quantum Algebra
 EPrint:
 67 pages, 13 figures, harvmac