Communications in Mathematical Physics

, Volume 255, Issue 3, pp 577–627

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

Authors

  • Sergei Gukov
    • Jefferson Physical LaboratoryHarvard University
Article

DOI: 10.1007/s00220-005-1312-y

Cite this article as:
Gukov, S. Commun. Math. Phys. (2005) 255: 577. 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2005