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Communications in Mathematical Physics

, Volume 255, Issue 3, pp 577–627 | Cite as

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

  • Sergei Gukov
Article

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.

Keywords

Partition Function Gauge Group Wilson Loop Quantum Computing Algebraic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Sergei Gukov
    • 1
  1. 1.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA

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