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


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.


Partition Function Gauge Group Wilson Loop Quantum Computing Algebraic Curve 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

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

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