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Letters in Mathematical Physics

, Volume 86, Issue 2–3, pp 79–98 | Cite as

\({SL(2, \mathbb{C})}\) Chern–Simons Theory and the Asymptotic Behavior of the Colored Jones Polynomial

  • Sergei GukovEmail author
  • Hitoshi Murakami
Article

Abstract

It has been proposed that the asymptotic behavior of the colored Jones polynomial is equal to the perturbative expansion of the Chern–Simons gauge theory with complex gauge group \({SL(2, \mathbb{C})}\) on the hyperbolic knot complement. In this note we make the first step toward verifying this relation beyond the semi-classical approximation. This requires a careful understanding of some delicate issues, such as normalization of the colored Jones polynomial and the choice of polarization in Chern–Simons theory. Addressing these issues allows us to go beyond the volume conjecture and to verify some predictions for the behavior of the subleading terms in the asymptotic expansion of the colored Jones polynomial.

Mathematics Subject Classification (2000)

Primary 57M27 57M25 57M50 58J28 58J52 

Keywords

colored Jones polynomial volume conjecture A-polynomial Chern–Simons theory 

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References

  1. 1.
    Axelrod, S., Singer, I.M.: Chern–Simons perturbation theory. In: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, vols. 1, 2 (New York, 1991) (River Edge, NJ), pp. 3–45. World Scientific Publishing, Singapore (1992)Google Scholar
  2. 2.
    Bar-Natan, D.: Perturbative aspects of the Chern–Simons topological quantum field theory. Ph.D. thesis, Princeton Univeristy (1991)Google Scholar
  3. 3.
    Boden H.U., Herald C.M., Kirk P.A., Klassen E.P.: Gauge theoretic invariants of Dehn surgeries on knots. Geom. Topol. 5, 143–226 (2001) electroniczbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cheeger J.: Analytic torsion and the heat equation (2). Ann. Math. 109(2), 259–322 (1979)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cooper D., Culler M., Gillet H., Long D.D., Shalen P.B.: Plane curves associated to character varieties of 3-manifolds. Invent. Math. 118(1), 47–84 (1994)zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Dubois J.: Non Abelian twisted Reidemeister torsion for fibered knots. Can. Math. Bull. 49(1), 55–71 (2006)zbMATHGoogle Scholar
  7. 7.
    Freed D.S., Gompf R.E.: Computer calculation of Witten’s 3-manifold invariant. Commun. Math. Phys. 141(1), 79–117 (1991)zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Garoufalidis, S., Le, T.T.Q.: An analytic version of the Melvin–Morton–Rozansky conjecture. arXiv:math.GT/0503641Google Scholar
  9. 9.
    Garoufalidis, S., Le, T.T.Q.: Asymptotics of the colored Jones function of a knot. arXiv:math/0508100Google Scholar
  10. 10.
    Gukov S.: Three-dimensional quantum gravity, Chern–Simons theory, and the A-polynomial. Commun. Math. Phys. 255(3), 577–627 (2005)zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Hodgson C.D., Kerckhoff S.P.: Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differ. Geom. 48(1), 1–59 (1998)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Kashaev R.M.: The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3), 269–275 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lickorish W.B.R.: An introduction to knot theory. Graduate Texts in Mathematics, vol. 175. Springer, New York (1997)Google Scholar
  14. 14.
    Müller W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28(3), 233–305 (1978)zbMATHCrossRefGoogle Scholar
  15. 15.
    Murakami H.: A version of the volume conjecture. Adv. Math. 211(2), 678–683 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Murakami H.: The colored Jones polynomials and the Alexander polynomial of the figure-eight knot. J.P.J. Geom. Topol. 7(2), 219–269 (2007)MathSciNetGoogle Scholar
  17. 17.
    Murakami H.: Some limits of the colored Jones polynomials of the figure-eight knot. Kyungpook Math. J. 44(3), 369–383 (2004)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Murakami H., Murakami J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Murakami, H., Yokota, Y.: The colored Jones polynomials of the figure-eight knot and its Dehn surgery spaces. J. Reine Angew. Math. (2008, in press). arXiv:math.GT/0401084Google Scholar
  20. 20.
    Neumann W.D., Zagier D.: Volumes of hyperbolic three-manifolds. Topology 24(3), 307–332 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Porti, J. (1997) Torsion de Reidemeister pour les variétés hyperboliques. Mem. Am. Math. Soc. 128(612), x + 139Google Scholar
  22. 22.
    Ray D.B., Singer I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Ray D.B., Singer I.M.: Analytic torsion for complex manifolds. Ann. Math. (2) 98, 154–177 (1973)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Weil A.: On discrete subgroups of Lie groups. Ann. Math. (2) 72, 369–384 (1960)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Weil A.: On discrete subgroups of Lie groups. II. Ann. Math (2) 75, 578–602 (1962)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Weil A.: Remarks on the cohomology of groups. Ann. Math. (2) 80, 149–157 (1964)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)zbMATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Woodhouse N.M.J.: Geometric quantization. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1992)Google Scholar
  29. 29.
    Zheng H.: Proof of the volume conjecture for Whitehead doubles of a family of torus knots. Chin. Ann. Math. Ser. B 28(4), 375–388 (2007)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Department of Physics and MathematicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of MathematicsTokyo Institute of TechnologyTokyoJapan

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