Communications in Mathematical Physics

, Volume 359, Issue 3, pp 915–936 | Cite as

All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern–Simons Theory

Article
  • 41 Downloads

Abstract

We propose an extension of the recently-proposed volume conjecture for closed hyperbolic 3-manifolds, to all orders in perturbative expansion. We first derive formulas for the perturbative expansion of the partition function of complex Chern–Simons theory around a hyperbolic flat connection, which produces infinitely-many perturbative invariants of the closed oriented 3-manifold. The conjecture is that this expansion coincides with the perturbative expansion of the Witten–Reshetikhin–Turaev invariants at roots of unity \({q=e^{2\pi i/r}}\) with r odd, in the limit \({r\to \infty}\). We provide numerical evidence for our conjecture.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Witten E.: Quantization of Chern–Simons gauge theory with complex gauge group. Commun. Math. Phys. 137, 29 (1991).  https://doi.org/10.1007/BF02099116 ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Witten E.: Analytic continuation of Chern–Simons theory. AMS/IP Stud. Adv. Math. 50, 347 (2011) arXiv:1001.2933 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, Q., Yang, T.: A volume conjecture for a family of Turaev–Viro type invariants of 3-manifolds with boundary. arXiv:1503.02547
  4. 4.
    Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989).  https://doi.org/10.1007/BF01217730 ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Reshetikhin N., Turaev V.G.: Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547 (1991).  https://doi.org/10.1007/BF01239527 ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kashaev R.M.: A link invariant from quantum dilogarithm. Mod. Phys. Lett. A 10, 1409 (1995).  https://doi.org/10.1142/S0217732395001526 ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Murakami H., Murakami J.: The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186, 85 (2001).  https://doi.org/10.1007/BF02392716 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Murakami, H., Murakami, J., Okamoto, M., Takata, T., Yokota, Y.: Kashaev’s conjecture and the Chern–Simons invariants of knots and links. Exp. Math. 11, 427 (2002) http://projecteuclid.org/euclid.em/1057777432
  9. 9.
    Murakami, H.: An introduction to the volume conjecture. In: Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory, pp. 1–40. American Mathematical Society, Providence, RI (2011) https://doi.org/10.1090/conm/541/10677
  10. 10.
    Jones V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc. (N. S.) 12, 103 (1985).  https://doi.org/10.1090/S0273-0979-1985-15304-2 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 12, 5 (1982) http://www.numdam.org/item?id=PMIHES_1982_56_5_0
  12. 12.
    Habiro K.: A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres. Invent. Math. 171, 1 (2008).  https://doi.org/10.1007/s00222-007-0071-0 ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Murakami, H.: Optimistic calculations about the Witten–Reshetikhin–Turaev invariants of closed three-manifolds obtained from the figure-eight knot by integral Dehn surgeries. In: Sūrikaisekikenkyūsho Kōkyūroku 171, 70 (2000). Recent progress towards the volume conjecture (Japanese) (Kyoto, 2000)Google Scholar
  14. 14.
    Costantino F.: 6j-symbols, hyperbolic structures and the volume conjecture. Geom. Topol. 11, 1831 (2007).  https://doi.org/10.2140/gt.2007.11.1831 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Costantino F., Murakami J.: On the \({{\rm SL}(2,{\mathbb{C}})}\) quantum 6j-symbols and their relation to the hyperbolic volume. Quantum Topol. 4, 303 (2013).  https://doi.org/10.4171/QT/41 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kirby R., Melvin P.: The 3-manifold invariants of Witten and Reshetikhin–Turaev for sl(2, C). Invent. Math. 105, 473 (1991).  https://doi.org/10.1007/BF01232277 ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Blanchet C., Habegger N., Masbaum G., Vogel P.: Three-manifold invariants derived from the Kauffman bracket. Topology 31, 685 (1992).  https://doi.org/10.1016/0040-9383(92)90002-Y MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lickorish W.B.R.: The skein method for three-manifold invariants. J. Knot Theory Ramif. 2, 171 (1993).  https://doi.org/10.1142/S0218216593000118 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lickorish, W.B.R.: Invariants for 3-manifolds from the combinatorics of the Jones polynomial. Pac. J. Math. 149, 337 (1991). http://projecteuclid.org/euclid.pjm/1102644467
  20. 20.
    Lickorish W.B.R.: Three-manifolds and the Temperley–Lieb algebra. Math. Ann. 290, 657 (1991).  https://doi.org/10.1007/BF01459265 MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lickorish W.B.R.: Calculations with the Temperley–Lieb algebra. Comment. Math. Helv. 67, 571 (1992).  https://doi.org/10.1007/BF02566519 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Blanchet, C., Habegger, N., Masbaum, G., Vogel, P.: Remarks on the three-manifold invariants \({\theta _p}\). In: Operator Algebras, Mathematical Physics, and Low-Dimensional Topology (Istanbul, 1991), pp. 39–59. A K Peters, Wellesley, MA (1993)Google Scholar
  23. 23.
    Bae J-B, Gang D, Lee J: 3d \({{\mathcal N} =2}\) minimal SCFTs from wrapped M5-branes. J. High Energy Phys. 2017(8), 118 (2017) arXiv:1610.09259 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Terashima Y., Yamazaki M.: SL(2,R) Chern–Simons, Liouville, and gauge theory on duality walls. JHEP 1108, 135 (2011).  https://doi.org/10.1007/JHEP08(2011)135 arXiv:1103.5748 ADSCrossRefMATHGoogle Scholar
  25. 25.
    Dimofte, T., Gaiotto, D., Gukov, S.: Gauge theories labelled by three-manifolds. arXiv:1108.4389
  26. 26.
    Ohtsuki, T.: On the asymptotic expansion of the quantum SU(2) invariant at \({q = \exp (4\pi \sqrt{-1}/N)}\) for closed hyperbolic 3-manifolds obtained by integral surgery along the figure-eight knot. PreprintGoogle Scholar
  27. 27.
    Lickorish W.B.R.: A representation of orientable combinatorial 3-manifolds. Ann. Math. 2(76), 531 (1962).  https://doi.org/10.2307/1970373 arXiv:1103.5748 MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wallace A.H.: Modifications and cobounding manifolds. Can. J. Math. 12, 503 (1960).  https://doi.org/10.4153/CJM-1960-045-7 MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Hikami K.: Generalized volume conjecture and the A-polynomials: the Neumann–Zagier potential function as a classical limit of the partition function. J. Geom. Phys. 57, 1895 (2007).  https://doi.org/10.1016/j.geomphys.2007.03.008 arXiv:math/0604094 ADSMathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Dimofte T.: Quantum riemann surfaces in Chern–Simons theory. Adv. Theor. Math. Phys. 17, 479 (2013).  https://doi.org/10.4310/ATMP.2013.v17.n3.a1 arXiv:1102.4847 MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Andersen J.E., Kashaev R.: A TQFT from quantum Teichmller theory. Commun. Math. Phys. 330, 887 (2014).  https://doi.org/10.1007/s00220-014-2073-2 arXiv:1109.6295 ADSCrossRefMATHGoogle Scholar
  32. 32.
    Dimofte T.D., Garoufalidis S.: The quantum content of the gluing equations. Geom. Topol. 17, 1253 (2013) arXiv:1202.6268 MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Ohtsuki T., Takata T.: On the Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots. Geom. Topol. 19, 853 (2015)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Ohtsuki T.: On the asymptotic expansion of the Kashaev invariant of the 52 knot. Quantum Topol. 7(4), 669–735 (2016)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ohtsuki, T.: On the asymptotic expansion of the Kashaev invariant of the hyperbolic knots with 7 crossings. PreprintGoogle Scholar
  36. 36.
    Ohtsuki, T., Yokota Y.: On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In: Mathematical Proceedings of the Cambridge Philosophical Society, pp. 1–53Google Scholar
  37. 37.
    Gukov S.: Three-dimensional quantum gravity, Chern–Simons theory, and the A polynomial. Commun. Math. Phys. 255, 577 (2005).  https://doi.org/10.1007/s00220-005-1312-y arXiv:hep-th/0306165 ADSMathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Faddeev L.D., Kashaev R.M.: Quantum dilogarithm. Mod. Phys. Lett. A 9, 427 (1994).  https://doi.org/10.1142/S0217732394000447 ADSMathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Neumann W.D.: Extended Bloch group and the Cheeger–Chern–Simons class. Geom. Topol. 8, 413 (2004).  https://doi.org/10.2140/gt.2004.8.413 arXiv:math/0307092 MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Alday L.F., Genolini P.B., Bullimore M., van Loon M.: Refined 3d–3d correspondence. J. High Energy Phys. 2017(4), 170 (2017) arXiv:1702.05045 MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Terashima Y., Yamazaki M.: 3d \({\mathcal{N}=2}\) theories from cluster algebras. PTEP 023, B01 (2014).  https://doi.org/10.1093/ptep/PTT115 arXiv:1301.5902 MATHGoogle Scholar
  42. 42.
    Gang D., Kim N., Romo M., Yamazaki M.: Taming supersymmetric defects in 3d–3d correspondence. J. Phys. A 49, 30LT02 (2016).  https://doi.org/10.1088/1751-8113/49/30/30LT02 arXiv:1510.03884 MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Gang D., Kim N., Romo M., Yamazaki M.: Aspects of defects in 3d–3d correspondence. JHEP 1610, 062 (2016).  https://doi.org/10.1007/JHEP10(2016)062 arXiv:1510.05011 ADSMathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Kirby R.: A calculus for framed links in S 3. Invent. Math. 45, 35 (1978).  https://doi.org/10.1007/BF01406222 ADSMathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Dimofte T., Gukov S., Lenells J., Zagier D.: Exact results for perturbative Chern–Simons theory with complex gauge group. Commun. Number Theory Phys. 3, 363 (2009) arXiv:0903.2472 MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Axelrod, S., Singer, I.M.: Chern–Simons perturbation theory. In: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991), pp. 3–45, World Scientific Publishing, River Edge, NJ (1992)Google Scholar
  47. 47.
    Axelrod, S., Singer, I.M.: Chern–Simons perturbation theory. II. J. Differen. Geom. 39, 173 (1994) http://projecteuclid.org/euclid.jdg/1214454681
  48. 48.
    Neumann WD, Zagier D: Volumes of hyperbolic three-manifolds. Topology 24, 307 (1985).  https://doi.org/10.1016/0040-9383(85)90004-7 MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Dimofte T., Gabella M., Goncharov A.B.: K-decompositions and 3d gauge theories. J. High Energy Phys. 2016(11), 151 (2016) arXiv:1301.0192 ADSMathSciNetCrossRefGoogle Scholar
  50. 50.
    Chung H.-J., Dimofte T., Gukov S., Sulkowski P.: 3d–3d Correspondence revisited. J. High Energy Phys. 2016(4), 140 (2016) arXiv:1405.3663 MathSciNetCrossRefGoogle Scholar
  51. 51.
    Porti, J.: Reidemeister torsion, hyperbolic three-manifolds, and character varieties. ArXiv e-prints 24, (2015). arXiv:1511.00400
  52. 52.
    Culler, M., Dunfield, N., Weeks, J.R.: SnapPy, a computer program for studying the geometry and topology of 3-manifolds. 24. http://snappy.computop.org
  53. 53.
    Habiro, K.: On the colored Jones polynomials of some simple links. Sūrikaisekikenkyūsho Kōkyūroku 24, 34 (2000). Recent progress towards the volume conjecture (Japanese) (Kyoto, 2000)Google Scholar
  54. 54.
    Masbaum G.: Skein-theoretical derivation of some formulas of Habiro. Algebr. Geom. Topol. 3, 537 (2003).  https://doi.org/10.2140/agt.2003.3.537 MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Rolfsen, D.: Surgery calculus: extension of Kirby’s theorem. Pac. J. Math. 110, 377 (1984) http://projecteuclid.org/euclid.pjm/1102710925

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Dongmin Gang
    • 1
  • Mauricio Romo
    • 2
  • Masahito Yamazaki
    • 1
  1. 1.Kavli IPMUUniversity of TokyoKashiwaJapan
  2. 2.School of Natural SciencesInstitute for Advanced StudyPrincetonUSA

Personalised recommendations