Inventiones mathematicae

, Volume 201, Issue 2, pp 519–559 | Cite as

Construction of the Witten–Reshetikhin–Turaev TQFT from conformal field theory

Article

Abstract

In Andersen and Ueno (J Knot Theory Ramif 16:127–202, 2007) we constructed the vacua modular functor based on the sheaf of vacua theory developed in Tsuchiya et al. (Adv Stud Pure Math 19:459–566, 1989) and the abelian analog in Andersen and Ueno (Int J Math 18:919–993, 2007). We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten–Reshetikhin–Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vogel to the vacua modular functor. This thus provides a geometric construction of the TQFT first proposed by Witten and constructed first by Reshetikhin–Turaev from the quantum group \(U_q(\text{ sl }(N))\).

References

  1. 1.
    Aiston, A.K.: Skein theoretic idempotents of Hecke algebras and quantum group invariants. Ph.D. Thesis, University of Liverpool, Liverpool (1996)Google Scholar
  2. 2.
    Aiston, A.K., Morton, H.R.: Idempotents of Hecke algebras of type A. J. Knot Theory Ramif. 7, 463–487 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andersen, J.E.: The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I. J. Reine und angevantes Matematik 681, 1–36 (2013). (Aarhus University preprint 1995)CrossRefGoogle Scholar
  4. 4.
    Andersen, J.E., Masbaum, G.: Involutions on moduli spaces and refinements of the Verlinde formula. Math. Ann. 314, 291–326 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Andersen, J.E.: Asymptotic faithfulness of the quantum \(SU(n)\) representations of the mapping class groups. Ann. Math. 163, 347–368 (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Andersen, J.E., Grove, J.: Fixed varieties in the moduli space of semi-stable vector bundles. Oxf. Q. J. Math. 57, 1–35 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Andersen, J.E., Masbaum, G., Ueno, K.: Topological quantum field theory and the Nielsen–Thurston classification of \(M(0,4)\). Math. Proc. Camb. Philos. Soc. 141, 477–488 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Andersen, J.E., Ueno, K.: Abelian conformal field theories and determinant bundles. Int. J. Math. 18, 919–993 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Andersen, J.E., Ueno, K.: Geometric construction of modular functors from conformal field theory. J. Knot Theory Ramif. 16, 127–202 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Andersen, J.E.: The Nielsen–Thurston classification of mapping classes is determined by TQFT. J. Math. Kyoto Univ. 48(2), 323–338 (2008)MathSciNetMATHGoogle Scholar
  11. 11.
    Andersen, J.E.: Asymptotics of the Hilbert–Schmidt norm in TQFT. Lett. Math. Phys. 91, 205–214 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Andersen, J.E.: Toeplitz operators and Hitchin’s connection. In: Bourguignon, J.P., Garcia-Prada, O., Salamon, S. (eds.) The many facets of geometry: a tribute to Nigel Hitchin. Oxford University Press, Oxford (2010)Google Scholar
  13. 13.
    Andersen, J.E., Gammelgaard, N.L.: Hitchin’ s projectively flat connection, Toeplitz operators and the asymptotic expansion of TQFT curve operators. In: Grassmannians, Moduli Spaces and Vector Bundles, pp. 1–24. Clay Mathematics Proceedings, vol. 14. American Mathematical Society, Providence (2011)Google Scholar
  14. 14.
    Andersen, J.E., Blaavand, J.: Asymptotics of Toeplitz operators and applications in TQFT. Traveaux Mathématiques 19, 167–201 (2011)MathSciNetGoogle Scholar
  15. 15.
    Andersen, J.E., Ueno, K.: Modular functors are determined by their genus zero data. Quantum Topol. 3(3–4), 255–291 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Andersen, J.E., Himpel, B.: The Witten–Reshetikhin–Turaev invariants of finite order mapping tori II. Quantum Topol. 3, 377–421 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Andersen, J.E.: Mapping Class Groups do not have Kazhdan’s Property (T). arXiv:0706.2184
  18. 18.
    Andersen, J.E.: Mapping class group invariant unitarity of the Hitchin connection over Teichmüller space. arXiv:1206.2635
  19. 19.
    Andersen, J.E.: A geometric formula for the Witten–Reshetikhin–Turaev quantum invariants and some applications. arXiv:1206.2785
  20. 20.
    Atiyah, M.F.: On framings of 3-manifolds. Topology 29, 1–7 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Atiyah, M.F.: The geometry and physics of knots. In: Lezioni Lincee [Lincei Lectures]. Cambridge University Press, Cambridge (1990)Google Scholar
  22. 22.
    Bakalov, B., Jr., Kirillov, A.A: Lectures on tensor categories and modular functors. In: University Lecture Series, vol. 21. American Mathematical Society, Providence (2001)Google Scholar
  23. 23.
    Blanchet, C.: Hecke algebras, modular categories and \(3\)-manifolds quantum invariants. Topology 39(1), 193–223 (2000)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Blanchet, C., Habegger, N., Masbaum, G., Vogel, P.: Three-manifold invariants derived from the Kauffman bracket. Topology 31(4), 685–699 (1992)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Blanchet, C., Habegger, N., Masbaum, G., Vogel, P.: Topological quantum field theories derived from the Kauffman bracket. Topology 34(4), 883–927 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Funar, L.: On the TQFT representations of the mapping class group. Pac. J. Math. 188, 251–274 (1999)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Freyd, P., Yetter, D., Horste, J., Lickorish, W.B.R., Millett, K.C., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. AMS 12, 239–335 (1985)CrossRefMATHGoogle Scholar
  28. 28.
    Grove, J.: Constructing TQFTs from modular functors. J. Knot Theory Ramif. 10(8), 1085–1131 (2001)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Gyoja, A.: A q-analogue of Young symmetriser. Osaka J. Math. 23, 841–852 (1986)MathSciNetMATHGoogle Scholar
  30. 30.
    Hitchin, N.J.: Flat connections and geometric quantization. Comm. Math. Phys. 131, 347–380 (1990)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jones, V.F.R.: Hecke algebra representations of braid groups. Ann. Math. 126, 335–388 (1987)CrossRefMATHGoogle Scholar
  32. 32.
    Kanie, Y.: Conformal field theory and the braid group. Bull. Fac. Edu. Mie Univ. 40, 1–43 (1989)Google Scholar
  33. 33.
    Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y.: Geometric realization of conformal field theory on Riemann surfaces. Comm. Math. Phys. 116, 247–308 (1988)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kontsevich, M.: Rational conformal field theory and invariants of 3-manifolds. Preprint of Centre de Physique Theorique Marseille, CPT-88/p2189 (1988)Google Scholar
  35. 35.
    Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess–Zumino model in two dimensions. Nucl. Phys. B 247, 83–103 (1984)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Kohno, T.: Three-manifold invariants derived from conformal field theory and projective representations of modular groups. Int. J. Mod. Phys. 6, 1795–1805 (1992)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Laszlo, Y.: Hitchin’s and WZW connections are the same. J. Diff. Geom. 49(3), 547–576 (1998)Google Scholar
  38. 38.
    Laszlo, Y., Pauly, C., Sorger, C.: On the monodromy of the Hitchin connection. J. Geom. Phys. 64, 64–78 (2013)Google Scholar
  39. 39.
    Lickorish, W.B.R.: Sampling the \(SU(N)\) invariants of three-manifolds. J. Knot Theory Ramif. 6, 45–60 (1997)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Lickorish, W.B.R.: Skeins, \(SU(N)\) three-manifold invariants and TQFT. Comment. Math. Helv. 75, 45–64 (2000)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Masbaum, G.: An element of infinite order in TQFT-representations of mapping class groups. In: Low-Dimensional Topology (Funchal, 1998), pp. 137–139. Contemporary Mathematics, vol. 233. American Mathematical Society, Providence (1999)Google Scholar
  42. 42.
    Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. Phys. Lett. B 212, 451–460 (1988)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Moore, G., Seiberg, N.: Classical and quantum conformal field theory. Comm. Math. Phys. 123, 177–254 (1989)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Reshetikhin, N., Turaev, V.: Ribbon graphs and their invariants derived fron quantum groups. Comm. Math. Phys. 127, 1–26 (1990)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Reshetikhin, N., Turaev, V.: Invariants of \(3\)-manifolds via link polynomials and quantum groups. Invent. Math. 103, 547–597 (1991)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Segal, G.: Topology, geometry and quantum field theory, pp. 421–577. In: Lecture Note Series, vol. 308. London Mathematical Society, London (2004)Google Scholar
  47. 47.
    Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on \( \mathbb{P}^1\) and monodromy representations of Braid group. Adv. Stud. Pure Math. 16, 297–326 (1988)MathSciNetMATHGoogle Scholar
  48. 48.
    Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure Math. 19, 459–566 (1989)MathSciNetGoogle Scholar
  49. 49.
    Turaev, V.: Quantum Invariants of Knots and 3-Manifolds. De Gruyter studies in Mathematics, vol. 18. W. de Gruyter, Berlin (1994)Google Scholar
  50. 50.
    Turaev, V., Wenzl, H.: Quantum invariants of 3-manifolds associated with classical simple Lie algebras. Int. J. Math. 4, 323–358 (1993)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Turaev, V., Wenzl, H.: Semisimple and modular categories from link invariants. Masthematische Annalen 309, 411–461 (1997)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Ueno, K.: On conformal field theory. Lecture Note, vol. 208, pp. 283–345. London Mathematical Society, London (1995)Google Scholar
  53. 53.
    Ueno, K.: Introduction to conformal field theory with gauge symmetries. In: Geometry and Physics (Aarhus, 1995). Lecture Notes in Pure and Applied Mathematics, vol. 184, pp. 603–745. Dekker, New York (1997)Google Scholar
  54. 54.
    Ueno, K.: Conformal field theory with gauge symmetry. In: Fields Institute Monographs, vol. 24. American Mathematical Society, Providence; Fields Institute for Research in Mathematical Sciences, Toronto (2008)Google Scholar
  55. 55.
    Ueno, K.: Conformal field theory and modular functor. In: Advances in Algebra and Combinatorics, pp. 335–352. World Scientific Publications, Hackensack (2008)Google Scholar
  56. 56.
    Verlinde, E.: Fusion rules and modular transformations in \(2\)d conformal field theory. Nucl. Phys. B 300(FS22), 360–376 (1988)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Walker, K.: On Witten’s 3-manifold invariants. Preliminary version # 2 (1991, preprint)Google Scholar
  58. 58.
    Wall, C.T.C.: Non-additivity of the signature. Invent. Math. 7, 269–274 (1969)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Wenzl, H.: Hecke algebra of type \(A_n\) and subfactors. Invent. Math. 92, 349–383 (1988)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Wenzl, H.: Braids and invariants of 3-manifolds. Invent. Math. 114, 235–275 (1993)MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Witten, E.: Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121, 351–399 (1989)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Yokota, Y.: Skeins and quantum \(SU(N)\) invariants of \(3\)-manifolds. Math. Ann. 307, 109–138 (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Center for Quantum Geometry of Moduli SpacesUniversity of AarhusAarhusDenmark
  2. 2.Faculty of Science and Engineering, Department of Industrial and System EngineeringHosei UniversityKoganeiJapan

Personalised recommendations