Skip to main content
SpringerLink
Log in

Faddeev’s Quantum Dilogarithm and State-Integrals on Shaped Triangulations

  • Chapter
  • First Online:
Mathematical Aspects of Quantum Field Theories

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

Using Faddeev’s quantum dilogarithm function, we review our description of a one parameter family of state-integrals on shaped triangulated pseudo 3-manifolds. This invariant is part of a certain TQFT, which we have constructed previously in a number of papers on the subject.

Supported by the center of excellence grant “Center for quantum geometry of Moduli Spaces” from the Danish National Research Foundation and Swiss National Science Foundation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J.E. Andersen, Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups. Ann. Math. (2) 163(1), 347–368 (2006)

    Article  MATH  Google Scholar 

  2. J.E. Andersen, Mapping class groups do not have Kazhdan’s property (T) (2007)

    Google Scholar 

  3. J.E. Andersen, The Nielsen-Thurston classification of mapping classes is determined by TQFT. J. Math. Kyoto Univ. 48(2), 323–338 (2008)

    MathSciNet  MATH  Google Scholar 

  4. J.E. Andersen, Hitchin’s connection, Toeplitz operators, and symmetry invariant deformation quantization. Quantum Topol. 3(3–4), 293–325 (2012)

    Google Scholar 

  5. J.E. Andersen, The Witten-Reshetikhin-Turaev invariants of finite order mapping tori I. J. Reine Angew. Math. 681, 1–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. J.E. Andersen, N.L. Gammelgaard. Hitchin’s projectively flat connection, Toeplitz operators and the asymptotic expansion of TQFT curve operators, in Grassmannians, moduli spaces and vector bundles, Clay Math. Proc. 14, 1–24. American Mathematical Society, Providence (2011)

    Google Scholar 

  7. J.E. Andersen, N.L. Gammelgaard, M.R. Lauridsen, Hitchin’s connection in metaplectic quantization. Quantum Topol. 3(3–4), 327–357 (2012)

    Google Scholar 

  8. J.E. Andersen, B. Himpel, The Witten-Reshetikhin-Turaev invariants of finite order mapping tori II. Quantum Topol. 3(3–4), 377–421 (2012)

    Google Scholar 

  9. J.E. Andersen, B. Himpel, S.F. Jørgensen, J. Martens, B. McLellan, The Witten-Reshetikhin-Turaev invariant for links in finite order mapping tori I (2014)

    Google Scholar 

  10. J.E. Andersen, R. Kashaev, A new formulation of the Teichmüller TQFT (2013). arXiv:1305.4291

  11. J.E. Andersen, R. Kashaev, Complex quantum chern-simons (2014). arXiv:1409.1208

  12. J.E. Andersen, R. Kashaev, A TQFT from quantum Teichmüller theory. Comm. Math. Phys. 330(3), 887–934 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. J.E. Andersen, K. Ueno, Geometric construction of modular functors from conformal field theory. J. Knot Theor. Ramifications 16(2), 127–202 (2007)

    Google Scholar 

  14. J.E. Andersen, K. Ueno, Abelian conformal field theories and determinant bundles. Int. J. Math. 18, 919–993 (2007)

    Google Scholar 

  15. J.E. Andersen, K. Ueno, Modular functors are determined by their genus zero data. Quantum Topol. 3, 255–291 (2012)

    Google Scholar 

  16. J.E. Andersen, K. Ueno, Construction of the Witten-Reshetikhin-Turaev TQFT from conformal field theory. Inventiones Math. (2014). doi:10.1007/s00222-014-0555-7

  17. J.E. Andersen, G. Masbaum, K. Ueno, Topological quantum field theory and the Nielsen-Thurston classification of M(0,4). Math. Proc. Cambridge Philos. Soc. 141(3), 477–488 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. S. Axelrod, S.D. Pietra, E. Witten, Geometric quantization of Chern-Simons gauge theory. J. Differ. Geom. 33(3), 787–902 (1991)

    MATH  Google Scholar 

  19. D. Bar-Natan, E. Witten, Perturbative expansion of Chern-Simons theory with noncompact gauge group. Comm. Math. Phys. 141(2), 423–440 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological quantum field theories derived from the Kauffman bracket. Topology 34(4), 883–927 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Casson, Private communication

    Google Scholar 

  22. R. Dijkgraaf, H. Fuji, M. Manabe, The volume conjecture, perturbative knot invariants, and recursion relations for topological strings. Nucl. Phys. B 849(1), 166–211 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. T. Dimofte, Quantum Riemann surfaces in Chern-Simons theory (2011). arXiv:1102.4847

  24. T. Dimofte, Quantum Riemann surfaces in Chern-Simons theory. Adv. Theor. Math. Phys. 17(3), 479–599 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. T. Dimofte, Complex Chern-Simons theory at level textitk via the 3d–3d correspondence (2014)

    Google Scholar 

  26. T. Dimofte, D. Gaiotto, S. Gukov, Gauge theories labelled by three-manifolds. Comm. Math. Phys. 325(2), 367–419 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. T. Dimofte, S. Garoufalidis, The quantum content of the gluing equations. Geom. Topol. 17(3), 1253–1315 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Dimofte, S. Gukov, Chern-Simons theory and S-duality. J. High Energy Phys. (5), 109, front matter+65 (2013)

    Google Scholar 

  29. T. Dimofte, S. Gukov, J. Lenells, D. Zagier, Exact results for perturbative Chern-Simons theory with complex gauge group. Commun. Number Theory Phys. 3(2), 363–443 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. L.D. Faddeev, Current-like variables in massive and massless integrable models (1994). arXiv:hep-th/9408041

  31. L.D. Faddeev, Discrete Heisenberg-Weyl group and modular group. Lett. Math. Phys. 34(3), 249–254 (1995)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. L.D. Faddeev, R.M. Kashaev, Quantum dilogarithm. Mod. Phys. Lett. A 9(5), 427–434 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. L.D. Faddeev, R.M. Kashaev, A.Yu. Volkov, Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality. Comm. Math. Phys. 219(1), 199–219 (2001)

    Google Scholar 

  34. V.V. Fock, L.O. Chekhov, Quantum Teichmüller spaces. Teoret. Mat. Fiz. 120(3), 511–528 (1999)

    Article  MathSciNet  Google Scholar 

  35. S. Garoufalidis, The 3D index of an ideal triangulation and angle structures (2012)

    Google Scholar 

  36. S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial. Comm. Math. Phys. 255(3), 577–627 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. S. Gukov, H. Murakami, SL(2, \(\mathbb{C}\)) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial. Lett. Math. Phys. 86(2–3), 79–98 (2008)

    Google Scholar 

  38. A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)

    MATH  Google Scholar 

  39. K. Hikami, Hyperbolicity of partition function and quantum gravity. Nucl. Phys. B 616(3), 537–548 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. K. Hikami, Generalized volume conjecture and the A-polynomials: the Neumann-Zagier potential function as a classical limit of the partition function. J. Geom. Phys. 57(9), 1895–1940 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. N.J. Hitchin, Flat connections and geometric quantization. Comm. Math. Phys. 131(2), 347–380 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. R.M. Kashaev, The hyperbolic volume of knots from the quantum dilogarithm. Lett. Math. Phys. 39(3), 269–275 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. R.M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm. Lett. Math. Phys. 43(2), 105–115 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  44. R.M. Kashaev, The Liouville central charge in quantum Teichmüller theory. Tr. Mat. Inst. Steklova 226, 72–81 (1999). (Mat. Fiz. Probl. Kvantovoi Teor. Polya)

    MathSciNet  Google Scholar 

  45. R.M. Kashaev, On the Spectrum of Dehn Twists in Quantum Teichmüller Theory, Physics and Combinatorics, 2000 (Nagoya) (World Scientific Publishing, River Edge, 2001)

    Google Scholar 

  46. M. Lackenby, Word hyperbolic Dehn surgery. Invent. Math. 140(2), 243–282 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Y. Laszlo, Hitchin’s and WZW connections are the same. J. Differ. Geom. 49(3), 547–576 (1998)

    MathSciNet  MATH  Google Scholar 

  48. J. Milnor, Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (N.S.) 6(1), 9–24 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  49. H. Murakami, J. Murakami, The colored Jones polynomials and the simplicial volume of a knot. Acta Math. 186(1), 85–104 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  50. W.D. Neumann, D. Zagier, Volumes of hyperbolic three-manifolds. Topology 24(3), 307–332 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  51. N. Reshetikhin, V.G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103(3), 547–597 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. I. Rivin, Combinatorial optimization in geometry. Adv. Appl. Math. 31(1), 242–271 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. E. Witten, 2+1-dimensional gravity as an exactly soluble system. Nucl. Phys. B 311(1):46–78 (1988/1989)

    Google Scholar 

  54. E. Witten, Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121(3), 351–399 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. E. Witten, Quantization of Chern-Simons gauge theory with complex gauge group. Comm. Math. Phys. 137(1), 29–66 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. E. Witten, Analytic continuation of Chern-Simons theory, in Chern-Simons gauge theory: 20 years after, AMS/IP Stud. Adv. Math. 50, 347–446. American Mathematical Society, Providence (2011)

    Google Scholar 

  57. S.L. Woronowicz, Quantum exponential function. Rev. Math. Phys. 12(6), 873–920 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Quantum Geometry of Moduli Spaces, University of Aarhus, 8000, Aarhus, Denmark

    Jørgen Ellegaard Andersen

  2. University of Geneva, 2-4 Rue du Lièvre, Case Postale 64, 1211, Genève 4, Switzerland

    Rinat Kashaev

Authors
  1. Jørgen Ellegaard Andersen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rinat Kashaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jørgen Ellegaard Andersen .

Editor information

Editors and Affiliations

  1. Université Montpellier 2, Montpellier, France

    Damien Calaque

  2. Université Claude Bernard Lyon 1, Villeurbanne Cedex, France

    Thomas Strobl

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Andersen, J.E., Kashaev, R. (2015). Faddeev’s Quantum Dilogarithm and State-Integrals on Shaped Triangulations. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-09949-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation