Skip to main content
SpringerLink
Log in

Phase constants in the Fock–Goncharov quantum cluster varieties

  • Published:
Analysis and Mathematical Physics Aims and scope Submit manuscript

Abstract

A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized to unitary maps between the Hilbert spaces intertwining the representations. These unitary intertwiners are described using the quantum dilogarithm function \(\Phi ^\hbar \). Algebraic relations among classical mutations are satisfied by the intertwiners up to complex constants. The present paper shows that these constants are 1. So the mapping class group representations resulting from the Chekhov–Fock–Goncharov quantum Teichmüller theory are genuine, not projective. During the course, the hexagon and the octagon operator identities for \(\Phi ^\hbar \) are derived.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Notes

  1. In the case of cluster variety coming from a punctured surface, this group coincides with the mapping class group of the surface.

References

  1. Barnes, E.W.: Theory of the double gamma function. Philos. Trans. R. Soc. A 196, 265–388 (1901)

    MATH  Google Scholar 

  2. Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math. 195(2), 405–455 (2005)

    Article  MathSciNet  Google Scholar 

  3. Derksen, H., Weyman, J., Zelevinsky, A.: Quivers with potentials and their representations II: applications to cluster algebras. J. Am. Math. Soc. 23(3), 749–790 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Faddeev, L.D., Kashaev, R.M., Volkov, A.Y.: Strongly coupled quantum discrete Liouville theory, I: algebraic approach and duality. Commun. Math. Phys. 219, 199–219 (2001)

    Article  MathSciNet  Google Scholar 

  7. Fock, V.V., Goncharov, A.B.: Cluster Ensembles, Quantization and the Dilogarithm II: The Intertwiner in Manin’s Festschrift. Birkhäuser, Boston (2007)

    Google Scholar 

  8. Fock, V.V., Goncharov, A.B.: The quantum dilogarithm and representations of the quantum cluster varieties. Invent. Math. 175, 223–286 (2009)

    Article  MathSciNet  Google Scholar 

  9. Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Norm. Supér. (4) 42(6), 865–930 (2009)

    Article  MathSciNet  Google Scholar 

  10. Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)

    Article  MathSciNet  Google Scholar 

  11. Fomin, S., Zelevinsky, A.: Cluster algebras IV: coefficients. Compos. Math. 143, 112–164 (2007)

    Article  MathSciNet  Google Scholar 

  12. Frenkel, I.B., Kim, H.: Quantum Teichmüller space from the quantum plane. Duke Math. J. 161(2), 305–366 (2012)

    Article  MathSciNet  Google Scholar 

  13. Funar, L., Sergiescu, V.: Central extensions of the Ptolemy–Thompson group and quantized Teichmüller theory. J. Topol. 3, 29–62 (2010)

    Article  MathSciNet  Google Scholar 

  14. Goncharov, A.B.: Pentagon relation for the quantum dilogarithm and quantized \(\cal{M}^{\rm cyc}_{0,5}\). In: Geometry and Dynamics of Groups and Spaces, Progress in Mathematics, vol. 265. Birkhäuser, Basel, pp. 415–428 (2008)

  15. Gross, M., Hacking, P., Keel, S., Kontsevich, M.: Canonical bases for cluster algebras. J. Am. Math. Soc. 31(2), 497–608 (2018). arXiv:1411.1394

    Article  MathSciNet  Google Scholar 

  16. Hall, B.C.: Quantum Theory for Mathematicians. GTM, vol. 267. Springer, New York (2013)

    Book  Google Scholar 

  17. Ip, I.C.: Positive representations of split real quantum groups: the universal R operator. Int. Math. Res. Not. IMRN 1, 240–287 (2015). arXiv:1212.5149

    Article  MathSciNet  Google Scholar 

  18. Kashaev, R.M., Nakanishi, T.: Classical and quantum dilogarithm identities. SIGMA Symmetry Integrability Geom. Methods Appl. 7, Paper 102, 29 pages (2011). arXiv:1104.4630

  19. Keller.: On cluster theory and quantum dilogarithm identities. In: Representations of Algebras and Related Topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, pp. 85—116 (2011). arXiv:1102.4148

  20. Kim, H.: The dilogarithmic central extension of the Ptolemy–Thompson group via the Kashaev quantization. Adv. Math. 293, 529–588 (2016). arXiv:1211.4300

    Article  MathSciNet  Google Scholar 

  21. Kim, H.: Phase constants in the Fock–Goncharov quantization of cluster varieties: long version. arXiv:1602.00361v2

  22. Kim, H.: Irreducible self-adjoint representation of quantum Teichmüller space and phase constants. arXiv:2001.06802

  23. Kim, H., Scarinci, C.: Quantum moduli spaces of \((2+1)\)-gravity (in preparation)

  24. Kim, H., Yamazaki, M.: Comments on exchange graphs in cluster algebras. Exp. Math. 29(1), 79–100 (2020). arXiv:1612.00145

    Article  MathSciNet  Google Scholar 

  25. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Revised and Enlarged Edition. Academic Press, San Diego (1980)

    Google Scholar 

  26. Schmüdgen, K.: Unbounded Self-Adjoint Operators on Hilbert Space, GTM, vol. 265. Springer, Dordrecht (2012)

    Book  Google Scholar 

  27. von Neumann, J.: Die Eindeutigkeit der Schrödingerschen Operatoren. Math. Ann. 104, 570–578 (1931)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. Xu, B.: Central extension of mapping class group via Chekhov–Fock quantization. J. Geom. Phys. 110, 9–24 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to thank Myungho Kim, Dylan Allegretti, Ivan Chi-ho Ip, Carlos Scarinci, Seung-Jo Jung, Woocheol Choi, Louis Funar, Vladimir V. Fock, and Alexander B. Goncharov for helpful discussions. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant Number 2017R1D1A1B03030230).

Author information

Authors and Affiliations

  1. Department of Mathematics, Ewha Womans University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul, 03760, Republic of Korea

    Hyun Kyu Kim

Authors
  1. Hyun Kyu Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyun Kyu Kim.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, H.K. Phase constants in the Fock–Goncharov quantum cluster varieties. Anal.Math.Phys. 11, 2 (2021). https://doi.org/10.1007/s13324-020-00439-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13324-020-00439-3

Search

Navigation