Letters in Mathematical Physics

, Volume 104, Issue 1, pp 75–87 | Cite as

From Hurwitz Numbers to Kontsevich–Witten Tau-Function: A Connection by Virasoro Operators



In this letter, we present our conjecture on the connection between the Kontsevich–Witten and the Hurwitz tau-functions. The conjectural formula connects these two tau-functions by means of the GL(∞) group element. An important feature of this group element is its simplicity: this is a group element of the Virasoro subalgebra of gl(∞). If proved, this conjecture would allow to derive the Virasoro constraints for the Hurwitz tau-function, which remain unknown in spite of existence of several matrix model representations, as well as to give an integrable operator description of the Kontsevich–Witten tau-function.

Mathematics Subject Classification (2010)

Primary 37K10 14N35 81R10 17B68 Secondary 81T30 


matrix models tau-functions Virasoro algebra Hurwitz numbers 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Okounkov, A., Pandharipande, R.: Gromov–Witten theory, Hurwitz numbers, and matrix models. 1. [math/0101147 [math.AG]]Google Scholar
  2. 2.
    Aganagic M., Dijkgraaf R., Klemm A., Marino M., Vafa C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451–516 (2006) [hep-th/0312085]ADSCrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1–23 (1992)ADSCrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Witten E.: Two-dimensional gravity and intersection theory on moduli space. Surveys Differ. Geom. 1, 243–310 (1991)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Kharchev S., Marshakov A., Mironov A., Morozov A., Zabrodin A.: Towards unified theory of 2-d gravity. Nucl. Phys. B 380, 181–240 (1992) [arXiv:hep-th/9201013 [hep-th]]ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Okounkov A., Pandharipande R.: The Equivariant Gromov–Witten theory of P 1. Ann. Math. 163, 561–605 (2006) [math/0207233 [math-ag]]CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Morozov A.: Integrability and matrix models. Phys. Usp. 37, 1–55 (1994) [hep-th/9303139]ADSCrossRefMATHGoogle Scholar
  8. 8.
    Kac V., Schwarz A.S.: Geometric interpretation of the partition function of 2-D gravity. Phys. Lett. B 257, 329 (1991)ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schwarz A.S.: On some mathematical problems of 2-D gravity and W(h) gravity. Mod. Phys. Lett. A 6, 611 (1991)ADSCrossRefMATHGoogle Scholar
  10. 10.
    Itzykson C., Zuber J.B.: Combinatorics of the modular group. 2. The Kontsevich integrals. Int. J. Mod. Phys. A 7, 5661–5705 (1992) [hep-th/9201001]ADSCrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Alexandrov A.S., Mironov A., Morozov A., Morozov A., Morozov A.: Partition functions of matrix models as the first special functions of string theory. II. Kontsevich Model. Int. J. Mod. Phys. A 24, 4939–4998 (2009) [arXiv:0811.2825 [hep-th]]ADSCrossRefMATHGoogle Scholar
  12. 12.
    Goulden I.P., Jackson D.M.: Transitive factorisations into transpositions and holomorphic mappings on the sphere. Proc. AMS 125, 51–60 (1997)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Vakil, R.: Enumerative Geometry of Curves via Degeneration Methods. Harvard Ph.D. thesis (1997)Google Scholar
  14. 14.
    Alexandrov A., Mironov A., Morozov A., Natanzon S.: Integrability of Hurwitz partition functions. I. Summary. J. Phys. A 45, 045209 (2012) [arXiv:1103.4100 [hep-th]]ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bouchard V., Marino M.: Hurwitz numbers, matrix models and enumerative geometry. Proc. Symp. Pure Math. 78, 263 (2008) [arXiv:0709.1458 [math.AG]]CrossRefMathSciNetGoogle Scholar
  16. 16.
    Borot G., Eynard B., Mulase M., Safnuk B.: A Matrix model for simple Hurwitz numbers, and topological recursion. J. Geom. Phys. 61, 522–540 (2011) [arXiv:0906.1206 [math-ph]]ADSCrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Morozov A., Shakirov S.: Generation of Matrix Models by W-operators. JHEP 0904, 064 (2009) [arXiv:0902.2627 [hep-th]]ADSCrossRefMathSciNetGoogle Scholar
  18. 18.
    Alexandrov A.: Matrix models for random partitions. Nucl. Phys. B 851, 620–650 (2011) [arXiv:1005.5715 [hep-th]]ADSCrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Goulden I., Jackson D., Vakil R.: The Gromov–Witten potential of a point, Hurwitz numbers, and Hodge integrals. Proc. Lond. Math. Soc. 83, 563–581 (2001) [math/9910004 [math.AG]]CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Kazarian M.: KP hierarchy for Hodge integrals. Adv. Math. 221, 1–21 (2009) [arXiv:0809.3263 [math.AG]]CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Mironov A., Morozov A.: Virasoro constraints for Kontsevich-Hurwitz partition function. JHEP 0902, 024 (2009) [arXiv:0807.2843 [hep-th]]ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ekedahl T., Lando S., Shapiro M., Vainshtein A.: Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146, 297–327 (2001) [math/0004096 [math.AG]]ADSCrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Mumford, D.: Towards enumerative geometry on the moduli space of curves. In: Artin, M., Tate, J. (eds.) Arithmetics and Geometry, vol. 2, pp. 271–328. Birkhauser, Basel (1983)Google Scholar
  24. 24.
    Faber C., Pandharipande R.: Hodge integrals and Gromov–Witten theory. Invent. Math. 139, 173–199 (2000) [arXiv:math/9810173v1 [math.AG]]ADSCrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Givental A.: Semisimple Frobenius structures at higher genus. Int. Math. Res. Notices 23, 1265–1286 (2001) [arXiv:math/0108100 [math.AG]]CrossRefMathSciNetGoogle Scholar
  26. 26.
    Givental A.: Gromov–Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1, 551–568 (2001) [arXiv:math/0108100 [math.AG]]MATHMathSciNetGoogle Scholar
  27. 27.
    Grinevich, P.G., Orlov, A.Y.: Flag spaces in KP theory and Virasoro action on determinant det D j and Segal–Wilson tau function [math-ph/9804019]Google Scholar
  28. 28.
    Shiota T.: Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83, 333–382 (1986)ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Alexandrov A.: Cut-and-Join operator representation for Kontsevich–Witten tau-function. Mod. Phys. Lett. A 26, 2193–2199 (2011) [arXiv:1009.4887 [hep-th]]ADSCrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Miwa T., Jimbo M., Date E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge University Press, Cambridge (2000)Google Scholar
  31. 31.
    Alexandrov A., Zabrodin A.: Free fermions and tau-functions. J. Geom. Phys. 67, 37 (2013) [arXiv:1212.6049 [math-ph]]ADSCrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Chekhov, L.: Matrix models and geometry of moduli spaces. [hep-th/9509001]Google Scholar
  33. 33.
    Kostov I.: Matrix models as CFT: Genus expansion. Nucl. Phys. B 837, 221–238 (2010) [arXiv:0912.2137 [hep-th]]ADSCrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kostov I., Orantin N.: CFT and topological recursion. JHEP 1011, 056 (2010) [arXiv:1006.2028 [hep-th]]ADSCrossRefMathSciNetGoogle Scholar
  35. 35.
    Alexandrov A.S., Mironov A., Morozov A.: Instantons and merons in matrix models. Physica D 235, 126–167 (2007) [hep-th/0608228]ADSCrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Alexandrov A.S., Mironov A., Morozov A.: M-theory of matrix models. Teor. Mat. Fiz. 150, 179–192 (2007) [hep-th/0605171]CrossRefGoogle Scholar
  37. 37.
    Alexandrov A., Mironov A., Morozov A.: BGWM as Second Constituent of Complex Matrix Model. JHEP 0912, 053 (2009) [arXiv:0906.3305 [hep-th]]ADSCrossRefMathSciNetGoogle Scholar
  38. 38.
    Givental A.: A n-1 singularities and nKdV hierarchie. Mosc. Math. J. 3(2), 475–505 (2003) [arXiv:math/0209205 [math.AG]]MATHMathSciNetGoogle Scholar
  39. 39.
    Mironov A., Morozov A., Natanzon S.: Complete Set of Cut-and-Join Operators in Hurwitz–Kontsevich Theory. Theor. Math. Phys. 166, 1–22 (2011) [arXiv:0904.4227 [hep-th]]CrossRefMATHGoogle Scholar
  40. 40.
    Mironov A., Morozov A., Natanzon S.: Integrability properties of Hurwitz partition functions. II. Multiplication of cut-and-join operators and WDVV equations. JHEP 1111, 097 (2011) [arXiv:1108.0885 [hep-th]]ADSCrossRefMathSciNetGoogle Scholar
  41. 41.
    Kharchev, S.: Kadomtsev–Petviashvili hierarchy and generalized Kontsevich model. [hep-th/9810091]Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of FreiburgFreiburgGermany
  2. 2.ITEPMoscowRussia

Personalised recommendations