Advertisement

Theoretical and Mathematical Physics

, Volume 197, Issue 2, pp 1535–1571 | Cite as

Discriminant Circle Bundles over Local Models of Strebel Graphs and Boutroux Curves

  • M. BertolaEmail author
  • D. A. Korotkin
Article

Abstract

We study special “discriminant” circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by Q 0 (−7) and Q 0 ([−3]2). The space Q 0 (−7) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order seven with real periods; it appears naturally in the study of a neighborhood of the Witten cycle W5 in the combinatorial model based on Jenkins–Strebel quadratic differentials of Mg,n. The space Q 0 ([−3]2) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most three with real periods; it appears in the description of a neighborhood of Kontsevich’s boundary W1,1 of the combinatorial model. Applying the formalism of the Bergman tau function to the combinatorial model (with the goal of analytically computing cycles Poincaré dual to certain combinations of tautological classes) requires studying special sections of circle bundles over Q 0 (−7) and Q 0 ([−3]2). In the Q 0 (−7) case, a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces Q 0 (−7) and Q 0 ([−3]2), also called the spaces of Boutroux curves, in detail together with the corresponding circle bundles.

Keywords

moduli space quadratic differential Boutroux curve tau function Jenkins–Strebel differential ribbon graph 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Jenkins, “On the existence of certain general extremal metrics,” Ann. of Math. (2), 66, 440–453 (1957).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    K. Strebel, Quadratic Differentials (Ergeb. Math. Grenzgeb., Vol. 5), Springer, Berlin (1984).CrossRefzbMATHGoogle Scholar
  3. 3.
    J. Harer, “The cohomology of the moduli space of curves,” in: Theory of Moduli (Lect. Notes Math., Vol. 1337, E. Sernesi, ed.), Springer, Berlin (1988), pp. 138–221.CrossRefGoogle Scholar
  4. 4.
    J. L. Harer, “The virtual cohomological dimension of the mapping class group of an orientable surface,” Invent. Math., 84, 157–176 (1986).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix Airy function,” Commun. Math. Phys., 147, 1–23 (1992).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Kontsevich, “Feynman diagrams and low-dimensional topology,” in: First European Congress of Mathematics (Progr. Math., Vol. 120, A. Joseph, F. Mignot, F. Murat, B. Prum, and R. Rentschler, eds.), Vol. 2, Invited Lectures (Part 2), Birkhaüser, Basel (1994), pp. 97–121.Google Scholar
  7. 7.
    E. Witten, “Two-dimensional gravity and intersection theory on moduli space,” Surveys Diff. Geom., 1, 243–310 (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    K. Igusa, “Graph cohomology and Kontsevich cycles,” Topology, 43, 1469–1510 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    K. Igusa, “Combinatorial Miller–Morita–Mumford classes and Witten cycles,” Algebr. Geom. Topol., 4, 473–520 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    G. Mondello, “Riemann surfaces, ribbon graphs, and combinatorial classes,” in: Handbook of Teichmüller Theory (IRMA Lect. Math. Theor. Phys., Vol. 13, A. Papadopoulos, ed.), Vol. 2, European Math. Soc. Publ. House, Zürich (2009), pp. 151–215.CrossRefGoogle Scholar
  11. 11.
    G. Mondello, “Combinatorial classes on Mg,n are tautological,” Int. Math. Res. Notices, 2004, 2329–2390 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R. Penner, “The simplicial compactification of Riemann’s moduli space,” in: Topology and Teichmüller spaces (Katinkulta, Finland, 24–28 July 1995, S. Kojima, Y. Matsumoto, K. Saito, and M. Seppälä, eds.), World Scientific, Singapore (1996), pp. 237–252.CrossRefGoogle Scholar
  13. 13.
    E. Arbarello and M. Cornalba, “Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves,” J. Algebraic Geom., 5, 705–749 (1996).MathSciNetzbMATHGoogle Scholar
  14. 14.
    D. Zvonkine, “Strebel differentials on stable curves and Kontsevich’s proof of Witten’s conjecture,” arXiv: math/0209071v2 (2002).Google Scholar
  15. 15.
    D. Korotkin, “Solution of matrix Riemann–Hilbert problems with quasi-permutation monodromy matrices,” Math. Ann., 329, 335–364 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    B. Dubrovin, “Geometry of 2D topological field theories,” in: Integrable Systems and Quantum Groups (Lect. Notes Math., Vol. 1620, M. Francaviglia and S. Greco, eds.), Springer, Berlin (1996), pp. 120–348.CrossRefGoogle Scholar
  17. 17.
    A. Kokotov and D. Korotkin, “Tau-functions on spaces of Abelian differentials and higher genus generalization of Ray–Singer formula,” J. Differ. Geom., 82, 35–100 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    A. Kokotov and D. Korotkin, “Tau-functions on spaces of Abelian and quadratic differentials and determinants of Laplacians in Strebel metrics of finite volume,” Preprint No. 46, Max-Planck Inst. Math. in Sci., Leipzig (2004); arXiv:math/0405042v5 (2004).zbMATHGoogle Scholar
  19. 19.
    A. Kokotov, D. Korotkin, and P. Zograf, “Isomonodromic tau function on the space of admissible covers,” Adv. Math., 227, 586–600 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D. Korotkin and P. Zograf, “Tau function and moduli of differentials,” Math. Res. Lett., 18, 447–458 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    D. Korotkin and P. Zograf, “Tau-function and Prym class,” in: Algebraic and Geometric Aspects of Integrable Systems and Random Matrices (Contemp. Math., Vol. 593, A. Dzhamay, K. Maruno, and V. U. Pierce, eds.), Amer. Math. Soc., Providence, R. I. (2013), pp. 241–261.Google Scholar
  22. 22.
    M. Bertola and D. Korotkin, “Hodge and Prym tau-functions, Jenkins–Strebel differentials, and combinatorial model of M g,n,” arXiv:1804.02495v2 [math-ph] (2018).Google Scholar
  23. 23.
    S.-Y. Lee, R. Teodorescu, and P. Wiegman, “Shocks and finite-time singularities in Hele–Shaw flow,” Phys. D, 238, 1113–1128 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    I. M. Krichever, “The t -function of the universal Whitham hierarchy, matrix models, and topological field theories,” Commun. Pure Appl. Math., 47, 437–475 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    M. Bertola, “Boutroux curves with external field: Equilibrium measures without a variational problem,” Anal. Math. Phys., 1, 167–211 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    P. Boutroux, “Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre,” Ann. Sci. École Norm. Sup. Ser. 3, 30, 255–375 (1913); “Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre (suite),” Ann. Sci. École Norm. Sup. Ser. 3, 31, 99–159 (1914).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    A. S. Fokas, A. R. Its, A. A. Kapaev, and V. Yu. Novokshenov, Painlevé Transcendents: The Riemann–Hilbert Approach (Math. Surv. Monogr., Vol. 128), Amer. Math. Soc., Providence, R. I. (2006).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityQuebecCanada
  2. 2.Area of MathematicsInternational School for Advanced Studies (SISSA)TriesteItaly

Personalised recommendations