Geodesic
Simple Hurwitz Numbers of a Disk
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 1, pp. 44-58.

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Let $D$ be the closed unit disk. We study the Hurwitz numbers corresponding to the coverings of $D$ whose only multiple critical value lies on the boundary of $D$ and find differential equations describing the generating function of these numbers.
Keywords: Hurwitz numbers, topological field theory, cut-and-join equation. 
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S. M. Natanzon. Simple Hurwitz Numbers of a Disk. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 1, pp. 44-58. http://geodesic.mathdoc.fr/item/FAA_2010_44_1_a3/

[1] A. Alexeevski, S. Natanzon, “Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves”, Selecta Math. (New Ser.), 12:3 (2006), 307–377, arXiv: math/0202164 | DOI | MR | Zbl

[2] N. L. Allin, N. Greenlef, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math., 219, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | DOI | MR | Zbl

[3] R. Dijkgraaf, “Mirror symmetry and elliptic curves”, The moduli spaces of curves, Progress in Math., 129, Brikhäuser, 1995, 149–163 | MR

[4] A. Dold, “Ramifield coverings, orbit projections and symmetric powers”, Math. Proc. Cambridge Philos. Soc., 99:1 (1986), 65–72 | DOI | MR | Zbl

[5] D. Goulden, D. M. Jackson, A. Vainshtein, “The number of ramified coverings of the sphere by torus and surfaces of higher genera”, Ann. Comb., 4:1 (2000), 27–46, Brikhäuser | DOI | MR

[6] A. Hurwitz, “Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten”, Math. Ann., 39:1 (1891), 1–61 | DOI | MR

[7] M. Kazarian, S. Lando, An algebro-geometric proof of Witten's conjecture, arXiv: math/0601760

[8] S. M. Natanzon, “Kleinovy poverkhnosti”, UMN, 45:6 (1990), 47–90 | MR | Zbl

[9] S. M. Natanzon, “Topology of 2-dimensional coverings and meromorphic functions on real and complex algebraic curves”, Selecta Math. Soviet., 12:3 (1993), 251–291 | MR

[10] S. M. Natanzon, Moduli rimanovykh poverkhnostei, veschestvennykh algebraicheskikh krivykh i ikh superanalogi, MTsNMO, M., 2003 | MR

[11] L. Smith, “Transfer and ramified coverings”, Math. Proc. Cambridge Philos. Soc., 93:3 (1983), 485–493 | DOI | MR | Zbl

[12] J. Zhou, Hodge integrals, Hurwitz numbers and symmetric groups, arXiv: math/0308024