Geodesic
The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces
Izvestiya. Mathematics, Tome 72 (2008) no. 4, pp. 627-646 Cet article a éte moissonné depuis la source Math-Net.Ru

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We extend the definition of Hurwitz numbers to the case of seamed surfaces, which arise in new models of mathematical physics, and prove that they form a system of correlators for a Klein topological field theory in the sense defined in [1]. We find the corresponding Cardy–Frobenius algebras, which yield a method for calculating the Hurwitz numbers. As a by-product, we prove that the vector space generated by the bipartite graphs with $n$ edges possesses a natural binary operation that makes this space into a non-commutative Frobenius algebra isomorphic to the algebra of intertwining operators for a representation of the symmetric group $S_n$ on the space generated by the set of all partitions of a set of $n$ elements. 
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A. V. Alekseevskii; S. M. Natanzon. The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces. Izvestiya. Mathematics, Tome 72 (2008) no. 4, pp. 627-646. http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a0/

[1] A. Alexeevski, S. Natanzon, “Noncommutative two-dimensional topological field theories and Hurwitz numbers for real algebraic curves”, Selecta Math. (N.S.), 12:3–4 (2006), 307–377 ; arXiv: math/0202164 | DOI | MR | Zbl

[2] A. Hurwitz, “Ueber Riemann'sche Flächen mit gegebenen Verzweigungspunkten”, Math. Ann., 39:1 (1891), 1–60 | DOI | MR

[3] S. Cordes, G. Moore, S. Ramgoolam, “Large $N$ $2\mathrm{D}$ Yang–Mills theory and topological string theory”, Commun. Math. Phys., 185:3 (1997), 543–619 | DOI | MR | Zbl

[4] D. J. Gross, W. Taylor, “Twists and Wilson loops in the string theory of two dimensional QCD”, Nuclear Phys. B, 403:1–2 (1993), 395–449 | DOI | MR | Zbl

[5] R. Dijkgraaf, “Mirror symmetry and elliptic curves”, The moduli spaces of curves (Texel Island, 1994), Progr. Math., 129, 1995, 149–163 | MR | Zbl

[6] V. I. Arnol'd, “Topological classification of trigonometric polynomials and combinatorics of graphs with an equal number of vertices and edges”, Funct. Anal. Appl., 30:1 (1996), 1–14 | DOI | MR | Zbl

[7] I. K. Kostov, M. Staudacher, T. Wynter, “Complex matrix models and statistics of branched coverings of 2D surfaces”, Comm. Math. Phys., 191:2 (1998), 283–298 | DOI | MR | Zbl

[8] A. Okounkov, R. Pandharipande, Gromov–Witten theory, Hurwitz numbers and matrix models, I, arXiv: math/0101147

[9] M. Khovanov, L. Rozansky, “Topological Landau–Ginzburg models on a world-sheet foam”, Adv. Theor. Math. Phys., 11:2 (2007), 233–260 ; arXiv: hep-th/0404189 | MR | Zbl

[10] L. Rozansky, “Topological A-models on seamed Riemann surfaces”, Adv. Theor. Math. Phys., 11:4 (2007), 517–529 ; arXiv: hep-th/0305205 | MR | Zbl

[11] N. L. Alling, N. Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Math., 219, Springer-Verlag, Berlin–New York, 1971 | MR | Zbl

[12] S. M. Natanzon, “Klein surfaces”, Russian Math. Surveys, 45:6 (1990), 53–108 | DOI | MR | Zbl

[13] S. M. Natanzon, Moduli rimanovykh poverkhnostei, veschestvennykh algebraicheskii krivykh i ikh superanalogi, Novye matematicheskie distsipliny, MTsNMO, M., 2003 | MR | Zbl

[14] C. I. Lazaroiu, “On the structure of open-closed topological field theory in two dimensions”, Nuclear Phys. B, 603:3 (2001), 497–530 | DOI | MR | Zbl

[15] R. Dijkgraaf, Geometrical approach to two-dimensional conformal field theory, Ph. D. Thesis, Utrecht, 1989

[16] C. Faith, Algebra. II: Ring theory, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin–Heidelberg–New York, 1976 | MR | Zbl