This article is cited in 16 papers

The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces

A. V. Alekseevskii^a, S. M. Natanzon<sup>bca

a</sup> A. N. Belozersky Institute of Physico-Chemical Biology, M. V. Lomonosov Moscow State University
^b Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
^c Independent University of Moscow

Abstract: 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.

UDC: 514.7+512.7

MSC: 30F50, 14H30, 20C05, 81T45

Received: 28.12.2005
Revised: 15.02.2007

DOI: 10.4213/im737

 English version: Izvestiya: Mathematics, 2008, 72:4, 627–646

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