Infinite-dimensional topological field theories from Hurwitz numbers

Mironov, Andrey; Morozov, Aleksey; Natanzon, Sergey

doi:10.1142/S0218216514500333

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Infinite-dimensional topological field theories from Hurwitz numbers

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Natanzon, Sergey
Max Planck Institute for Mathematics, Max Planck Society;

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http://dx.doi.org/10.1142/S0218216514500333
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Mironov, A., Morozov, A., & Natanzon, S. (2014). Infinite-dimensional topological field theories from Hurwitz numbers. Journal of Knot Theory and its Ramifications, 23(6): 1450033. doi:10.1142/S0218216514500333.

Cite as: https://hdl.handle.net/21.11116/0000-0004-DC2F-1

Abstract

Classical Hurwitz numbers of a fixed degree together with Hurwitz numbers of seamed surfaces give rise to a Klein topological field theory (see [A. Alexeevski and S. Natanzon, The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces, Izv. Math.
72(4) (2008) 627–646]). We extend this construction to Hurwitz numbers of all degrees simultaneously. The corresponding infinite-dimensional Cardy–Frobenius algebra is computed in terms of Young diagrams and bipartite graphs. This algebra turns out to be isomorphic to the algebra of differential operators introduced in [A. Mironov, A. Morozov and S. Natanzon, Cardy–Frobenius extension of algebra of cut-and-join operators, J. Geom. Phys. 73(2012) 243–251, arXiv:1210.6955; A Hurwitz theory avatar of open-closed string,
Eur. Phys. J. C 73(2) (2013) 1–10, arXiv:1208.5057], which serves a model for open-closed string theory. We prove that the operators corresponding to Young diagrams and bipartite graphs give rise to relations between Hurwitz numbers.