Infinite-dimensional topological field theories from Hurwitz numbers

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