Cardy–Frobenius extension of the algebra of cut-and-join operators

Motivated by the algebraic open–closed string models, we introduce and discuss an infinite-dimensional counterpart of the open–closed Hurwitz theory describing branching coverings generated both by the compact oriented surfaces and by the foam surfaces. We manifestly construct the corresponding infinite-dimensional equipped Cardy–Frobenius algebra, with the closed and open sectors being represented by the conjugation classes of permutations and of pairs of permutations, i.e. by the algebras of Young diagrams and of bipartite graphs respectively.