A Hurwitz theory avatar of open-closed strings

We review and explain an infinite-dimensional counterpart of the Hurwitz theory realization of algebraic open-closed string model a la Moore and Lizaroiu, where the closed and open sectors are represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphes respectively. An intriguing feature of this Hurwitz string model is coexistence of two different multiplications, reflecting the deep interrelation between the theory of symmetric and linear groups S_\infty and GL(\infty).Comment: 11 page

Cut-and-join structure and integrability for spin Hurwitz numbers

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators W_\Delta with \Delta\in OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a \tau-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.Comment: 22 page

Cardy-Frobenius extension of 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 are represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphes respectively.Comment: 12 page