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

TL;DR

This paper develops an infinite-dimensional algebraic framework inspired by open-closed string models, extending Hurwitz theory to include foam surfaces and constructing a corresponding Cardy-Frobenius algebra with applications to permutation and bipartite graph structures.

Contribution

It introduces an infinite-dimensional extension of open-closed Hurwitz theory and constructs a new Cardy-Frobenius algebra incorporating foam surfaces and permutation-based sectors.

Findings

Constructed an infinite-dimensional Cardy-Frobenius algebra.

Unified algebraic description of open and closed sectors.

Connected algebraic structures to permutation classes and bipartite graphs.

Abstract

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.