Symmetric Group Gauge Theories and Simple Gauge/String Dualities

We study two-dimensional topological gauge theories with gauge group equal to the symmetric group $S_n$ and their string theory duals. The simplest such theory is the topological quantum field theory of principal $S_n$ fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of $n$ are coded in a partial permutation algebra. We propose a generalization of the partial permutation algebra to any symmetric orbifold topological quantum field theory and show that the latter theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov-Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles.