Defects in Topological Symmetric Orbifolds and their String Dual

Abstract

We review defect topological field theories with finite gauge groups. We then concentrate on restriction and induction defects between symmetric orbifold theories and recall that the defects form an algebra. This requires identifying the features of restriction and induction defects that are independent of the order of the symmetric group, which is natural in a grand canonical framework. The construction leads to Khovanov's categorification of the Heisenberg algebra. The bulk algebra of partial permutations corresponds to the endomorphisms of the identity defect in the category. We also interpret the Hurwitz theory defects in the dual Gromov-Witten theory. This gives a defect descendant interpretation to higher level operators that arise in the integrable hierarchy of the Gromov-Witten theory of curves. In further applications, we relate known defects to our symmetric group theoretic description. For seamed surfaces, we determine the boundary operator algebra for higher values of the covering degree n and provide asymptotics for the large n growth of the number of boundary operators.