Categorified Open Topological Field Theories

In this short note, we classify linear categorified open topological field theories in dimension two by pivotal Grothendieck-Verdier categories. In combination with recently developed string-net techniques, this leads to a new description of the spaces of conformal blocks of Drinfeld centers Z(C) of pivotal finite tensor categories C in terms of the modular envelope of the cyclic associative operad. If C is unimodular, we prove that the space of conformal blocks inherits the structure of a module over the algebra of class functions of C for every free boundary component. As a further application, we prove that the sewing along a boundary circle for the modular functor for Z(C) can be decomposed into a sewing procedure along an interval and the application of the partial trace. Finally, we construct mapping class group representations from Grothendieck-Verdier categories that are not necessarily rigid and make precise how these generalize existing constructions.