QM Research Seminar: Defects in skein theory and TQFT

Speaker: ​Patrick Kinnear (University of Hamburg).

Abstract: 
​​​​​​​In the study of topological symmetry, defects are embedded submanifolds labelled by algebraic data; the evaluation of defect TQFTs (i.e. TQFTs which make sense for bordisms supporting defects) implements topological symmetry. The (2+1) TQFT of Reshetikhin and Turaev was enhanced to a defect TQFT in work of Carqueville--Runkel--Schaumann. This enhancement is a generalized state sum construction, and requires that the defects are labelled by semisimple data; the state spaces for surfaces are given by a projector from the state space of the original RT theory. On the other hand, given a 3-manifold whose boundary is a surface with defects, the usual definitions of skein theory can be extended to define a defect skein module. In this talk I will report on joint work with Ingo Runkel, where we define the defect skein module and prove that it is isomorphic to the defect RT state space of its boundary. This generalizes the well-known fact that the state spaces of RT theory are skein modules. Our work points to non-semisimple generalizations of defect RT theories using skein methods, while in the other direction it gives a state sum construction for defect skein modules labelled by semisimple data.