A general state-sum construction of 2-dimensional topological quantum field theories with defects

Abstract

We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda \& Pfeiffer \cite{LP}). From the extended Pachner moves (Crane \& Yetter \cite{CY}), we derive equations that we subsequently translate into string diagrams so that we can easily observe their properties. As in Dougherty, Park and Yetter \cite{DPY}, we require that triangulations be flag-like, meaning that each simplex of the triangulation is either disjoint from the defect curve, or intersects it in a closed face, and that the extended Pachner moves preserve flag-likeness. This research was conducted under the mentorship of Prof. David Yetter at Kansas State University with the support of NSF grant DMS-1262877.