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

TL;DR

This paper develops a comprehensive state-sum framework for 2D topological quantum field theories with defects, extending previous models to include source defects on oriented curves and ensuring invariance under extended Pachner moves.

Contribution

It introduces a general state-sum construction for 2D TQFTs with defects, extending the algebraic framework and ensuring invariance under flag-like triangulations and extended Pachner moves.

Findings

Derived equations from extended Pachner moves

Translated equations into string diagrams for clarity

Ensured invariance under flag-like triangulations

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.