A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories
Jurgen Fuchs, Christoph Schweigert, Alessandro Valentino
TL;DR
The paper develops a geometric, Lagrangian-based framework to study boundaries and surface defects in Dijkgraaf-Witten theories by linearizing relative bundles. It constructs categories of generalized Wilson lines for decorated one-dimensional manifolds and shows these agree with the boundary/defect Wilson line formalism of Fus-Virasoro and with Ostrik's module-category description for $G$-graded vector spaces twisted by $\omega$. Central to the approach are relative bundles, groupoid cohomology, and a graphical calculus that yields explicit 2-cocycles governing twisted linearizations; the interval and circle building blocks are computed and shown to reproduce the expected bulk and boundary Wilson line categories, including the transparent defect as the unit for defect fusion. The results provide a concrete geometric realization of DW theories with boundaries/defects and connect locality, groupoid cohomology, and module-category theory in a unified extended TFT framework.
Abstract
Dijkgraaf-Witten theories are extended three-dimensional topological field theories of Turaev-Viro type. They can be constructed geometrically from categories of bundles via linearization. Boundaries and surface defects or interfaces in quantum field theories are of interest in various applications and provide structural insight. We perform a geometric study of boundary conditions and surface defects in Dijkgraaf-Witten theories. A crucial tool is the linearization of categories of relative bundles. We present the categories of generalized Wilson lines produced by such a linearization procedure. We establish that they agree with the Wilson line categories that are predicted by the general formalism for boundary conditions and surface defects in three-dimensional topological field theories that has been developed in arXive:1203.4568.