Gerbe-holonomy for surfaces with defect networks

TL;DR

This paper introduces a new framework for analyzing surfaces with defect networks in sigma-models, deriving defect conditions, and exploring their topological and conformal properties, with applications to the WZW model and 3-cocycles.

Contribution

It defines the sigma-model action for surfaces with defect networks, derives defect gluing conditions, and connects classical and quantum cocycles in the WZW model.

Findings

Defect gluing conditions distinguish conformal and topological defects.

Holonomy for defect networks yields a 3-cocycle on Z(G).

Classical and quantum cocycles are cohomologous.

Abstract

We define the sigma-model action for world-sheets with embedded defect networks in the presence of a three-form field strength. We derive the defect gluing condition for the sigma-model fields and their derivatives, and use it to distinguish between conformal and topological defects. As an example, we treat the WZW model with defects labelled by elements of the centre Z(G) of the target Lie group G; comparing the holonomy for different defect networks gives rise to a 3-cocycle on Z(G). Next, we describe the factorisation properties of two-dimensional quantum field theories in the presence of defects and compare the correlators for different defect networks in the quantum WZW model. This, again, results in a 3-cocycle on Z(G). We observe that the cocycles obtained in the classical and in the quantum computation are cohomologous.