Gerbe-holonomy for surfaces with defect networks

TL;DR

This work develops a comprehensive framework for gerbe-based holonomy on world-sheets with defect networks, including defect-junctions, by extending the Deligne cohomology description to abelian bi-branes and inter-bi-branes. It derives defect gluing conditions that distinguish conformal from topological defects and demonstrates how, in the WZW model, defect networks produce a 3-cocycle on the center Z(G) both at the classical (geometric) and quantum (CFT/TFT) levels. The central result is that the classical and quantum 3-cocycles are cohomologous, implying that the obstruction data governing orbifolds and simple-current symmetries are preserved under quantisation. The paper connects geometric structures of gerbes and their higher morphisms with CFT data, providing a nonperturbative handle on defect-induced associativity and orbifold consistency. This framework has direct relevance for understanding defect-based symmetries, orbifold constructions, and the interplay between classical geometry and quantum conformal field theory in two dimensions.

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.

Figures (8)

• Figure 1: When a sum over intermediate states is inserted on a circle that intersects defect lines, the intermediate states lie in a twisted space of states.
• Figure 2: States $\,|\phi\rangle\,$ in a twisted space of states correspond to fields $\,\phi\,$ at defect junctions via the state-field correspondence.
• Figure 3: The relevant part of the two world-sheets $\,\Sigma_L\,$ and $\,\Sigma_R\,$ used in the definition of the 3-cocycle on $\,Z(\textrm{G})$. The field jumps by multiplication with the indicated element of $\,Z(\textrm{G})\,$ when crossing the defect line. The values of the field on $\,\Sigma_L\,$ and $\,\Sigma_R\,$ differ only in the shaded region.
• Figure 4: When a triangle $\,t\,$ shares an edge $\,e\,$ with a defect line, the orientation of the defect either agrees with that of $\,\partial t\,$ or not. This decides which pullback map to apply to the connection 1-forms $\,A_{ij}\,$ and the transition functions $\,g_{ijk}\,$ on $\,M$.
• Figure 5: The four-valent defect vertex in $\,\Sigma_{L|R}\,$ obtained as a result of collapsing a pair of three-valent vertices in two inequivalent ways, whereby the two 2-morphisms $\,\widetilde{\varphi}^L\,$ and $\,\widetilde{\varphi}^R\,$ are induced at the vertex.