Gerbe-holonomy for surfaces with defect networks

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. *

The Gauging of Two-Dimensional Bosonic Sigma Models on World-Sheets With Defects

“…[GSW10] that accommodates the full-blown 2-categorial background of a general σ-model characterised in Refs. [FSW08,RS09,Sus11]. In application to boundary defects, it extends the results of Ref.…”

“…We start by identifying, after Ref. [FSW08,RS09], the differential-geometric structure on the target space underlying the definition of the Feynman amplitudes of the field theory of interest.…”

“…Ref. [RS09]. Indeed, upon taking the target space in the special form M ′ ∶= M ⊔ {•} for an arbitrary singleton {•}, alongside any manifold Q equipped with a pair of smooth maps ι 1 ∶ Q → M ⊂ M ′ and ι 2 ∶ Q → {•} ⊂ M ′ (the constant map) as a bi-brane world-volume, we readily recover the standard notion in a new guise: The patch on one side of the defect line carrying the data of the (boundary) bi-brane is mapped to a single point, thereby mimicking an interface between a nontrivial phase of the σ-model and the trivial (void) σ-model.…”

“…is given by the decorated-surface holonomy Hol G,Φ,(ϕn) (ϕ Γ) for the network-field configuration (ϕ Γ), described in detail in Ref. [RS09] and generalising the holonomy of Ref. [FSW08] to the case with defect networks.…”

“…A special class of such defects, the central-jump defects for which g −1 1 g 2 ∈ Z, where Z is the center of G , were considered at length in Ref. [RS09]. The more general jump defects, with the jump given by g −1 1 ⋅ g 2 ∈ C λ , were first considered in Ref.…”