Finite-group gauging wall conventions

Use the interface data package cited below: $C$ supplies bulk field-label spaces and correlators; its topological interfaces have labelled-network distributional correlators, specified one-sided stress traces, isotopy invariance, coherent junction sewing, and specified fusion limits. An endodefect has $C$ on both sides. A normalized action of a finite abelian group $G$ consists of such invertible topological lines $L_g$, indexed by $g\in G$, with $L_e\cong \mathbf{1}$ and chosen invertible multiplication junctions $L_g\otimes L_h\cong L_{gh}$ satisfying the unit and associativity coherence conditions. The action need not be faithful unless this is separately assumed.

An anomaly-free finite gauging of this action is a chosen consistent gauging: the local trivializations and junctions can be summed over flat principal $G$-bundles so that the resulting gauge-invariant correlators define the gauged CFT, denoted $C\mathrm{/}G$, and a topological gauging interface $W_G:C\to C\mathrm{/}G$. Its orientation-reversed interface is ${\bar{W}}_G:C\mathrm{/}G\to C$. This is an adopted existence condition; it does not assert that every finite group action is anomaly-free. Throughout, $G$ is regarded as a finite discrete group; a flat principal $G$-bundle has locally constant transition functions and is classified by its monodromy representation up to gauge equivalence, with any stated boundary frames held fixed.

For a finite groupoid $\mathcal{G}$ (a category in which every morphism is invertible), its groupoid cardinality is $\mathrm{\mid }\mathcal{G}\mathrm{\mid }={\sum }_{[x]\in {\pi }_0(\mathcal{G})}\frac{1}{\mathrm{\mid }{\mathrm{Aut}}_{\mathcal{G}}(x)\mathrm{\mid }},$ where ${\pi }_0(\mathcal{G})$ is the set of isomorphism classes of objects and ${\mathrm{Aut}}_{\mathcal{G}}(x)$ is the finite group of automorphisms of $x$. The finite-gauging path integral uses this weight for the groupoid of flat bundles with the boundary data fixed by the wall construction.

The gauging in this definition is untwisted: no discrete-torsion phase or other topological action multiplies the weight of a flat-bundle sector.

In the standard endpoint-framed wall normalization used here, the composite $W_G\otimes {\bar{W}}_G$ is evaluated on a thin strip whose transverse interval is $[0,1]$, with a gauge trivialization fixed at each endpoint; gauge isomorphisms must preserve both endpoint trivializations. Each flat bundle on this interval is classified by its relative parallel transport $g\in G$ between the two fixed trivializations. Every such framed bundle has trivial endpoint-preserving automorphism group, so its groupoid-cardinality weight is $1\mathrm{/}1=1$. No additional division by $\mathrm{\mid }G\mathrm{\mid }$ is made at the framed endpoints. This fixes the finite-wall normalization used by the fusion result; it does not fix the absolute normalization of a continuous-gauging wall.

With this gauging-wall convention, an oriented strip sector of relative transport $g$ contributes the action line $L_g$ on the $C$ side; reversing the strip orientation replaces $g$ by $g^{-1}$ and $L_g$ by its orientation reverse. Subdividing and fusing strips uses the chosen junctions $L_g\otimes L_h\cong L_{gh}$. For the trivial group, the single framed sector has weight one and $W_{\{e\}}$ is the identity interface.

Topological defects, interfaces, orientation, and fusion conventions

Argurio et al., §3.1 and §4.2

Arias-Tamargo et al., §§2.3–2.5, 3.1–3.4

The finite gauging condition includes the absence of the relevant 't Hooft anomaly. In the compact-boson application the two cyclic factors have a mixed anomaly, and the source permits their joint gauging when their orders are coprime. The definition does not assert anomaly cancellation for an arbitrary group action.