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Fusion of a normalized finite-group gauging wall with its reverse

Physical scope

Conditional finite-state fusion rule for the normalized gauging interface of an anomaly-free finite abelian symmetry in a two-dimensional CFT.

Statement

Let a finite abelian group $G$ act on a two-dimensional CFT $C$ by normalized, anomaly-free topological symmetry lines, with action and wall conventions as specified below. Let $W_{G} : C \to C / G$ be the endpoint-framed gauging interface. Assume the full $C$ -linear subcategory generated from the $L_{g}$ by fusion, orientation reversal, and finite direct sums, with topological junctions as morphisms, unit $1_{C}$ , and tensor product defect fusion, is a finite semisimple fusion category. Also assume the $L_{g}$ are pairwise non-isomorphic simple objects. Fuse $W_{G}$ with its orientation reverse using the framed finite-groupoid-cardinality convention. Then, as a direct sum of topological lines on $C$ , $W_{G} \otimes {\overset{ˉ}{W}}_{G} = ⨁_{g \in G} L_{g} .$ Each $L_{g}$ occurs once as a simple summand. This is a finite-gauge counting statement under the stated line-sector hypothesis and makes no assertion about continuous groups.

Statement references

Finite-group gauging wall conventions

Fusion and multifusion categories

Argurio et al., §§3.1, 4.2

Arias-Tamargo et al., §3.4

Facts & Assumptions

Given: A finite abelian anomaly-free group $G$ , its normalized symmetry lines $L_{g}$ , and the gauging interface $W_{G}$ with the endpoint convention in the statement.

[F1]

Finite-group gauging wall conventions: $G$ acts through normalized lines and coherent multiplication junctions; the sum is untwisted and uses the endpoint-framed flat-bundle weight $1 / ∣ Aut ∣$ , so each relative-holonomy sector with trivial framed automorphism group has weight one.

[F2]

Fusion and multifusion categories: the assumed topological-line category is a finite semisimple fusion category; each $L_{g}$ is a pairwise distinct simple object, so a finite direct-sum decomposition has a well-defined simple-summand multiplicity.

Proof

1.1givenF1algebra

Let $I = [0, 1]$ be the interval transverse to the fused wall. A flat principal $G$ -bundle on $I$ with chosen endpoint trivializations has relative parallel transport $g \in G$ . Since $G$ is finite and discrete and $I$ is contractible, an endpoint-preserving gauge transformation cannot change this transport; conversely, each $g \in G$ is realized by a framed flat bundle. Thus the isomorphism classes in the framed groupoid are in bijection with $G$ .

2.1givenF1F2step 1.1algebra∎

By the oriented-strip convention in [F1], the sector with relative transport $g$ contributes the action line $L_{g}$ on the $C$ side. Its coefficient is $1 / ∣ Aut ∣ = 1$ because its framed automorphism group is trivial. There is exactly one such sector for each $g$ by step 1.1. The assumed finite semisimple line category and pairwise non-isomorphic simplicity in [F2] therefore give one copy of each $L_{g}$ and no identifications between distinct summands, proving the displayed fusion rule.

Remarks

The proof uses only finite flat connections on an interval, where a fixed endpoint framing removes conjugacy ambiguity. It does not justify the analogous manipulation for an $R$ -valued gauge field or its continuous measure.

Fusion of a normalized finite-group gauging wall with its reverse

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources