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A nontrivial cocycle weight on the finite two-element gauge groupoid

Example

Let $G = C_{2} = {1, s}$ and let the finite background groupoid be the conjugation action groupoid $G / / G$ : its objects are $g \in G$ , its arrows $g \to g^{'}$ are labels $h \in G$ with $g^{'} = h g h^{- 1}$ , and composition of labels $h$ then $k$ is $k h$ . Give $U (1)$ the trivial $G$ -action. Define $χ : G \to U (1)$ by $χ (1) = 1$ and $χ (s) = - 1$ ; for this action the normalized group $1$ -cocycle condition is $χ (g h) = χ (g) χ (h)$ , which these values satisfy. Set $W (g) = χ (g)$ . The action weight is invariant under every arrow, and the normalized gauging sum is zero.

Example references

Finite background groupoid and cocycle-weighted gauging data

Facts & Assumptions

Given: The conjugation action groupoid $G / / G$ for $G = C_{2}$ , the trivial action of $G$ on $U (1)$ , and the function $χ$ specified in the Example.

[F1]

Finite background groupoid and cocycle-weighted gauging data: a cocycle/action weight is a $U (1)$ -valued function constant on every gauge arrow, and the normalized sum divides each isomorphism-class weight by its automorphism-group order.

Proof

1.1givenF1algebra

Since $C_{2}$ is abelian, conjugation fixes each of its two elements. Thus the groupoid has two isomorphism classes, represented by $1$ and $s$ , and each automorphism group is all of $C_{2}$ .

1.2givenF1algebra

The map $χ$ is a group homomorphism: its values satisfy $χ (1) = 1$ and $χ (s)^{2} = 1 = χ (s^{2})$ . Hence it is a normalized group 1-cocycle. Since conjugation is trivial, $W (g) = χ (g)$ is constant on each arrow and is an admissible action weight.

2.1givenF1step 1.1step 1.2algebra∎

The normalized sum is $Z (G / / G, W) = χ (1) / 2 + χ (s) / 2 = 1 / 2 - 1 / 2 = 0$ .

Depends on

Finite background groupoid and cocycle-weighted gauging data