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Central stabilizer and based stabilizer comparison

Example

Use the configuration fixed in the earlier definition Smooth and Sobolev configuration and reduced gauge spaces: $M$ is a compact, connected, oriented, boundaryless smooth Riemannian manifold of dimension $d \in {3, 4}$ ; $N \geq 2$ ; $π : P \to M$ is a smooth right principal $S U (N)$ -bundle; and $p_{m} \in P$ is the fixed lift over the chosen base point. There $G_{k + 1} = H^{k + 1} (Ad P)$ , $G_{k + 1, m} = {u : u (p_{m}) = e}$ , $Z_{N} = {z I_{N} : z^{N} = 1}$ , and ${\overset{ˉ}{G}}_{k + 1} = G_{k + 1} / Z_{N}$ , with holonomy acting in the standard representation on $C^{N}$ . Let $A$ be a smooth connection and $k$ an integer with $k > d / 2 + 1$ . The evaluation map $u \mapsto u (p_{m})$ identifies the stabilizer ${u \in G_{k + 1} : A^{u} = A}$ with the centralizer $C_{S U (N)} ({Hol}_{p_{m}} (A))$ . Consequently the based stabilizer is trivial for every $A$ . If $A$ is irreducible, meaning its holonomy leaves no nonzero proper complex subspace of $C^{N}$ invariant, its full stabilizer is $Z_{N}$ , so its stabilizer under the reduced group is trivial as well.

Example references

Walpuski, holonomy centralizers and gauge stabilizers

Williams, based and unbased gauge stabilizers in the abelian case

Facts & Assumptions

Given: Full AC; the configuration conventions above; and a smooth connection $A$ on the fixed connected base.

[A1]

Full AC is inherited through the configuration and gauge-stabilizer suppliers. The Axiom of Choice

[F1]

Smooth and Sobolev configuration and reduced gauge spaces fixes $p_{m}$ , $G_{k + 1, m}$ , $Z_{N}$ , ${\overset{ˉ}{G}}_{k + 1}$ , and the right-action convention $A^{u} = u^{- 1} A u + u^{- 1} d u$ .

[F2]

Gauge stabilizers and properness of the Sobolev action identifies the stabilizer by evaluation with the holonomy centralizer and gives centralizer $Z_{N}$ for irreducible $S U (N)$ holonomy.

Proof

Given: The hypotheses of the Example.

1.1A1F1F2algebra

If $u$ is in the stabilizer, its value $u (p_{m})$ is in the holonomy centralizer by [F2]. Conversely, [F2] says each centralizing value extends uniquely by parallel transport to a stabilizing gauge section. Thus evaluation is a bijection between the stabilizer and the centralizer. In particular, when $u (p_{m}) = e$ , parallel transport gives $u = e$ everywhere, proving the based stabilizer is trivial without irreducibility.

2.1F1F2step 1.1algebra∎

If $A$ is irreducible, [F2] identifies the full stabilizer with $Z_{N}$ . Every element of this center is divided out in ${\overset{ˉ}{G}}_{k + 1}$ , so the reduced stabilizer consists only of the identity coset.

Central stabilizer and based stabilizer comparison

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources