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Coulomb chart linearization and derivative loss

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 $d \in {3, 4}$ , $π : P \to M$ is a smooth right principal $S U (N)$ -bundle with $N \geq 2$ , and $A_{0}$ is the fixed smooth reference connection. There $A_{k} = A_{0} + H^{k} (T^{*} M \otimes ad P)$ , $d_{A} ξ = d ξ + [A, ξ]$ , the right action is $B^{u} = u^{- 1} B u + u^{- 1} d u$ , and $d_{A}^{*}$ is the formal adjoint with its stated local coefficient formula. Let $k$ be an integer with $k > d / 2 + 1$ and let $A \in A_{k}^{*}$ , meaning its holonomy representation on $C^{N}$ is irreducible. The differential at $(0, 0)$ of the Coulomb-gauge action map

$Φ_{A} : \ker (d_{A}^{*} : H^{k} (T^{*} M \otimes ad P) \to H^{k - 1} (ad P)) \times H^{k + 1} (ad P) ⟶ A_{k}, (a, ξ) ⟼ (A + a)^{\exp ξ},$

is the bounded isomorphism

$L_{A} (a, ξ) = a + d_{A} ξ .$

Its inverse, for $v \in H^{k} (T^{*} M \otimes ad P)$ , is

$L_{A}^{- 1} v = (v - d_{A} G_{A} d_{A}^{*} v, G_{A} d_{A}^{*} v),$

where $G_{A} = (d_{A}^{*} d_{A})^{- 1} : H^{k - 1} (ad P) \to H^{k + 1} (ad P)$ is the bounded inverse from the preceding covariant-Laplacian result. Thus the connection variation lies in $H^{k}$ , the infinitesimal gauge parameter lies in $H^{k + 1}$ , and the gauge term $d_{A} ξ$ loses one derivative. This is a linearized chart statement and asserts no determinant or measure formula.

Example references

Walpuski, Coulomb slice and linearized gauge action

Williams, elliptic slice operator in the abelian case

Facts & Assumptions

Given: $A \in A_{k}^{*}$ and the configuration data with $k > d / 2 + 1$ .

[A1]

Full AC is inherited through the configuration, Sobolev, and elliptic suppliers. The Axiom of Choice

[F1]

Smooth and Sobolev configuration and reduced gauge spaces fixes the Sobolev spaces and right-action convention. Sobolev gauge calculus and smooth gauge action gives the differential $D Φ_{A} (0, 0) (a, ξ) = a + d_{A} ξ$ .

[F2]

Rough-coefficient covariant Green inverse and Coulomb splitting gives the bounded inverse $G_{A}$ and decomposition $H^{k} = \ker d_{A}^{*} \oplus im d_{A}$ .

[F3]

Sobolev and smooth irreducible gauge-orbit principal bundles identifies this derivative as the tangent map of the local Coulomb-slice principal-bundle chart.

Proof

Given: The hypotheses of the Example.

1.1A1F1algebra

The calculus lemma [F1] differentiates the right-action formula at the identity, giving $D Φ_{A} (0, 0) (a, ξ) = a + d_{A} ξ$ . Its domain is $\ker d_{A}^{*} \oplus H^{k + 1} (ad P)$ and its codomain is $H^{k} (T^{*} M \otimes ad P)$ . The gauge derivative $d_{A} : H^{k + 1} \to H^{k}$ explains the one-derivative loss.

2.1F2F3step 1.1algebra∎

Set $Q_{A} = d_{A} G_{A} d_{A}^{*}$ , the bounded projection onto $im d_{A}$ from [F2]. For any $v \in H^{k}$ , $d_{A}^{*} (v - Q_{A} v) = d_{A}^{*} v - d_{A}^{*} d_{A} G_{A} d_{A}^{*} v = 0$ and $L_{A} (v - Q_{A} v, G_{A} d_{A}^{*} v) = v$ . Conversely, for $a \in \ker d_{A}^{*}$ and $ξ \in H^{k + 1}$ , applying the proposed inverse to $a + d_{A} ξ$ gives $G_{A} d_{A}^{*} (a + d_{A} ξ) = G_{A} (d_{A}^{*} a + d_{A}^{*} d_{A} ξ) = ξ$ and first component $a$ . Both composites are therefore identities with the Sobolev orders supplied by [F2], proving the stated bounded inverse. The chart interpretation is the one established in [F3].

Coulomb chart linearization and derivative loss

Example

Example references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources