Manuscript VI@1 · VI.tex · theorem · b5b60afb

Theorem: Exact first-order switch defect

An exact compact-switch obstruction › The candidate and the tested identification · lines 1422–1481 · proof included or linked

VI@1#thm:switch-defect, VI@1#eq:switch-defect, VI@1#eq:P-g

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For every smooth free scalar $ϕ_{0}$ and every $ξ \in g_{T}$ ,

A_{f, g}^{(1)} (h + K ξ, ϕ_{0}) - A_{f, g}^{(1)} (h, ϕ_{0}) = \int f [(g - 1) X_{ξ} ϕ_{0} + G_{R} ((\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} - [P, g] X_{ξ} ϕ_{0})] d^{4} x,

where

[P, g] u = (□_{η} g) u + 2 (\partial^{μ} g) \partial_{μ} u .

Proof. Since $ξ$ has zero Cauchy data at $t = 0$ , equations (eq:DK) and (eq:w) imply

w [h + K ξ] - w [h] = G_{□, 0} □_{η} ξ = ξ .

The dressing variation is therefore $- X_{ξ} ϕ_{0}$ .

Equations (eq:pure-gauge-operator) and (eq:Q-switch) give

Q_{g K ξ} ϕ_{0} = - g P (X_{ξ} ϕ_{0}) - (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} .

Consequently,

δ_{ξ} ϕ_{1, g} = G_{R} (g P (X_{ξ} ϕ_{0}) + (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0}) .

Use

g P (X_{ξ} ϕ_{0}) = P (g X_{ξ} ϕ_{0}) - [P, g] X_{ξ} ϕ_{0} .

The function $g X_{ξ} ϕ_{0}$ is compactly supported, so $G_{R} P (g X_{ξ} ϕ_{0}) = g X_{ξ} ϕ_{0}$ . Subtracting the dressing variation gives (eq:switch-defect).

Displayed formulas in this passage

VI@1#eq:switch-defect · line 1426
$A_{f, g}^{(1)} (h + K ξ, ϕ_{0}) - A_{f, g}^{(1)} (h, ϕ_{0}) = \int f [(g - 1) X_{ξ} ϕ_{0} + G_{R} ((\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} - [P, g] X_{ξ} ϕ_{0})] d^{4} x,$
VI@1#eq:P-g · line 1441
$[P, g] u = (□_{η} g) u + 2 (\partial^{μ} g) \partial_{μ} u .$
unlabeled · line 1450
$w [h + K ξ] - w [h] = G_{□, 0} □_{η} ξ = ξ .$
unlabeled · line 1456
$Q_{g K ξ} ϕ_{0} = - g P (X_{ξ} ϕ_{0}) - (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0} .$
unlabeled · line 1464
$δ_{ξ} ϕ_{1, g} = G_{R} (g P (X_{ξ} ϕ_{0}) + (\partial_{μ} g) K ξ^{μν} \partial_{ν} ϕ_{0}) .$
unlabeled · line 1474
$g P (X_{ξ} ϕ_{0}) = P (g X_{ξ} ϕ_{0}) - [P, g] X_{ξ} ϕ_{0} .$

Theorem: Exact first-order switch defect

Displayed formulas in this passage

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