Manuscript X-C@3 · X-C-v3.tex · lemma · 2260f5f8

Lemma: Switch defect and its retained derivatives

C4: completed test, three-term variation, and cancellation › Switch factors displayed through second order · lines 1102–1160 · proof included or linked

X-C@3#lem:sigma, X-C@3#eq:sigma, X-C@3#eq:sigma-expanded

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The switch defect has coefficients supported in $C = supp ξ \cup supp ζ$ , and

Σ^{(1)} Σ^{(2)} = - N_{1} [K ξ] = (g - 1) Q_{K ξ} + (\partial_{μ} g) a_{1}^{μν} [K ξ] \partial_{ν}, = - (N_{2} [h^{'}, k^{'}] - N_{2} [h, k]) - N_{1} [h^{'}] X_{ξ} + D_{ξ} N_{1} [h] .

If $g = 1$ on a neighbourhood of $C$ , then $Σ_{F} = 0$ .

Proof. The copied naturality of the unswitched curved operator $E_{1} [q]$ , and $E_{g} [q] = E_{1} [q] - κ N_{1} - κ^{2} N_{2}$ , give

Σ_{F} = - (κ N_{1} [h^{'}] + κ^{2} N_{2} [h^{'}, k^{'}]) F^{*} + F^{♯} (κ N_{1} [h] + κ^{2} N_{2} [h, k]) .

Expansion gives (eq:sigma). Outside $C$ , the transformation is locally the identity, so the operator coefficients vanish there.

For an explicit display, put $X = X_{ξ}$ , $D = D_{ξ}$ , and $Δ A_{2} = A_{2} [h^{'}, k^{'}] - A_{2} [h, k]$ . The second line of (eq:sigma), acting on $u$ , is

Σ^{(2)} u = (1 - g) [- (Q_{2} [h^{'}, k^{'}] - Q_{2} [h, k]) u - Q_{1} [h^{'}] X u + D Q_{1} [h] u] + (\partial_{μ} g) Δ A_{2}^{μν} \partial_{ν} u + (\partial_{μ} g) a_{1}^{μν} [h^{'}] \partial_{ν} (X u) - (X g) Q_{1} [h] u - \partial_{ρ} [ξ^{ρ} (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u] .

The last divergence retains derivatives of the switch; expanding it can produce second derivatives of $g$ . No such term is silently removed.

On a neighbourhood where $g = 1$ , $1 - g$ and all positive-order derivatives of $g$ vanish. Every displayed term is therefore zero on $C$ , and the operator is zero outside $C$ already.

Displayed formulas in this passage

X-C@3#eq:sigma · line 1106
$Σ^{(1)} Σ^{(2)} = - N_{1} [K ξ] = (g - 1) Q_{K ξ} + (\partial_{μ} g) a_{1}^{μν} [K ξ] \partial_{ν}, = - (N_{2} [h^{'}, k^{'}] - N_{2} [h, k]) - N_{1} [h^{'}] X_{ξ} + D_{ξ} N_{1} [h] .$
X-C@3#eq:sigma-expanded · line 1136
$Σ^{(2)} u = (1 - g) [- (Q_{2} [h^{'}, k^{'}] - Q_{2} [h, k]) u - Q_{1} [h^{'}] X u + D Q_{1} [h] u] + (\partial_{μ} g) Δ A_{2}^{μν} \partial_{ν} u + (\partial_{μ} g) a_{1}^{μν} [h^{'}] \partial_{ν} (X u) - (X g) Q_{1} [h] u - \partial_{ρ} [ξ^{ρ} (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u] .$
unlabeled · line 1124
$Σ_{F} = - (κ N_{1} [h^{'}] + κ^{2} N_{2} [h^{'}, k^{'}]) F^{*} + F^{♯} (κ N_{1} [h] + κ^{2} N_{2} [h, k]) .$

Lemma: Switch defect and its retained derivatives

Displayed formulas in this passage

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