Manuscript X-C-ST@1 · X-C-ST.tex · lemma · b679954e

Lemma: Switch defect with all retained derivatives

Formal duality and the two defects › Switch derivatives before restriction · lines 1403–1478 · proof included or linked

X-C-ST@1#lem:sigma, X-C-ST@1#eq:sigma, X-C-ST@1#eq:sigma-expanded

Let $K_{F} = supp ξ \cup supp ζ$ . The coefficients of $S_{1}, S_{2}$ are smooth and supported in $K_{F}$ . They are

S_{1} S_{2} = - N_{1} [K ξ] = (g - 1) Q_{1} [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 $K_{F}$ , then $Σ_{F} = 0$ .

Proof. Write

E_{g} [q] = E_{unsw} [q] - κ N_{1} [h] - κ^{2} N_{2} [h, k] .

Unswitched naturality cancels its contribution to (eq:defects). The remainder is

- (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), using the linearity of $N_{1}$ . Outside $K_{F}$ , every transformation coefficient is locally the identity, and the differences vanish.

To check the switch dependence without suppressing derivatives, put

X = X_{ξ}, D = D_{ξ}, Δ A_{2} = A_{2} [h^{'}, k^{'}] - A_{2} [h, k], Δ Q_{2} = Q_{2} [h^{'}, k^{'}] - Q_{2} [h, k] .

Equation (eq:sigma) gives

S_{2} u = (1 - g) (- Δ Q_{2} 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) .

In the final line, the last divergence includes

- (\partial_{ρ} ξ^{ρ}) (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u - ξ^{ρ} (\partial_{ρ} \partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u - ξ^{ρ} (\partial_{μ} g) (\partial_{ρ} a_{1}^{μν} [h]) \partial_{ν} u - ξ^{ρ} (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ρ} \partial_{ν} u .

Thus the possible second derivatives of $g$ have not been dropped.

When $g = 1$ near $K_{F}$ , $1 - g$ and all its positive-order derivatives vanish there. The displayed operators vanish there, and already vanish outside $K_{F}$ .

Displayed formulas in this passage

X-C-ST@1#eq:sigma · line 1408
$S_{1} S_{2} = - N_{1} [K ξ] = (g - 1) Q_{1} [K ξ] + (\partial_{μ} g) a_{1}^{μν} [K ξ] \partial_{ν}, = - (N_{2} [h^{'}, k^{'}] - N_{2} [h, k]) - N_{1} [h^{'}] X_{ξ} + D_{ξ} N_{1} [h] .$
X-C-ST@1#eq:sigma-expanded · line 1446
$S_{2} u = (1 - g) (- Δ Q_{2} u - Q_{1} [h^{'}] X u + D Q_{1} [h] u) + (\partial_{μ} g) Δ A_{2}^{μν} \partial_{ν} u + (\partial_{μ} g) a_{1}^{μν} [h^{'}] \partial_{ν} (X u) - (X g) Q_{1} [h] u - \partial_{ρ} (ξ^{ρ} (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u) .$
unlabeled · line 1460
$- (\partial_{ρ} ξ^{ρ}) (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u - ξ^{ρ} (\partial_{ρ} \partial_{μ} g) a_{1}^{μν} [h] \partial_{ν} u - ξ^{ρ} (\partial_{μ} g) (\partial_{ρ} a_{1}^{μν} [h]) \partial_{ν} u - ξ^{ρ} (\partial_{μ} g) a_{1}^{μν} [h] \partial_{ρ} \partial_{ν} u .$
unlabeled · line 1423
$E_{g} [q] = E_{unsw} [q] - κ N_{1} [h] - κ^{2} N_{2} [h, k] .$
unlabeled · line 1429
$- (κ N_{1} [h^{'}] + κ^{2} N_{2} [h^{'}, k^{'}]) F^{*} + F^{♯} (κ N_{1} [h] + κ^{2} N_{2} [h, k]) .$
unlabeled · line 1440
$X = X_{ξ}, D = D_{ξ}, Δ A_{2} = A_{2} [h^{'}, k^{'}] - A_{2} [h, k], Δ Q_{2} = Q_{2} [h^{'}, k^{'}] - Q_{2} [h, k] .$

Lemma: Switch defect with all retained derivatives

Displayed formulas in this passage

Nearby in this section