Manuscript XII-D@1 · XII-D.tex · section · 308a2080

Defects before imposing plateau support

D5: the exact signed plateau variation › Defects before imposing plateau support · lines 1040–1079

XII-D@1#eq:layer-variation

For reference, XII-C's changed action identity is

N_{ν} (e_{D}) = κ [E_{ϕ} \partial_{ν} ϕ - B_{g, ν} - (σ_{ε} - 1) Q_{ν} (A γ) + R_{σ_{ε}}_{ν} + κ Q_{ν} (Z_{σ}_{ε})],

where

Z_{σ}_{ε} = (σ_{ε} - 1) J_{3} + D_{σ}_{ε},

B_{g, ν} = (\partial_{μ} g) T_{0}^{μ}_{ν} - (1 - g) (P_{m} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) (L_{η} - L_{q}) .

The complete ramp defect is (eq:ramp-defect). The gauge-fixed identity has the additional term $N_{ν} (e_{gf})$ . These terms are displayed before setting $g = σ_{ε} = 1$ on the parameter support.

For $ξ \in g_{U}$ , locality gives

J_{D} [γ + K ξ] - J_{D} [γ] = J_{3} [γ + K ξ] - J_{3} [γ] .

Indeed every term in the difference contains a derivative of $ξ$ or $ξ$ itself, hence is supported in $U$ . There $σ_{ε} = 1$ and all ramp derivatives vanish. The layer has not been discarded from the response: its variation under these parameters is zero.

Displayed formulas in this passage

XII-D@1#eq:layer-variation · line 1069
$J_{D} [γ + K ξ] - J_{D} [γ] = J_{3} [γ + K ξ] - J_{3} [γ] .$
unlabeled · line 1043
$N_{ν} (e_{D}) = κ [E_{ϕ} \partial_{ν} ϕ - B_{g, ν} - (σ_{ε} - 1) Q_{ν} (A γ) + R_{σ_{ε}}_{ν} + κ Q_{ν} (Z_{σ}_{ε})],$
unlabeled · line 1052
$Z_{σ}_{ε} = (σ_{ε} - 1) J_{3} + D_{σ}_{ε},$
unlabeled · line 1055
$B_{g, ν} = (\partial_{μ} g) T_{0}^{μ}_{ν} - (1 - g) (P_{m} ϕ) \partial_{ν} ϕ + (\partial_{ν} g) (L_{η} - L_{q}) .$