Manuscript XI-E@1 · XI-E.tex · section · 46e55514

Full auxiliary and ghost blocks

Route 1: a named polynomial Hadamard extension › Full auxiliary and ghost blocks · lines 599–634

XI-E@1#eq:Heven, XI-E@1#eq:Haux, XI-E@1#eq:Hghost

Let $A = 2 G_{lin}$ . The metric–auxiliary Hessian is

(A 2 D 2 D^{*} - 2 id), A + 2 D^{*} D = - □ P,

with

(D^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} .

Flat tensor and vector pairings are understood throughout.

Define the nonzero blocks of a bidistribution matrix $H$ by

H_{γ γ} H_{γ b} H_{c c ˉ} = - P h_{0}, = D_{y} H_{γ γ}, = - i h_{0}, H_{ϕϕ} H_{bγ} H_{c ˉ c} = - h_{m}, = D_{x} H_{γ γ}, = i h_{0} . H_{bb} = D_{x} D_{y} H_{γ γ} = 0,

Vector identities accompany the ghost blocks. All other cross-blocks vanish.

Displayed formulas in this passage

XI-E@1#eq:Heven · line 616
$H_{γ γ} H_{γ b} H_{c c ˉ} = - P h_{0}, = D_{y} H_{γ γ}, = - i h_{0}, H_{ϕϕ} H_{bγ} H_{c ˉ c} = - h_{m}, = D_{x} H_{γ γ}, = i h_{0} . H_{bb} = D_{x} D_{y} H_{γ γ} = 0,$
XI-E@1#eq:Haux · line 616
$H_{γ γ} H_{γ b} H_{c c ˉ} = - P h_{0}, = D_{y} H_{γ γ}, = - i h_{0}, H_{ϕϕ} H_{bγ} H_{c ˉ c} = - h_{m}, = D_{x} H_{γ γ}, = i h_{0} . H_{bb} = D_{x} D_{y} H_{γ γ} = 0,$
XI-E@1#eq:Hghost · line 616
$H_{γ γ} H_{γ b} H_{c c ˉ} = - P h_{0}, = D_{y} H_{γ γ}, = - i h_{0}, H_{ϕϕ} H_{bγ} H_{c ˉ c} = - h_{m}, = D_{x} H_{γ γ}, = i h_{0} . H_{bb} = D_{x} D_{y} H_{γ γ} = 0,$
unlabeled · line 602
$(A 2 D 2 D^{*} - 2 id), A + 2 D^{*} D = - □ P,$
unlabeled · line 608
$(D^{*} v)_{μν} = - \partial_{(μ} v_{ν)} + \frac{1}{2} η_{μν} \partial_{ρ} v^{ρ} .$