Manuscript XIV-A@1 · XIV-A.tex · section · 71020abe

The observable

A0: the frozen sheet › The observable · lines 560–582

XIV-A@1#eq:O

The observable is the full displayed gauge-fixed second-order relational coefficient of XII-E,

O (f, g) [γ, ϕ] = Φ^{(2)}_{D} (f, g) [γ, ϕ] + A_{1} [f, g] (r_{g} [ϕ]; ϕ), Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D} (f, g) [γ, ϕ] .

It has metric and scalar arguments only, degree $(2, 1)$ plus the degree- $(0, 3)$ scalar-sourced slot, and $Q_{H} O \in E_{H}$ by XII-E. The sign $- 1$ on $C_{3, D}$ is XII-E's and is not reopened. The evaluation domains are

C_{sol}^{bos} = {(γ, ϕ, b) : □ γ = 0, P_{m} ϕ = 0, b = D γ}, C_{W}^{bos} = {u \in C_{sol}^{bos} : D^{*} b = 0 on W} .

Since $O$ has no ghost argument, its evaluations depend only on the bosonic part of a configuration.

Displayed formulas in this passage

XIV-A@1#eq:O · line 564
$O (f, g) [γ, ϕ] = Φ^{(2)}_{D} (f, g) [γ, ϕ] + A_{1} [f, g] (r_{g} [ϕ]; ϕ), Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D} (f, g) [γ, ϕ] .$
unlabeled · line 575
$C_{sol}^{bos} = {(γ, ϕ, b) : □ γ = 0, P_{m} ϕ = 0, b = D γ}, C_{W}^{bos} = {u \in C_{sol}^{bos} : D^{*} b = 0 on W} .$