Manuscript XIII-A@1 · XIII-A.tex · proposition · f56e91c7

Proposition: Exact free-differential display

A2: the free differential, tangency, and boundary terms › Terms remaining for homogeneous ghosts · lines 905–933 · proof included or linked

XIII-A@1#prop:sO, XIII-A@1#eq:sO

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With the preceding notation,

s_{0} O = \int f {p_{2, z} - Z^{μ} \partial_{μ} p - X^{μ} \partial_{μ} p_{z} + (Z^{ν} \partial_{ν} X^{μ} + X^{ν} \partial_{ν} Z^{μ} - Y^{μ}) \partial_{μ} ϕ + \frac{1}{2} (Z^{μ} X^{ν} + X^{μ} Z^{ν}) \partial_{μ} \partial_{ν} ϕ} d^{4} x .

It is an admitted odd, ghost-number-one polynomial. No vanishing on $C_{W}$ is asserted.

Proof. Differentiate each of the five terms of (eq:O-expanded). All undifferentiated coefficient fields are even, so the displayed first variations acquire no additional Koszul signs. Equations (eq:ghost-X2)–(eq:ghost-p2) are the product rule applied to (eq:completion-display). The shell equations justify only the simplifications stated above. Admission follows from the inherited preservation of $F_{mc, pol}$ by $s_{0}$ XIIA.

Displayed formulas in this passage

XIII-A@1#eq:sO · line 908
$s_{0} O = \int f {p_{2, z} - Z^{μ} \partial_{μ} p - X^{μ} \partial_{μ} p_{z} + (Z^{ν} \partial_{ν} X^{μ} + X^{ν} \partial_{ν} Z^{μ} - Y^{μ}) \partial_{μ} ϕ + \frac{1}{2} (Z^{μ} X^{ν} + X^{μ} Z^{ν}) \partial_{μ} \partial_{ν} ϕ} d^{4} x .$

Proposition: Exact free-differential display

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