Manuscript XIII-B@1 · XIII-B.tex · proposition · b8a5c0f5

Proposition: Entry differential

B5: the free differential, displayed without a defect › The composite entry · lines 1295–1319 · proof included or linked

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

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The free differential of the composite is

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 odd, has ghost number one, and belongs to the inherited polynomial domain.

Proof. Apply the graded product rule to the displayed completion. All undifferentiated coefficient factors are even. The first and second variations above are their ordinary directional derivatives in the odd direction $z$ . Admission follows from XII-A's preservation of $F_{mc, pol}$ by the frozen differential XIIA.

Displayed formulas in this passage

XIII-B@1#eq:sO · line 1298
$s_{0} O = \int f (p_{2, z} - Z^{μ} \partial_{μ} p - X^{μ} \partial_{μ} p_{z} + (Z^{ν} \partial_{ν} X^{μ} + X^{ν} \partial_{ν} Z^{μ} - Y^{μ}) \partial_{μ} ϕ + \frac{1}{2} (Z^{μ} X^{ν} + X^{μ} Z^{ν}) \partial_{μ} \partial_{ν} ϕ) d^{4} x .$

Proposition: Entry differential

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