Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

Perpendicular and reality checks

Complete conversion and the real counterterm class › Perpendicular and reality checks · lines 1987–2005

Choose a real compact test $ψ$ with $⟨ r, ψ ⟩ = 1$ . Such a test exists since $r$ is a nonzero positive distribution. Then

⟨ L_{ab}, ψ ⟩ = c_{ab} \in R, L_{ab}^{⊥} = L_{ab} - ⟨ L_{ab}, ψ ⟩ r = 0.

Thus both conditions for the prescribed real class $C = R r$ are checked. The weighted-pushforward representation is completely identified:

L_{ab} = c_{ab} c_{0} Ψ_{*} ν .

Its weight is constant, as a consequence of the full conversion limit rather than only the leading logarithmic coefficient.

Displayed formulas in this passage

unlabeled · line 1992
$⟨ L_{ab}, ψ ⟩ = c_{ab} \in R, L_{ab}^{⊥} = L_{ab} - ⟨ L_{ab}, ψ ⟩ r = 0.$
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$L_{ab} = c_{ab} c_{0} Ψ_{*} ν .$