Manuscript XI-D@1 · XI-D.tex · proposition · 994a5c1b
Proposition: Free variation and ghost number
A necessary free-order closure condition › The free variation is an actual classical functional · lines 687–712 · proof included or linked
XI-D@1#prop:variation, XI-D@1#eq:sB, XI-D@1#eq:sT
The bare insertion has ghost number zero. Its copied free variation, defined on smooth graded configurations, is
It has ghost number one. Moreover,
as a classical identity.
Proof. The coordinate operator is continuous and linear on smooth tensors at fixed width. Apply the chain rule to (eq:B), using (eq:sfree). The scalar has zero free variation. This gives (eq:sB) and the stated ghost numbers. The tree depends only on , so (eq:sT) follows.
Displayed formulas in this passage
XI-D@1#eq:sB · line 691
XI-D@1#eq:sT · line 699