Manuscript X-G-ST@1 · X-G-ST.tex · proposition · 3c97567f

Proposition: Natural background ghost operator

Natural ghost equations without an interacting quantum differential · lines 1014–1052 · proof included or linked

X-G-ST@1#prop:ghost, X-G-ST@1#eq:ghostcomm

Define the vector-density Euler operator

(P_{gh} [q ˉ] c)^{μ} = - J_{q ˉ} C_{q ˉ}^{μ} [L_{c} q ˉ] .

For the retained curvature convention,

C_{q ˉ}^{μ} [L_{c} q ˉ] = □ c ˉ^{μ} + R ˉ^{μ}_{ν} c^{ν} .

The operator is natural. Its first background coefficient satisfies

P_{gh}^{(1)} [K ξ] = L_{ξ}^{vd} P_{gh}^{(0)} - P_{gh}^{(0)} L_{ξ}^{v}, P_{gh}^{(0)} = - □,

where the two Lie derivatives act on vector densities and vectors, respectively.

Proof. Trace reversal of $L_{c} q ˉ$ , with its indices raised by $q ˉ$ , is

\nabla ˉ^{μ} c^{ν} + \nabla ˉ^{ν} c^{μ} - q ˉ^{μν} \nabla ˉ_{ρ} c^{ρ} .

Take the divergence. The commutator of the two covariant derivatives contributes the displayed Ricci term. Every operation is natural under simultaneous pullback, including the volume density. Differentiate the finite naturality identity at $η$ to obtain (eq:ghostcomm).

Displayed formulas in this passage

X-G-ST@1#eq:ghostcomm · line 1028
$P_{gh}^{(1)} [K ξ] = L_{ξ}^{vd} P_{gh}^{(0)} - P_{gh}^{(0)} L_{ξ}^{v}, P_{gh}^{(0)} = - □,$
unlabeled · line 1017
$(P_{gh} [q ˉ] c)^{μ} = - J_{q ˉ} C_{q ˉ}^{μ} [L_{c} q ˉ] .$
unlabeled · line 1022
$C_{q ˉ}^{μ} [L_{c} q ˉ] = □ c ˉ^{μ} + R ˉ^{μ}_{ν} c^{ν} .$
unlabeled · line 1042
$\nabla ˉ^{μ} c^{ν} + \nabla ˉ^{ν} c^{μ} - q ˉ^{μν} \nabla ˉ_{ρ} c^{ρ} .$

Proposition: Natural background ghost operator

Displayed formulas in this passage

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