Manuscript X-G@1 · X-G.tex · proposition · 01566e14

Proposition: Ghost operator and naturality

The nonlinear kinematics and background ghost operator › The natural background fermion · lines 873–919 · proof included or linked

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

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With X-E's curvature conventions,

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

Moreover,

P_{gh} [F^{*} q ˉ] (F^{*} c) = F^{*} (P_{gh} [q ˉ] c) .

P_{gh} [η + κ w] = P_{gh}^{(0)} + κ P_{gh}^{(1)} [w] + O (κ^{2}), P_{gh}^{(0)} = - □,

then

P_{gh}^{(1)} [K ξ] = L_{ξ} P_{gh}^{(0)} - P_{gh}^{(0)} L_{ξ} .

The left Lie derivative acts on vector densities and the right one on vectors.

Proof. Trace reversal of $L_{c} q ˉ$ gives

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

Take its divergence and commute the two derivatives on the vector. The commutator contraction is $R ˉ^{μ}_{ν} c^{ν}$ , giving (eq:ghostclosed).

The metric connection, Lie derivative, contractions, and volume density are natural. This proves the finite identity. Differentiate it along a compact flow through $η$ to obtain (eq:ghostcomm); locality removes the compactness restriction from the resulting differential identity.

Displayed formulas in this passage

X-G@1#eq:ghostclosed · line 876
$C^{μ}_{q ˉ} [L_{c} q ˉ] = □ c ˉ^{μ} + R ˉ^{μ}_{ν} c^{ν} .$
X-G@1#eq:ghostcomm · line 894
$P_{gh}^{(1)} [K ξ] = L_{ξ} P_{gh}^{(0)} - P_{gh}^{(0)} L_{ξ} .$
unlabeled · line 882
$P_{gh} [F^{*} q ˉ] (F^{*} c) = F^{*} (P_{gh} [q ˉ] c) .$
unlabeled · line 887
$P_{gh} [η + κ w] = P_{gh}^{(0)} + κ P_{gh}^{(1)} [w] + O (κ^{2}), P_{gh}^{(0)} = - □,$
unlabeled · line 906
$\nabla ˉ^{ν} c^{μ} + \nabla ˉ^{μ} c^{ν} - q ˉ^{ν μ} \nabla ˉ_{ρ} c^{ρ} .$

Proposition: Ghost operator and naturality

Displayed formulas in this passage

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