Manuscript X-E-ST@1 · X-E-ST.tex · section · 44614221

Transformation with the free fields held fixed

The one-tree background identity and its nonzero defect › Transformation with the free fields held fixed · lines 1902–1932

X-E-ST@1#eq:LieGamma

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This is the X-E background defect for the same one-tree truncation XE (Background gauge identity). It is not a statement about the all-tree expression studied on another sheet.

The useful differential decomposition is

(L_{ξ} r)_{μν} = (K (r \cdot ξ))_{μν} - 2 ξ^{λ} Γ_{1, λ μν} [r], Γ_{1, λ μν} = \frac{1}{2} (\partial_{μ} r_{λ ν} + \partial_{ν} r_{λ μ} - \partial_{λ} r_{μν}) .

Since $r \cdot ξ$ has support in $U$ , (eq:scalaridentity) removes its $K$ -part modulo $J$ . Therefore

- A_{1} [L_{ξ} r] \equiv 2 A_{1} [ξ^{λ} Γ_{1, λ \cdot \cdot} [r]] (mod J) .

This identifies the missing tensorial two-jet term; it does not construct it.

Displayed formulas in this passage

X-E-ST@1#eq:LieGamma · line 1910
$(L_{ξ} r)_{μν} = (K (r \cdot ξ))_{μν} - 2 ξ^{λ} Γ_{1, λ μν} [r], Γ_{1, λ μν} = \frac{1}{2} (\partial_{μ} r_{λ ν} + \partial_{ν} r_{λ μ} - \partial_{λ} r_{μν}) .$
unlabeled · line 1924
$- A_{1} [L_{ξ} r] \equiv 2 A_{1} [ξ^{λ} Γ_{1, λ \cdot \cdot} [r]] (mod J) .$

Transformation with the free fields held fixed

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