Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Coordinate invariance of the local Beltrami antighost pairing

Physical scope

Local tensorial coordinate covariance of the bosonic bc/Beltrami pairing on an overlap; it supplies no global gauge slice or determinant line.

Statement

Let $z$ and $w = f (z)$ be holomorphic coordinates with $f^{'} (z) \neq 0$ on an overlap. If $b = b_{z z} (z) d z^{2}$ is the antighost quadratic differential and $μ = μ_{\overset{ˉ}{z}}^{z} d \overset{ˉ}{z} \otimes \partial_{z}$ is a Beltrami differential, then the local density $b_{z z} μ_{\overset{ˉ}{z}}^{z} d x d y$ is invariant under this coordinate change. Hence, for compactly supported local data, $(b, μ) = \int b_{z z} μ_{\overset{ˉ}{z}}^{z} d x d y$ is independent of the chosen holomorphic chart.

Facts & Assumptions

Given: A complex coordinate overlap $w = f (z)$ with nonvanishing derivative and compactly supported local $b, μ$ .

[F1]

Bosonic Bc Ghost System: $b$ has holomorphic conformal weight $2$ and transforms as a quadratic differential. The Beltrami tensor type is the explicit hypothesis $μ = μ_{\overset{ˉ}{z}}^{z} d \overset{ˉ}{z} \otimes \partial_{z}$ in the Statement, not a conclusion of this ghost supplier.

Proof

1.1givenF1algebra

By [F1], $b_{w w} = b_{z z} (d z / d w)^{2} = b_{z z} / f^{'} (z)^{2}$ . For the tensor $μ$ specified in the Statement, $d \overset{ˉ}{z} = d \overset{ˉ}{w} / \overline{f^{'} (z)}$ and $\partial_{z} = f^{'} (z) \partial_{w}$ , so $μ_{\overset{ˉ}{w}}^{w} = f^{'} (z) μ_{\overset{ˉ}{z}}^{z} / \overline{f^{'} (z)}$ . The area elements satisfy $d x_{w} d y_{w} = ∣ f^{'} (z) ∣^{2} d x_{z} d y_{z}$ .

2.1step 1.1algebra∎

Multiplying these three factors gives $b_{w w} μ_{\overset{ˉ}{w}}^{w} d x_{w} d y_{w} = \frac{b_{z z}}{f^{' 2}} \frac{f^{'}}{\overline{f^{'}}} ∣ f^{'} ∣^{2} d x_{z} d y_{z} = b_{z z} μ_{\overset{ˉ}{z}}^{z} d x_{z} d y_{z} .$ Because the support is compact and the density agrees pointwise on each overlap, a partition into coordinate patches gives the same integral in either atlas. No statement about a global gauge slice or determinant trivialization follows.

Depends on

Bosonic Bc Ghost System

Coordinate invariance of the local Beltrami antighost pairing

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources