PostulateStatement: AI-adaptedProof: Not applicablePipeline-generatedprecheck pass‡ sources checked 2026-11-04

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Buscher Covariant Measure Dilaton Shift

Physical scope

Adopted local covariantly regularized determinant contribution for abelian worldsheet isometry gauging at leading sigma-model order; it is not a derivation of global path-integral equivalence or higher-loop corrections.

Postulate

Adopt the standard covariantly regularized local measure for gauging one free spacelike circle isometry in the Euclidean NS–NS sigma model on a compact closed worldsheet. Here a free circle isometry means a smooth $S^{1}$ action with trivial stabilizer at every target point (if $u \cdot p = p$ for $u \in S^{1}$ , then $u = 1$ ), generated by a nowhere-vanishing vector field $k$ satisfying $L_{k} G^{(s)} = 0$ . Choose an adapted dimensionless angular target-space coordinate $y \sim y + 2 π$ in which $k = \partial_{y}$ ; require $\partial_{y} G^{(s)} = 0$ and $\partial_{y} Φ = 0$ , and choose a local gauge for the two-form potential $B_{2}$ in which $\partial_{y} B_{2} = 0$ . In this chart let $G_{y y} > 0$ be the length-squared component of the string-frame metric and set ${\hat{G}}_{y y} = G_{y y} / α^{'}$ . Integrating out the worldsheet gauge field contributes

- \frac{1}{8 π} \int_{Σ} \sqrt{h} R^{(2)} \log {\hat{G}}_{y y}

to the effective curvature coupling. With the Fradkin–Tseytlin coefficient $\frac{1}{4 π} \int_{Σ} \sqrt{h} Φ R^{(2)}$ , this measure convention gives the local rule $Φ^{'} = Φ - \frac{1}{2} \log {\hat{G}}_{y y}$ . This is adopted leading-order measure data, not a proof of global path-integral or exact conformal-field-theory equivalence.

Postulate references

Alvarez, Barbón, and Borlaf, §2

Alvarez, Alvarez-Gaumé, and Lozano, §2.1 equation (2.1.6) and §4

Aldazabal, Marqués, and Núñez, §3.1 equations (3.11) and (3.13)

Facts & Assumptions

Given: The local Euclidean NS–NS sigma-model curvature coupling, the covariant gauge-field measure prescription stated here, the free circle action and invariant local target data in the Postulate, and ${\hat{G}}_{y y} > 0$ .

[F1]

RNS NS Background Sigma Model: the Fradkin–Tseytlin term is $\frac{1}{4 π} \int_{Σ} \sqrt{h} Φ R^{(2)}$ .

Remarks

The determinant is included as adopted physical measure data, not proved locally. Alvarez–Alvarez-Gaumé–Lozano define the regularized determinant by a ratio of differential-operator determinants in equation (2.1.7), identify the gauge-field measure factor in (4.0.9), and evaluate its curvature contribution with the heat-kernel expansion in (4.0.10)–(4.0.16). Their determinant formulas use the sigma-model-normalized, dimensionless isometry metric coefficient: in an $α^{'} = 1$ string-unit notation it is $g_{00}$ , while in this item's dimensionful string-frame chart it is ${\hat{G}}_{y y} = G_{y y}^{(s)} / α^{'}$ . Accordingly the logarithm is of the dimensionless ratio $G_{y y}^{(s)} / α^{'}$ , not of a dimensionful metric component. The source's alternative complex-coordinate normalization is stated separately in its §2.1; neither convention changes this conversion. No global gauge-bundle sum, higher-derivative term, string-loop correction, or exact CFT statement is supplied by this postulate.

Depends on

RNS NS Background Sigma Model