DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6-luna)

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String Coupling Genus Weight

Physical scope

The formal Euclidean topology factor for a connected compact oriented closed worldsheet with constant dilaton in the adopted local NS–NS model; not a complete amplitude normalization, measure construction, or nonperturbative definition.

Definition

Let $Σ$ be a compact connected Riemann surface with complex structure $J$ , compatible Euclidean metric $h$ , supplied Spin structure, and genus $g$ . Assume the Axiom of Choice used by the exact genus and Gauss–Bonnet suppliers. For a constant real dimensionless dilaton $Φ_{0}$ , define the dimensionless string coupling $g_{s} : = e^{Φ_{0}} > 0$ . In the adopted Euclidean convention, a topology sector is formally weighted by $e^{- S_{E}}$ . The constant Fradkin–Tseytlin term contributes the factor $e^{- Φ_{0} χ (Σ)} = g_{s}^{- χ (Σ)} .$ Using $χ (Σ) = 2 - 2 g$ gives the equivalent factor $g_{s}^{2 g - 2}$ . This is only the topology factor: vertex normalizations, Chan–Paton traces, external-state normalization, and amplitude integrals are not included. “Formally weighted” refers to the exponential in the adopted action and does not assert that a functional integration measure exists.

Definition references

RNS NS Background Sigma Model

Genus and Euler characteristic of a compact Riemann surface

Global Gauss-Bonnet for closed oriented surfaces

Facts & Assumptions

Given: A compact connected Riemann surface $Σ$ with complex structure $J$ , compatible Euclidean metric $h$ , supplied Spin structure, genus $g$ , and constant $Φ_{0} \in R$ . Adopt the Euclidean sector convention $e^{- S_{E}}$ and define $g_{s} = e^{Φ_{0}} > 0$ ; assume the Axiom of Choice.

[F1]

RNS NS Background Sigma Model adopts the Euclidean curvature term $S_{E}^{F T} = (4 π)^{- 1} \int_{Σ} \sqrt{h} Φ (x) R^{(2)} (h) d^{2} σ$ , with $R^{(2)} = 2 K_{h}$ .

[F2]

Global Gauss-Bonnet for closed oriented surfaces: for a compact oriented closed Riemannian surface, $\int_{Σ} K_{h} d A_{h} = 2 π χ (Σ)$ ; the supplier states its full Axiom-of-Choice premise through its triangulation arguments.

[F3]

Euler characteristic of an orientable compact surface: under the full Axiom of Choice, the underlying nonempty compact connected orientable boundaryless topological surface has a unique genus $g \geq 0$ and $χ (Σ) = 2 - 2 g$ .

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function; this full choice assumption is carried from both mathematical suppliers.

Verification

1.1givenA1F1F2algebra

Since $Φ (x) = Φ_{0}$ is constant, [F1] gives $S_{E}^{F T} = \frac{Φ_{0}}{4 π} \int_{Σ} \sqrt{h} R^{(2)} d^{2} σ$ . The convention $R^{(2)} = 2 K_{h}$ and [F2] yield $S_{E}^{F T} = \frac{Φ_{0}}{2 π} \int_{Σ} K_{h} d A_{h} = Φ_{0} χ (Σ)$ .

2.1givenF1step 1.1algebra

Splitting this term from the remaining action in the formal Euclidean factor $e^{- S_{E}}$ gives the multiplicative topology factor $e^{- S_{E}^{F T}} = e^{- Φ_{0} χ (Σ)} = (e^{Φ_{0}})^{- χ (Σ)} = g_{s}^{- χ (Σ)}$ .

3.1F3A1step 2.1algebra∎

By [F3] and [A1], $χ (Σ) = 2 - 2 g$ under the exact full-choice hypothesis. Therefore $g_{s}^{- χ (Σ)} = g_{s}^{- (2 - 2 g)} = g_{s}^{2 g - 2}$ , with no additional normalization included.

Remarks

The Weigand notes call the curvature-integral identity “Riemann–Roch”; the local dependency and proof use the correctly named global Gauss–Bonnet theorem. The use of the genus-power form carries the exact full-choice premise from Genus and Euler characteristic of a compact Riemann surface; the curvature identity also carries the full-choice premise declared by Global Gauss-Bonnet for closed oriented surfaces.

String Coupling Genus Weight

Definition

Definition references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources