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

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Closed string genus-n-point coupling power

Physical scope

Normalization identity for connected oriented closed surfaces and canonically sphere-normalized closed vertices; it is not an integral, convergence theorem, or open-string coupling rule.

Statement

Let $Σ$ be a connected compact oriented closed worldsheet of genus $g$ , and assume full AC as required by the supplied genus classification. If its adopted constant-dilaton topology factor is $g_{s}^{2 g - 2}$ and each of $n \geq 0$ canonically sphere-normalized closed external vertices contributes $g_{s}$ , then the explicit coupling factor is $g_{s}^{2 g - 2 + n}$ . This is a multiplication identity only; it supplies no convergence of the genus series. The formula does not apply to a boundary worldsheet.

Facts & Assumptions

Given: $g_{s} = e^{Φ_{0}} > 0$ , the exact R2 closed constant-dilaton convention, and $n$ external closed vertices in the stated sphere normalization.

[F1]

String Coupling Genus Weight derives the weight $g_{s}^{2 g - 2}$ for a compact connected Riemann surface with complex structure, compatible Euclidean metric, supplied Spin structure, and full AC. The present Statement explicitly assumes that numerical topology factor; it does not extend the supplier's action derivation to surfaces lacking those structures.

[F2]

Genus and Euler characteristic of a compact Riemann surface defines genus using the topological classification of a compact Riemann surface under full AC. This supplier fixes the intended worldsheet genus convention; the multiplication below does not require a new classification theorem or use a curvature integral.

[F3]

String vertex datum and covariant insertion distinguishes the local vertex datum from its separately declared external-state normalization; the present given fixes that normalization as one factor $g_{s}$ per closed insertion.

[A1]

The Axiom of Choice is assumed because [F2] requires full AC.

Proof

1.1givenF1F3algebra

By the explicit topology-factor hypothesis in the Statement and Given, the contribution is $g_{s}^{2 g - 2}$ ; [F1] identifies the structured worldsheet model in which that factor was derived. By [F3] and the explicit normalization in Given, the product of the $n$ closed external factors is $\prod_{i = 1}^{n} g_{s} = g_{s}^{n}$ (the empty product is $1$ when $n = 0$ ).

2.1F1F2A1step 1.1algebra∎

Multiplication gives $g_{s}^{2 g - 2} g_{s}^{n} = g_{s}^{2 g - 2 + n}$ because $g_{s} > 0$ and exponents add. The argument uses only the closed topology convention; a worldsheet with boundary has a different Euler characteristic and its own open-vertex normalization, so it is outside this statement.

Remarks

The local vertex supplier depends on stale CFT readiness records, so this result remains conditional on its interface. The formula is not inferred for the disk and does not imply that the coefficient series in $g_{s}$ converges.

Depends on

Dependency tree · 2 levels

Sources

David Tong, String Theory, Cambridge Part III notes