Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-11-04 (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

RNS chiral picture-charge selection rule

Physical scope

Formal chiral free-field charge neutrality for a fixed genus-g RNS sector with the stated bosonized background-charge Ward convention; not a superstring measure.

Statement

Fix a closed genus- $g$ RNS worldsheet and one chiral sector with the bosonized background convention of RNS bosonized picture charge convention. Here a scalar-picture eigenoperator $O$ of charge $p$ means an insertion with the constant-shift law $O [ϕ + a] = e^{p a} O [ϕ]$ for every scalar constant $a$ , with all other fields fixed. Equivalently, in the split $ϕ = ϕ_{0} + \tilde{ϕ}$ , $O [ϕ] = e^{p ϕ_{0}} O [\tilde{ϕ}]$ ; $e^{p ϕ}$ times shift-invariant derivative or spectator fields is included. This is the scalar-exponent picture convention, not an unspecified total grading involving $η, ξ$ . For a formal chiral correlator with such picture eigenoperators of charges $p_{1}, \dots, p_{n}$ , a nonzero correlator requires $\sum_{i} p_{i} = 2 g - 2.$ The assertion is conditional on the stated bosonized zero-mode Ward convention and does not define a path-integral measure over moduli.

Statement references

RNS bosonized picture charge convention

Facts & Assumptions

Given: A compact connected closed genus- $g$ surface, one fixed RNS chirality, the bosonized curvature coupling and formal zero-mode Ward functional of RNS bosonized picture charge convention. Full AC is assumed as required by the genus and Gauss–Bonnet suppliers.

[F1]

RNS bosonized picture charge convention: for $Q_{ϕ} = 2$ , the curvature term is $S_{ϕ, c u r v} = - (4 π)^{- 1} \int R^{(2)} ϕ d A$ ; insertions of $e^{p_{i} ϕ}$ contribute zero-mode exponent $\sum_{i} p_{i}$ ; the formal correlator is neutral only when the total coefficient vanishes.

[F2]

Genus and Euler characteristic of a compact Riemann surface: under full AC, $χ = 2 - 2 g$ .

[F3]

Global Gauss-Bonnet for closed oriented surfaces: $\int_{Σ} K_{h} d A_{h} = 2 π χ (Σ)$ ; with the convention $R^{(2)} = 2 K_{h}$ fixed in [F1], this gives $\int_{Σ} R^{(2)} d A_{h} = 4 π χ (Σ)$ . The supplier proof carries full AC.

[A1]

The Axiom of Choice is assumed because [F2] and [F3] require it.

Proof

1.1givenF1F2F3A1algebra

In the zero-mode split $ϕ = ϕ_{0} + \tilde{ϕ}$ , [F3] and [F2] give the curvature term $S_{ϕ, c u r v} = - \frac{2}{8 π} (4 π χ) ϕ_{0} = - χ ϕ_{0} .$ Therefore $e^{- S_{ϕ, c u r v}}$ contributes $e^{χ ϕ_{0}}$ , and each insertion contributes $e^{p_{i} ϕ_{0}}$ by the eigenoperator shift law in the Statement (take $a = ϕ_{0}$ ). Their product therefore contributes $e^{(\sum_{i} p_{i}) ϕ_{0}}$ , including derivative and spectator factors invariant under that shift. The total coefficient is $\sum_{i} p_{i} + χ = \sum_{i} p_{i} + 2 - 2 g$ .

2.1F1step 1.1algebra∎

By the neutral zero-mode functional in [F1], a nonzero chiral correlator requires this coefficient to vanish. Solving $\sum_{i} p_{i} + 2 - 2 g = 0$ gives $\sum_{i} p_{i} = 2 g - 2$ . The calculation applies independently to the right-moving chiral copy.

Remarks

Picture charge is not beta-gamma ghost number. The result follows from the explicit curvature coupling and neutral zero-mode convention; it neither chooses PCO locations nor constructs vertical chains or a global superstring integration cycle.

RNS chiral picture-charge selection rule

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources