Physics theoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (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.

Polyakov Equations and Virasoro Constraints

Physical scope

Classical local field and metric equations of the free tensile flat-target Polyakov model in a Lorentzian conformal-gauge patch; no quantum Weyl-invariance claim.

Statement

For the postulated free flat Polyakov model on a local conformal-gauge patch with $h_{a b} = e^{2 ϕ} diag (- 1, + 1)$ , write $\dot{X} = \partial_{τ} X$ , $X^{'} = \partial_{σ} X$ , $σ^{\pm} = τ \pm σ$ , and $\partial_{\pm} = \frac{1}{2} (\partial_{τ} \pm \partial_{σ})$ . Define $T_{a b} = γ_{a b} - \frac{1}{2} h_{a b} h^{c d} γ_{c d}$ , where $γ_{a b} = \partial_{a} X \cdot \partial_{b} X$ . Variation of $X$ gives $\ddot{X} - X^{''} = 0$ . The metric equation $T_{a b} = 0$ gives $T_{+ +} = T_{- -} = 0$ , equivalently $\dot{X} \cdot X^{'} = 0$ and ${\dot{X}}^{2} + X^{' 2} = 0$ ; these constraints remain imposed after choosing conformal gauge and setting $ϕ = 0$ .

Statement references

Flat Bosonic String Model

Bosonic Polyakov Model

Conformal Gauge and Residual Symmetry

Facts & Assumptions

Given: A $C^{2}$ flat-target configuration $X$ , the adopted Polyakov action, and a local Lorentzian conformal-gauge patch with $h_{a b} = e^{2 ϕ} η_{a b}$ and $η_{a b} = diag (- 1, + 1)$ .

[F1]

Flat Bosonic String Model: $X$ maps a two-dimensional Lorentzian worldsheet into $R^{1, D - 1}$ with target metric $η = diag (- 1, + 1, \dots, + 1)$ .

[A1]

Bosonic Polyakov Model: $S_{P} = - \frac{T}{2} \int \sqrt{- h} h^{a b} \partial_{a} X \cdot \partial_{b} X d^{2} σ$ , and $h$ is varied as an auxiliary metric.

[F2]

Conformal Gauge and Residual Symmetry: in the selected patch $h_{a b} = e^{2 ϕ} η_{a b}$ , $σ^{\pm} = τ \pm σ$ , and $\partial_{\pm} = \frac{1}{2} (\partial_{τ} \pm \partial_{σ})$ ; in two dimensions $\sqrt{- h} h^{a b} = η^{a b}$ , so the conformal factor drops out of the classical $X$ -equation.

Proof

1.1givenF1A1F2algebra

Varying $X$ in [A1] and integrating by parts against compactly supported variations gives $\partial_{a} (\sqrt{- h} h^{a b} \partial_{b} X^{μ}) = 0$ . By [F2], this is $η^{a b} \partial_{a} \partial_{b} X^{μ} = - {\ddot{X}}^{μ} + X^{μ''} = 0$ , equivalent to the stated wave equation.

1.2givenF1A1algebra

Varying the inverse metric $h^{a b}$ in [A1] gives the coefficient $γ_{a b} - \frac{1}{2} h_{a b} h^{c d} γ_{c d}$ , where $γ_{a b} = \partial_{a} X \cdot \partial_{b} X$ ; this is $T_{a b}$ as defined in the Statement. Thus the metric equation is $T_{a b} = 0$ ; gauge fixing changes its coordinate representation but does not remove this Euler–Lagrange equation.

2.1givenF1step 1.2algebra

In the flat conformal representative $η_{a b} = diag (- 1, + 1)$ , direct substitution into [F1]'s induced-metric expression gives $T_{00} = \frac{1}{2} ({\dot{X}}^{2} + X^{' 2})$ , $T_{11} = \frac{1}{2} ({\dot{X}}^{2} + X^{' 2})$ , and $T_{01} = \dot{X} \cdot X^{'}$ . Hence $T_{a b} = 0$ is equivalent to ${\dot{X}}^{2} + X^{' 2} = 0$ and $\dot{X} \cdot X^{'} = 0$ .

3.1F2step 2.1algebra∎

With the light-cone convention in [F2], $T_{+ +} = \partial_{+} X \cdot \partial_{+} X = \frac{1}{4} ({\dot{X}}^{2} + X^{' 2} + 2 \dot{X} \cdot X^{'})$ , and $T_{- -} = \partial_{-} X \cdot \partial_{-} X = \frac{1}{4} ({\dot{X}}^{2} + X^{' 2} - 2 \dot{X} \cdot X^{'})$ . Their vanishing is equivalent to the two equations in step 2.1 by adding and subtracting.

Polyakov Equations and Virasoro Constraints

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources