ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Static Straight Dirichlet String

Physical scope

One explicit classical solution of the flat tensile bosonic Polyakov model on an open strip with fixed spatial endpoint data; not an empirical observation and not a dynamical D-brane model.

Example

On the strip $Σ = R \times [0, π]$ , fix a length $ℓ > 0$ , choose conformal metric $h = diag (- 1, + 1)$ , and set $X^{0} = ℓ τ$ , $X^{1} = ℓ σ$ , and $X^{μ} = 0$ for $μ \geq 2$ . Fix $X^{1}$ and all $X^{μ}$ with $μ \geq 2$ at both endpoints by Dirichlet data; $X^{0}$ satisfies Neumann data. Then $\ddot{X} - X^{''} = 0$ , ${\dot{X}}^{2} + X^{' 2} = 0$ , $\dot{X} \cdot X^{'} = 0$ , and $γ = diag (- ℓ^{2}, ℓ^{2})$ , so this is a nondegenerate Lorentzian classical Polyakov/Nambu–Goto solution on the positive branch. The fixed endpoints are prescribed boundary data, not dynamical D-branes.

Example references

Flat Bosonic String Model

String Tension Alpha Prime

Bosonic Polyakov Model

Polyakov Equations and Virasoro Constraints

Open String Boundary Variation

Nambu Goto Polyakov Equivalence

Facts & Assumptions

Given: $ℓ > 0$ ; the flat target metric is $η = diag (- 1, + 1, \dots, + 1)$ ; $τ, σ$ are dimensionless; $h = diag (- 1, + 1)$ ; the stated endpoint variations; and the adopted free Polyakov model.

[F1]

Flat Bosonic String Model defines $C^{2}$ Polyakov maps on the open strip $0 \leq σ \leq π$ and the additional nondegenerate Lorentzian immersion branch. Membership in that branch is checked for this example in step 2.2.

[F2]

String Tension Alpha Prime: $T = (2 π α^{'})^{- 1} > 0$ , and target coordinates have length dimension.

[A1]

Bosonic Polyakov Model: the free tensile flat-target Polyakov action is adopted without endpoint interactions.

[F3]

Polyakov Equations and Virasoro Constraints: a conformal-gauge solution must satisfy $\ddot{X} - X^{''} = 0$ , ${\dot{X}}^{2} + X^{' 2} = 0$ , and $\dot{X} \cdot X^{'} = 0$ .

[F4]

Open String Boundary Variation: the strip variation is stationary when each endpoint coordinate is fixed (Dirichlet) or has vanishing normal flux (Neumann).

[F5]

Nambu Goto Polyakov Equivalence: on the positive branch, matched classical Polyakov and Nambu–Goto stationary configurations correspond.

Verification

Given: The fields and data in [A1]–[F5].

1.1givenF1A1algebra

Differentiating the stated map gives $\dot{X} = (ℓ, 0, \dots, 0)$ and $X^{'} = (0, ℓ, 0, \dots, 0)$ . Both second derivatives vanish, so $\ddot{X} - X^{''} = 0$ .

2.1F3step 1.1algebra

With the mostly-plus target metric, ${\dot{X}}^{2} = - ℓ^{2}$ , $X^{' 2} = ℓ^{2}$ , and $\dot{X} \cdot X^{'} = 0$ . Thus both metric constraints in [F3] hold.

2.2givenF1step 1.1algebra

The induced metric has components $γ_{00} = - ℓ^{2}$ , $γ_{01} = 0$ , and $γ_{11} = ℓ^{2}$ , so $\det γ = - ℓ^{4} < 0$ ; the map has rank two and lies on $γ = ℓ^{2} h$ , hence $λ = ℓ^{2} > 0$ .

2.3givenF4step 1.1

At $σ = 0, π$ , $X^{1}$ and every $X^{μ}$ with $μ \geq 2$ are fixed, so their variations vanish; $X^{' 0} = 0$ , so the time component has zero Neumann flux. By [F4], the boundary term vanishes with these coordinatewise conditions.

3.1A1F2F5step 1.1step 2.1step 2.2step 2.3∎

Steps 1.1, 2.1, 2.2, and 2.3 verify the Polyakov field equation, both metric constraints, nondegeneracy, and boundary stationarity. Since $λ > 0$ , $T > 0$ by [F2], and the same boundary data apply, [F5] gives the classical Nambu–Goto solution as well; no endpoint object is introduced.

Static Straight Dirichlet String

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources