ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Symbolic monopole vacuum in the local dual U(1) model

Physical scope

Symbolic local vacuum calculation for the source-motivated dual U(1) effective model near one monopole singularity with nonzero small N=1 perturbation; it asserts no flux-tube existence, confinement, or UV continuum theory.

Example

In the local dual $U (1)_{D}$ model, let $u (A_{D})$ be holomorphic near $A_{D} = 0$ , let $c = u^{'} (0) \neq 0$ , and choose a neighborhood small enough that $u^{'} (A_{D}) \neq 0$ throughout. Choose a nonzero small perturbation $μ$ for which $(∣ μ c ∣ / \sqrt{2})^{1 / 2} < ε Λ_{E F T}$ , using the supplied cutoff, margin, background-validity domain and application excitation bound of the local model. These are physical applicability inputs, not a microscopic error estimate. The supersymmetric vacuum equations have the gauge orbit represented by

$A_{D} = 0, M = v > 0, \tilde{M} = - \frac{μ c}{\sqrt{2} v}, v = {(\frac{∣ μ c ∣}{\sqrt{2}})}^{1 / 2} .$

At this representative $∣ M ∣ = ∣ \tilde{M} ∣ = v$ , so the D equation vanishes. The dual $U (1)_{D}$ is fully Higgsed and the canonical dual vector mass is $m_{D}^{2} = 4 g_{D}^{2} v^{2} > 0$ . This is a symbolic local effective-model calculation. The example does not assert existence of a finite-energy vortex or linear confinement; the zero-perturbation case $μ = 0$ is outside its scope.

Example references

Seiberg and Witten, §5.6, equations (5.17)–(5.21).

Bilal, §5.3.

Local dual U(1) field content, potential, and mass normalization: Local dual U(1) monopole-Higgs effective data.

F/D-flat vacuum and compact-gauge quotient definition: Classical and quantum supersymmetric vacuum-moduli data.

Facts & Assumptions

Given: The local dual-Higgs fields, charge assignments, superpotential, D map, kinetic normalization, and cutoff regime in the Example.

[F1]

Local dual U(1) monopole-Higgs effective data: $W_{μ} = \sqrt{2} A_{D} M \tilde{M} + μ u (A_{D})$ , $D = g_{D} (∣ M ∣^{2} - ∣ \tilde{M} ∣^{2})$ , the F/D potential is a sum of squares, and the charged hypermultiplet kinetic terms are canonical while $A_{D}$ has kinetic coefficient $g_{D}^{- 2}$ , and the gauge kinetic term fixes the dual vector mass normalization.

[F2]

Classical and quantum supersymmetric vacuum-moduli data supplies the convention of taking compact-group orbits of simultaneous F/D zeros. Its global polynomial-superpotential model is not assumed to include this local holomorphic model; the latter's potential and vacuum equations are supplied directly by [F1].

[A1]

Assume $μ \neq 0$ , $c = u^{'} (0) \neq 0$ , $u^{'} (A_{D}) \neq 0$ throughout the selected neighborhood, and the induced scale obeys $v < ε Λ_{E F T}$ with the supplied local-model applicability inputs.

Verification

Given: [A1], the local F/D data [F1], and the vacuum definition [F2].

1.1A1F1F2givenalgebra

Differentiating $W_{μ} = \sqrt{2} A_{D} M \tilde{M} + μ u (A_{D})$ gives $F_{A_{D}} = \sqrt{2} M \tilde{M} + μ u^{'} (A_{D})$ , $F_{M} = \sqrt{2} A_{D} \tilde{M}$ , and $F_{\tilde{M}} = \sqrt{2} A_{D} M$ . The neutral kinetic term in [F1] is $g_{D}^{- 2} ∣ \partial A_{D} ∣^{2}$ , so set $A_{c a n} = A_{D} / g_{D}$ . By the chain rule its canonical auxiliary field is $F_{A_{c a n}} = \partial W_{μ} / \partial A_{c a n} = g_{D} \partial W_{μ} / \partial A_{D} = g_{D} F_{A_{D}}$ . Since $g_{D} > 0$ , the canonical and connection-normalized neutral F equations have the same zero set. Directly from [F1], the potential is $∣ g_{D} F_{A_{D}} ∣^{2} + ∣ F_{M} ∣^{2} + ∣ F_{\tilde{M}} ∣^{2} + D^{2} / 2$ ; positivity makes its zero locus exactly these four simultaneous zeros. This argument uses only local holomorphy of $u$ , without applying the global polynomial hypothesis of [F2]. The zero locus is invariant under the stated compact $U (1)_{D}$ action, and we use its orbit convention. At $A_{D} = 0$ , the first equation is $\sqrt{2} M \tilde{M} + μ c = 0$ , while the other two vanish.

2.1A1F1F2step 1.1constructalgebra

Set $v = (∣ μ c ∣ / \sqrt{2})^{1 / 2} > 0$ and $\tilde{M} = - μ c / (\sqrt{2} v)$ , with $M = v$ . Then $M \tilde{M} = - μ c / \sqrt{2}$ and $∣ \tilde{M} ∣ = ∣ μ c ∣ / (\sqrt{2} v) = v$ , so all F equations and $D = g_{D} (∣ M ∣^{2} - ∣ \tilde{M} ∣^{2}) = 0$ hold.

3.1A1F1F2step 1.1step 2.1algebra

Any solution in this patch with $A_{D} \neq 0$ would have $M = \tilde{M} = 0$ by the last two F equations, contradicting $μ u^{'} (A_{D}) \neq 0$ in the first. Thus $A_{D} = 0$ ; the first F equation fixes the nonzero product and the D equation fixes equal magnitudes. The U(1) phase can then make $M = v$ and fixes $\tilde{M}$ as in the Example, so the solution is one gauge orbit.

4.1F1step 2.1algebra∎

Expanding the canonical kinetic terms around $M = v$ and $∣ \tilde{M} ∣ = v$ , and canonically normalizing ${\tilde{A}}_{c} = \tilde{A} / g_{D}$ , gives $m_{D}^{2} = 2 g_{D}^{2} (∣ M ∣^{2} + ∣ \tilde{M} ∣^{2}) = 4 g_{D}^{2} v^{2} > 0$ . This verifies the local Higgs mass only; it supplies no vortex solution.

Symbolic monopole vacuum in the local dual U(1) model

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources