The result
The central result is a sharp universal lower bound for the conformally invariant norm of a magnetic field whenever a non-trivial Dirac zero mode exists on the sphere. The constant is displayed in the paper's main theorem. In plain terms, the paper asks whether the presence of the mode imposes a definite minimum size on the field, and answers that question within its stated geometric setting.
The theorem concerns the n-dimensional sphere in dimensions three and above. It assumes non-trivial zero modes, spinors with the paper's stated integrability, and a magnetic two-form derived from a vector potential, with the field in an integrability class whose exponent is half the sphere's dimension. Those conditions define the mathematical setting in which the lower bound is established.
The equality case is fully characterized. According to the theorem, equality occurs if and only if the dimension is odd. Modulo conformal and gauge transformations, the spinor is a Killing spinor and the vector potential is a real multiple of the associated Reeb field.
How the proof works
One of the proof's main moves is an orthogonal decomposition, combined with a two-form induced by the spinor. The construction is used to estimate the magnetic-field norm and connects the zero mode to the stated lower bound.
For dimensions five and above, the argument reduces to an improved Sobolev inequality on the sphere under a barycenter constraint. This is the part of the proof that supplies the auxiliary analytic estimate behind the main theorem.
In even dimensions, the proof uses reducibility of the spinor bundle through a chirality splitting, a decomposition based on chirality. The argument also extends from spinors that never vanish to spinors with zeros by a regularization argument.
A gain, with an open question
The Sobolev step yields a separate strict statement for dimensions five and above. If a_n denotes the barycenter-constrained improved Sobolev constant and n denotes the dimension of the sphere, Theorem 7.4 gives . The strict sign means the constant lies above the displayed dimension-dependent fraction, but its exact value remains unresolved.
Under the paper's stated upper condition on that constant, the constrained infimum is attained for dimensions five and above by a non-constant positive smooth function on the sphere. The result supplies an actual minimizer for that constrained problem, not just a numerical lower estimate.
In four dimensions, the inequality is strict. The proof says equality would make the quantity called beta define a smooth almost complex structure on the four-dimensional sphere, which the paper rules out as impossible. That obstruction excludes equality in this case.
Where the result stops
The authors interpret the main theorem as an affirmative answer to an open problem posed by Frank-Loss. They also present it as another sharp criterion for Dirac zero modes on the sphere, while saying that the lower bound for the improved Sobolev constant is sufficient for their purpose.
Several boundaries remain. The exact value of the improved Sobolev constant is still open, and the dimension-three proof is only briefly sketched in this document, with details deferred to cited prior work.
The supplied document is an arXiv version 1 preprint dated 28 September 2026.
Paper data and sources
Original title: Another sharp criterion for Dirac zero modes
Authors: Guofang Wang, Mingwei Zhang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-28
DOI: Not available
Original paper · Full text