For every integer n≥ 3, every varphi∈ L^p(S^n) with fracnn-1<p<∞, and every vector field A such that dA^flat∈ L^n/2(S^n) and varphi notequiv 0 satisfies D varphi=iA·varphi, prove the sharp lower bound |dA^flat|_L^n/2(S^n)≥ 2 lfloorfrac n2 rfloor^-1/2fracn-1n-2S_n=lfloor frac n2 rfloor^-1/2fracn(n-1)2,omega_n^2/n. Equality is attainable if and only if n is odd; when equality holds, modulo conformal and gauge transformations, varphi is a Killing spinor and A is a real multiple of the Reeb field associated with varphi.

Frank–Loss sharp Dirac zero-mode problem

For every integer $n\ge 3$ , every $\varphi\in L^p(\mathbb{S}^n)$ with $\frac{n}{n-1}<p<\infty$ , and every vector field $A$ such that $\mathrm{d}A^\flat\in L^{n/2}(\mathbb{S}^n)$ and $\varphi\not\equiv 0$ satisfies $D\varphi=iA\cdot\varphi$ , prove the sharp lower bound

$\|\mathrm{d}A^\flat\|_{L^{n/2}(\mathbb{S}^n)}\ge 2\left\lfloor\frac n2\right\rfloor^{-1/2}\frac{n-1}{n-2}S_n=\left\lfloor\frac n2\right\rfloor^{-1/2}\frac{n(n-1)}{2}\,\omega_n^{2/n}.$

Equality is attainable if and only if $n$ is odd; when equality holds, modulo conformal and gauge transformations, $\varphi$ is a Killing spinor and $A$ is a real multiple of the Reeb field associated with $\varphi$ .

References

Primary source

arXiv

Additional references

Another sharp criterion for Dirac zero modes — arXiv — Guofang Wang, Mingwei Zhang

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A September 2026 paper claims to settle the sharp magnetic-field bound in three or more dimensions, but the claim has not been independently verified and an auxiliary issue remains open.

The Frank–Loss problem asks for the optimal $L^{n/2}$ lower bound for magnetic fields supporting Dirac zero modes in dimension $n \ge 3$ . Frank and Loss posed the sharp-constant question after proving a nonsharp bound in dimension $3$ .

Known results

Frank–Loss, 2020/2021: proved a nonsharp dimension- $3$ bound, $\|B\|_{3/2} \ge 2S_{3}$ .
Frank–Loss, 2022: proved the sharp vector-potential bound $\|A\|_{L^{d}}^{2} \ge \frac{d}{d-2}S_{d}$ , with equality classified in odd dimensions; this is not the magnetic-field problem.
Wang–Zhang, 2026: proved the sharp magnetic-field inequality on $\mathbb{S}^{3}$ , $\|\operatorname{curl}A\|_{3/2} \ge 3\omega_{3}^{2/3}$ , with equality characterized.

September 2026 claimed resolution

Guofang Wang and Mingwei Zhang’s paper Another sharp criterion for Dirac zero modes claims the sharp $L^{n/2}$ magnetic-field bound for $n \ge 3$ , with equality characterized in odd dimensions up to conformal and gauge transformations. The claim is reported as resolving the problem, but the retrieved material does not provide independent verification; its abstract also leaves an auxiliary Sobolev constant open.

Current status (as of September 2026): the full sharp result is claimed in a new paper but remains unverified; the earlier vector-potential theorem and the sharp $\mathbb{S}^{3}$ case are established, while an auxiliary Sobolev constant remains open.

Sources

No solutions have been posted yet.