Another sharp criterion for Dirac zero modes
Abstract
We resolve an open problem posed by Frank--Loss. Let n≥ 3 and let φ∈ Lp(𝕊n), with n/(n−1)<p<∞, be a nontrivial Dirac zero mode on 𝕊n, i.e. a nonzero spinor satisfying Dφ= iA·φ, where D is the Dirac operator and A is a vector field with d A♭∈ Ln/2. We prove the sharp lower bound |d A♭|n/2 ≥ 2[n/2]−1/2(n−1)/(n−2)Sn = [n/2]−1/2(n(n−1))/2ωn2/n. Equality is attainable if and only if n is odd; in that case, modulo conformal and gauge transformations, φ is a Killing spinor and A is a real multiple of the Reeb field associated with φ. The case n=3 was proved in our recent paper using a different method. In the paper we divide the remaining cases into 3 cases: i) n≥ 5 is odd, ii) n≥ 5 is even and iii) n=4. All these cases need to use different methods. For n≥ 5, the argument crucially reduces to an improved Sobolev inequality on 𝕊n under a barycenter constraint. Specifically, for u∈ W1,2(𝕊n) we consider an := inf{ ( ∫ (|∇ u|2 + (n(n−2))/4u2) − (n(n−2))/4ωn2/n|u|2n/(n−2)2 )/( ωn2/n|u|2n/(n−2)2 − ∫ u2 ) | ∫ x|u|2n/(n−2)=0, ωn2/n|u|2n/(n−2)2 − ∫ u2>0 }, and we obtain the following universal estimate an > (n(n−2))/(4(n2−3n+1)), which is enough for our aim. Determining the exact value of an remains open.
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BibTeX
@misc{wang2026another,
title = {{Another sharp criterion for Dirac zero modes}},
author = {Guofang Wang and Mingwei Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34958},
eprint = {2609.34958},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34958},
}
RIS
TY - UNPB
TI - Another sharp criterion for Dirac zero modes
AU - Guofang Wang
AU - Mingwei Zhang
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34958
UR - https://arxiv.org/abs/2609.34958
AB - We resolve an open problem posed by Frank--Loss. Let n≥ 3 and let φ∈ L^p(𝕊^n), with n/(n−1)<p<∞, be a nontrivial Dirac zero mode on 𝕊^n, i.e. a nonzero spinor satisfying Dφ= iA·φ, where D is the Dirac operator and A is a vector field with d A^♭∈ L^n/2. We prove the sharp lower bound |d A^♭|_n/2 ≥ 2[n/2]^−1/2(n−1)/(n−2)S_n = [n/2]^−1/2(n(n−1))/2ω_n^2/n. Equality is attainable if and only if n is odd; in that case, modulo conformal and gauge transformations, φ is a Killing spinor and A is a real multiple of the Reeb field associated with φ. The case n=3 was proved in our recent paper using a different method. In the paper we divide the remaining cases into 3 cases: i) n≥ 5 is odd, ii) n≥ 5 is even and iii) n=4. All these cases need to use different methods. For n≥ 5, the argument crucially reduces to an improved Sobolev inequality on 𝕊^n under a barycenter constraint. Specifically, for u∈ W^1,2(𝕊^n) we consider a_n := inf{ ( ∫ (|∇ u|^2 + (n(n−2))/4u^2) − (n(n−2))/4ω_n^2/n|u|_2n/(n−2)^2 )/( ω_n^2/n|u|_2n/(n−2)^2 − ∫ u^2 ) | ∫ x|u|^2n/(n−2)=0, ω_n^2/n|u|_2n/(n−2)^2 − ∫ u^2>0 }, and we obtain the following universal estimate a_n > (n(n−2))/(4(n^2−3n+1)), which is enough for our aim. Determining the exact value of a_n remains open.
ER -