Rigorous establishment of the spin–three-dimensionality link

Establish rigorously whether the proposed geometric role of the free-electron Fermi surface S^2, as the unique sphere admitting a symplectic structure, implies a link between the existence of spin and the three-dimensionality of momentum space, including its proposed use as a counterargument to string theories.

Background

The paper develops a geometric framework in which spin and spin-dependent transport are related to the geometry of electronic bands and Fermi surfaces. It gives a proved representation of the spin operator through the exterior algebra of a band's tangent space under time-reversal symmetry, but presents the symplectic and contact-geometric extensions beyond that setting as conjectural.

The authors specifically associate the free-electron Fermi surface with the two-sphere S2 and suggest that its symplectic structure may connect the existence of spin to the three-dimensionality of momentum space. They explicitly state that rigorously establishing this proposed connection remains future work.

References

The symplectic and contact-geometric constructions we introduce to extend spin-separated transport beyond the time-reversal-symmetric case are, at this stage, conjectural, and we present them as a candidate framework rather than a completed theory. In particular, the special role played by the sphere $S2$ --- the Fermi surface of free electrons --- suggests a possible link between the existence of spin and the three-dimensionality of momentum space, because $S2$ is the only sphere out of all n-dimensional spheres admitting the symplectic structure. However, establishing this rigorously, as a counterargument to the string theories is left for future work.

— Band's Geometry Origin of Quantum Spin Transport Phenomena  (2608.27445 - Derunova et al., 27 Aug 2026) in Discussion and Conclusion