Bulk Dislocation Zero Mode Conjecture under Lattice Gradient Flow

Establish whether lattice gradient flows that shrink instantons in the five-dimensional extensions used by the Grabowska–Kaplan slab framework and the Kaplan–Sen disk framework generate fermionic zero modes localized at the bulk dislocation where the instanton disappears; if such modes exist, characterize their role in compensating four-dimensional instanton zero modes so as to preserve the separate U(1) fermion-number symmetries of each five-dimensional field.

Background

To reconcile exact U(1) fermion-number symmetries in the five-dimensional constructions with the usual ’t Hooft vertices of instantons, the authors propose a dynamical mechanism: under lattice gradient flows (defined in the appendix) instantons shrink and eventually disappear, leaving a dislocation in the five-dimensional flowed gauge field.

They conjecture that new bulk zero modes, localized at such dislocations, may arise to compensate the four-dimensional zero modes, thereby restoring conservation of individual five-dimensional fermion numbers. This conjecture aims to explain how ’t Hooft vertices are modified in these frameworks to preserve the exact U(1) symmetries.