Spinors in periodic self-dual gauge field backgrounds

Abstract

Normalizable zero modes of the Dirac operator are constructed for a class of self-dual, periodic SU(2) gauge field backgrounds characterized by two independent integer invariants. The integers are (ST/8π2), where ST is the action over one period T, and the asymptotic winding number (q) in R3, the solutions reducing to static ‘‘monopoles’’ for large spatial distances independently of the time. The spinor solutions are obtained for the simplest class of the hierarchy presented in Chakrabarti [Phys. Rev. D 35, 696 (1987)], corresponding to q=1 and ST=8π2⋅2n (n=1,2,3,...). The full number of zero modes for such backgrounds is ((ST/8π2)−q)=(2n−1). They are all constructed explicitly. It is shown how these results can be obtained through a simple scaling limit by starting with special classes of instantons with finite action over R4. A derivation of ST is also given.