Antiperiodic spinor zero-modes for periodic SU(2) instanton backgrounds

Abstract

Regular, normalizable, antiperiodic zero-modes are constructed explicitly for Dirac spinors in periodic, selfdual SU(2) gauge field backgrounds. This is done for isospin 1 2 and 1 . The results are compared to the periodic spinor solutions presented in previous papers. The interests of both types for semiclassical development are pointed out. It is also interesting to compare them from a topological point of view. Our periodic gauge fields have two indices P T =(8 π 2) −1X (action over one period T) and q(⩽ P T ) a monopole-like winding number. The number of periodic (antiperiodic) spinor solutions, denoted by N ( I) (+) ( N ( I) (-)) for isospin I, is found to be N ( 1 2 ) (+)=P T−q, N (1) (+)=4P T−2q and N ( 1 2 ) (-)=P T, N (1) (-)=4P T . A general explanation of these features is indicated though explicit solutions are presented only for q=0, 1.