Weyl-Dirac zero mode for calorons

In this paper, we give the exact expression for the SU(2) fermion zero mode in the field of the infinite volume caloron with a nontrivial holonomy and unit charge. Study of the gauge field configurations had, somewhat surprisingly, revealed that at nontrivial holonomy calorons have two Bogomol’nyi-Prasad-Sommerfield ~BPS! monopoles @N for SU(N)# as their constituents @1,2,3#. For the HarringtonShepard @4# solution with trivial holonomy, this is hidden because one of the constituents is massless ~it can be removed by a singular gauge transformation to show that the caloron for a large scale parameter becomes a single BPS monopole @5#!. We find that, for calorons with wellseparated constituents, the fermion zero mode is entirely supported on one of them. In itself it is not surprising that the zero mode is correlated to the monopole constituents. Independently, this observation was recently also made for gluino zero modes in the context of supersymmetric gauge theories @6#. Gluinos are in the adjoint representation of the gauge group, such that there are four zero modes that can be split in pairs associated with each of the two constituents @6#. However, for the Dirac fermion, there is only one zero mode. To understand the ‘‘affinity’’ of the zero mode to only one of the two monopoles, we will analyze in some detail what distinguishes them. Calorons are characterized by the ~fixed! holonomy @1,7#. In the gauge in which Am(x) is periodic, this holonomy is given by