Abstract
The linear modes (fluctuations self-dual up to first order) of a class of periodic self-dual SU(2) gauge fields are constructed explicitly. These periodic fields have two topological indices. One is ${P}_{T}={(8{\ensuremath{\pi}}^{2})}^{\ensuremath{-}1}{S}_{T}, {S}_{T}$ being the action over one period $T$. The other is $q (<{P}_{T})$, a monopolelike winding number in ${R}^{3}$. The number of periodic modes turns out to be ($8{P}_{T}\ensuremath{-}4q$), where $q=1$ for our particular class. The solutions are obtained by constructing the periodic zero modes of spinors of unit isospin in such gauge-field backgrounds. Our results are compared to those of Jackiw and Rebbi for aperiodic instantons. This exhibits clearly the role of the second index $q$ present in our case. Quasiperiodicity is approached as a limit of successive periodic approximations. The number of modes diverges in this limit. The possible consequences of quasiperiodicity are discussed.
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