Non-Abelian Berry phase, instantons, and ${\mathcal N} = (0,4)$N=(,4) supersymmetry

Abstract

In supersymmetric quantum mechanics, the non-Abelian Berry phase is known to obey certain differential equations. Here we study \documentclass[12pt]{minimal}\begin{document}${\cal N}=(0,4)$\end{document}N=(0,4) systems and show that the non-Abelian Berry connection over \documentclass[12pt]{minimal}\begin{document}${\bf R}^{4n}$\end{document}R4n satisfies a generalization of the self-dual Yang–Mills equations. Upon dimensional reduction, these become the tt* equations. We further study the Berry connection in \documentclass[12pt]{minimal}\begin{document}${\cal N}=(4,4)$\end{document}N=(4,4) theories and show that the curvature is covariantly constant.