Non-Abelian Berry Phase, Instantons and N=(0,4) Supersymmetry

TL;DR

This paper explores the mathematical structure of non-Abelian Berry phases in supersymmetric quantum mechanics, revealing that in N=(0,4) systems, the Berry connection satisfies generalized self-dual equations, with implications for lower-dimensional theories.

Contribution

It demonstrates that the non-Abelian Berry connection in N=(0,4) supersymmetric systems obeys a generalized self-dual Yang-Mills equation, extending understanding of Berry phases in supersymmetry.

Findings

Berry connection satisfies generalized self-dual equations

Curvature in N=(4,4) theories is covariantly constant

Dimensional reduction leads to tt* equations

Abstract

In supersymmetric quantum mechanics, the non-Abelian Berry phase is known to obey certain differential equations. Here we study N=(0,4) systems and show that the non-Abelian Berry connection over R^{4n} 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 N=(4,4) theories and show that the curvature is covariantly constant.