Berry Phase and Supersymmetry

We study the constraints of supersymmetry on the non-Abelian holonomy given by U = P exp(i ∫ A), the path-ordered exponential of a connection A. For theories with four supercharges, we show that A satisfies the tt* equations if it is a function of chiral multiplets. In contrast, when A is a function of vector multiplets, it satisfies the Bogomolnyi monopole equations. We describe applications of these results to the Berry connection in supersymmetric quantum mechanics. ∗On leave from DAMTP and Trinity College, Cambridge, [email protected] Introduction Often in physics we are interested in the holonomy of a state as we move along a path Γ in some space M. Such a holonomy is typically governed by the path-ordered exponential of a non-Abelian connection A over M,