On the eigenvalues of the twisted Dirac operator

Abstract

Given a compact Riemannian spin manifold whose untwisted Dirac operator has trivial kernel, we find a family of connections ∇At for t∊[0,1] on a trivial vector bundle of rank no larger than dim M+1, such that the first eigenvalue of the twisted Dirac operator DAt is nonzero for t≠1 and vanishes for t=1. However, if one restricts the class of twisting connections considered, then nonzero lower bounds do exist. We illustrate this fact by establishing a nonzero lower bound for the Dirac operator twisted by Hermitian–Einstein connections over Riemann surfaces.