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 dimM+1, such that the first eigenvalue of the twisted Dirac operator DAt is nonzero for t1 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.

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