On the eigenvalues of the twisted Dirac operator

Given a compact Riemannian spin manifold with positive scalar curvature, we find a family of connections ${\mathrm{\nabla }}^{A_t}$ for $t\in [0,1]$ on a trivial vector bundle of sufficiently high rank, such that the first eigenvalue of the twisted Dirac operator $D_{A_t}$ is nonzero and becomes arbitrarily small as $t\to 1$. 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.