  The existence of normalizable zero modes of the twisted Dirac operator is
proven for a class of static Einstein-Yang-Mills background fields with a
half-integer Chern-Simons number. The proof holds for any gauge group and
applies to Dirac spinors in an arbitrary representation of the gauge group. The
class of background fields contains all regular, asymptotically flat,
CP-symmetric configurations with a connection that is globally described by a
time-independent spatial one-form which vanishes sufficiently fast at infinity.
A subset is provided by all neutral, spherically symmetric configurations which
satisfy a certain genericity condition, and for which the gauge potential is
purely magnetic with real magnetic amplitudes.
