  We provide a semiclassical description of framed BPS states in
four-dimensional N = 2 super Yang-Mills theories probed by 't Hooft defects, in
terms of a supersymmetric quantum mechanics on the moduli space of singular
monopoles. Framed BPS states, like their ordinary counterparts in the theory
without defects, are associated with the L^2 kernel of certain Dirac operators
on moduli space, or equivalently with the L^2 cohomology of related Dolbeault
operators. The Dirac/Dolbeault operators depend on two Cartan-valued Higgs
vevs. We conjecture a map between these vevs and the Seiberg-Witten special
coordinates, consistent with a one-loop analysis and checked in examples. The
map incorporates all perturbative and nonperturbative corrections that are
relevant for the semiclassical construction of BPS states, over a suitably
defined weak coupling regime of the Coulomb branch. We use this map to
translate wall crossing formulae and the no-exotics theorem to statements about
the Dirac/Dolbeault operators. The no-exotics theorem, concerning the absence
of nontrivial SU(2)_R representations in the BPS spectrum, implies that the
kernel of the Dirac operator is chiral, and further translates into a statement
that all L^2 cohomology of the Dolbeault operator is concentrated in the middle
degree. Wall crossing formulae lead to detailed predictions for where the Dirac
operators fail to be Fredholm and how their kernels jump. We explore these
predictions in nontrivial examples. This paper explains the background and
arguments behind the results announced in a short accompanying note.
