  The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of
stable Higgs bundles of coprime rank and degree. We provide an algebraic
generalization to the case of trivial degree and the rank higher than $1$. This
allow us to generalize to this case the Nahm transform defined by Frejlich and
the second named author, which, out of a stable Higgs bundle, produces a vector
bundle with connection over the moduli space of rank 1 Higgs bundles. By
performing the higher rank Nahm transform we obtain a hyperholomorphic bundle
with connection over the moduli space of stable Higgs bundles of rank $n$ and
degree 0, twisted by the gerbe of liftings of the projective universal bundle.
  Such hyperholomorphic vector bundles over the moduli space of stable Higgs
bundles can be seen, in the physicist's language, as BBB-branes twisted by the
above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we
study the behaviour of Nahm branes under Fourier--Mukai transform over the
smooth locus of the Hitchin fibration, checking that the resulting objects are
supported on a Lagrangian multisection of the Hitchin fibration, so they
describe partial data of BAA-branes.
