  We obtain regular deformed D2-brane solutions with fractional D2-branes
arising as wrapped D4-branes. The space transverse to the D2-brane is a
complete Ricci-flat 7-manifold of G_2 holonomy, which is asymptotically conical
with principal orbits that are topologically CP^3 or the flag manifold
SU(3)/(U(1) x U(1)). We obtain the solution by first constructing an L^2
normalisable harmonic 3-form. We also review a previously-obtained regular
deformed D2-brane whose transverse space is a different 7-manifold of G_2
holonomy, with principal orbits that are topologically S^3 x S^3. This
describes D2-branes with fractional NS-NS 2-branes coming from the wrapping of
5-branes, which is supported by a non-normalisable harmonic 3-form on the
7-manifold. We prove that both types of solutions are supersymmetric,
preserving 1/16 of the maximal supersymmetry and hence that they are dual to
{\cal N}=1 three-dimensional gauge theories. In each case, the spectrum for
minimally-coupled scalars is discrete, indicating confinement in the infrared
region of the dual gauge theories. We examine resolutions of other branes, and
obtain necessary conditions for their regularity. The resolution of many of
these seems to lie beyond supergravity. In the process of studying these
questions, we construct new explicit examples of complete Ricci-flat metrics.
