  Modifications to the singularity structure of D3-branes that result from
turning on a flux for the R-R and NS-NS 3-forms (fractional D3-branes) provide
important gravity duals of four-dimensional N=1 super-Yang-Mills theories. We
construct generalisations of these modified p-brane solutions in a variety of
other cases, including heterotic 5-branes, dyonic strings, M2-branes,
D2-branes, D4-branes and type IIA and type IIB strings, by replacing the flat
transverse space with a Ricci-flat manifold M_n that admits covariantly
constant spinors, and turning on a flux built from a harmonic form in M_n, thus
deforming the original solution and introducing fractional branes. The
construction makes essential use of the Chern-Simons or ``transgression'' terms
in the Bianchi-identity or equation of motion of the field strength that
supports the original undeformed solution. If the harmonic form is L^2
normalisable, this can result in a deformation of the brane solution that is
free of singularities, thus providing viable gravity duals of field theories in
diverse dimensions that have less than maximal supersymmetry. We obtain
examples of non-singular heterotic 5-branes, dyonic strings, M2-branes, type
IIA strings, and D2-branes.
