Ricci-flat Metrics, Harmonic Forms and Brane Resolutions
  We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of S^{n+1}. We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic self-dual (p,q)-forms in the middle dimension p+q=(n+1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p,p)-forms are L^2-normalisable, while for (p,q)-forms the degree of divergence grows with |p-q|. We also construct a set of Ricci-flat metrics whose level surfaces are U(1) bundles over a product of N Einstein-Kahler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2,1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions. 
M. Cvetic, G.W. Gibbons, H. Lu and C.N. Pope