Ricci-flat Metrics, Harmonic Forms and Brane Resolutions

TL;DR

The paper provides an explicit construction of self-dual harmonic forms on complete non-compact Ricci-flat Kähler manifolds, focusing on Stenzel spaces $T^*S^{n+1}$ and their deformations. It shows how first-order equations derived from a superpotential yield Ricci-flat metrics and analyzes the middle-dimension harmonic forms $G_{(p,q)}$, establishing $L^2$-normalisability only for $(p,p)$ with $d=4p$; non-normalisable cases occur for $|p-q|>0$. These mathematical results are then applied to brane physics: deformed D3-branes on 6D Stenzel space and deformed M2-branes on 8D Stenzel spaces, with the KS fractional D3-brane supported by a pure $(2,1)$ form (supersymmetric) whereas some other constructions involve mixed $(p,q)$-types and can break supersymmetry. The authors extend the construction to Ricci-flat Kähler metrics on ${f C}^k$ bundles over products of Einstein–Kähler manifolds, yielding regular deformed brane solutions and providing a framework for analyzing dual field theories via AdS/CFT, Higgs vs. Coulomb branches, and IR conformal symmetry breaking. Overall, the work links precise geometric control of harmonic forms to explicitly smooth, supersymmetric brane configurations and their field-theory duals.

Abstract

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.