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Ricci-flat metrics, harmonic forms and brane resolutions. (English) Zbl 1027.53044
For each $$n \geq 1$$, Stenzel constructed a complete Ricci-flat Kähler metric on the cotangent bundle $$T^*S^{n+1}$$ of the sphere $$S^{n+1}$$, which for $$n=1$$ is just the Eguchi-Hanson metric on $$T^*S^2$$. The authors discuss aspects of the geometry and topology of these metrics and give some applications.
The special orthogonal group $$SO(n+2)$$ acts on $$T^*S^{n+1}$$ in the canonical way with generic orbits of codimension one. Using coset techniques one can therefore rewrite Ricci flatness in terms of a system of ordinary differential equations. The authors obtain the Stenzel metrics as an explicit solution to this system. Then the authors present a general construction of harmonic $$(p,q)$$-forms in the middle dimension $$p+q = n+1$$ of the Stenzel metric on $$T^*S^{n+1}$$. For $$p=q$$ these forms are $$L^2$$-normalisable, but not for $$p \neq q$$, where the degree of divergence grows with $$|p-q |$$. Then they use again the cohomogeneity one ansatz to construct Ricci-flat metrics where the level surfaces are $$U(1)$$-bundles over products of Einstein-Kähler manifolds, and construct examples of harmonic forms on these spaces as well.
As an application, the authors review the construction of a deformed fractional D3-brane based on the $$6$$-dimensional Stenzel metric and show that the D3-brane found by Klebanov and Strassler is supported by a $$(2,1)$$-form and hence supersymmetric. They also show that the fractional D3-brane discussed by Pando Zayas and Tseytlin is supported by a mixture of $$(1,2)$$ and $$(2,1)$$-forms. The authors then construct new examples of supersymmetric nonsingular M2-branes using the $$8$$-dimensional Stenzel metric. Finally, they comment on the implications for the corresponding dual field theories of the resolved brane solutions.

##### MSC:
 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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