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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.

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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