Ricci-flat metrics, harmonic forms and brane resolutions.

*(English)*Zbl 1027.53044For 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.

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.

Reviewer: Jürgen Berndt (Hull)