Hyper-Kähler Calabi metrics, L2 harmonic forms, resolved M2-branes, and AdS 4/CFT 3 correspondence

Abstract

We obtain a simple explicit expression for the hyper-Kähler Calabi metric on the cotangent bundle of CP n+1 , for all n, in which it is constructed as a metric of cohomogeneity one with SU( n+2)/ U( n) principal orbits. These results enable us to obtain explicit expressions for an L 2-normalisable harmonic 4-form in D=8, and an L 2-normalisable harmonic 6-form in D=12. We use the former in order to obtain an explicit resolved M2-brane solution, and we show that this solution is invariant under all three of the supersymmetries associated with the covariantly-constant spinors in the 8-dimensional Calabi metric. We give some discussion of the corresponding dual N=3 three-dimensional field theory. Various other topics are also addressed, including superpotentials for the Calabi metrics and the metrics of exceptional G 2 and Spin(7) holonomy in D=7 and D=8. We also present complex and quaternionic conifold constructions, associated with the cone metrics whose resolutions are provided by the Stenzel T ∗S n+1 and Calabi T ∗ CP n+1 metrics. In the latter case we relate the construction to the hyper-Kähler quotient. We then use the hyper-Kähler quotient to give a quaternionic rederivation of the Calabi metrics.