Hyper-Kahler Calabi Metrics, L^2 Harmonic Forms, Resolved M2-branes, and AdS_4/CFT_3 Correspondence

TL;DR

This work provides an explicit construction of the hyper-Kähler Calabi metric on $T^*\CP^{n+1}$ for all $n$, presenting a cohomogeneity-one metric with principal orbits $SU(n+2)/U(n)$ and a simple $n$-independent Calabi form. It then derives $L^2$-normalisable middle-dimension harmonic forms, including a $D=8$ harmonic 4-form and a $D=12$ harmonic 6-form, and uses the former to build a fully supersymmetric resolved M2-brane whose supersymmetry matches all covariantly-constant spinors of the eight-dimensional Calabi metric, yielding a dual ${\cal N}=3$ CFT in three dimensions. The paper develops first-order superpotential descriptions for the Einstein equations, obtaining Spin(7) holonomy in $D=8$, hyper-Kähler and Ricci-flat Kähler geometries, and connects these Calabi metrics to complex and quaternionic conifolds via hyper-Kähler quotient. It analyzes the holographic implications for AdS$_4$/CFT$_3$, including confinement mechanisms in the IR and the role of flux in breaking conformality, with discussions of D2-brane QFT$_3$ duals and related brane constructions. Overall, it provides explicit geometric tools for brane resolutions and their holographic interpretations, advancing both the geometry of Calabi metrics and their applications in M-theory and AdS/CFT$_3$.

Abstract

We obtain a simple explicit expression for the hyper-Kahler Calabi metric on the co-tangent 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-Kahler quotient. We then use the hyper-Kähler quotient to give a quaternionic rederivation of the Calabi metrics.