On Sasakian–einstein Geometry

Boyer, Charles P.; Galicki, Krzysztof

doi:10.1142/s0129167x00000477

Cited by 149 publications

(182 citation statements)

References 41 publications

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“…Recently Boyer, Galicki et al [18,19,20,21] have constructed many inhomogeneous Einstein-Sasaki (2n − 1) metrics on the links L f = C f ∩ S 2n+1 of weighted homogeneous polynomials f on C n+1 . The notation is as follows: C f ∈ C n+1 is the zero set f (z) = 0 of the polynomial, and S 2n+1 is the standard sphere.…”

Section: Introduction and Definitionmentioning

confidence: 99%

Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons

2003

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We study physical applications of the Bohm metrics, which are infinite sequences of inhomogeneous Einstein metrics on spheres and products of spheres of dimension 5 ≤ d ≤ 9.We prove that all the Bohm metrics on S 3 × S 2 and S 3 × S 3 have negative eigenvalue modes of the Lichnerowicz operator acting on transverse traceless symmetric tensors, and by numerical methods we establish that Bohm metrics on S 5 have negative eigenvalues too. General arguments suggest that all the Bohm metrics will have negative Lichnerowicz modes. These results imply that generalised higher-dimensional black-hole spacetimes, in which the Bohm metric replaces the usual round sphere metric, are classically unstable.We also show that the classical stability criterion for Freund-Rubin solutions, which are products of Einstein metrics with anti-de Sitter spacetimes, is the same in all dimensions as that for black-hole stability, and hence such solutions based on the Bohm metrics will also be unstable. We consider possible endpoints of the instabilities, and in particular we show that all Einstein-Sasaki manifolds give stable solutions. Next, we show how analytic continuation of Bohm metrics gives Lorentzian metrics that provide counterexamples to a strict form of the Cosmic Baldness conjecture, but they are nevertheless consistent with the intuition behind the cosmic No-Hair conjectures. We indicate how these Lorentzian metrics may be created "from nothing" in a no-boundary setting. We argue that Lorentzian Bohm metrics are unstable to decay to de Sitter spacetime. Finally, we argue that noncompact versions of the Bohm metrics have infinitely many negative Lichernowicz modes, and we conjecture a general relationship between Lichnerowicz eigenvalues and non-uniqueness of the Dirichlet problem for Einstein's equations.

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Section: Introduction and Definitionmentioning

confidence: 99%

Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons

2003

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“…A Sasakian manifold M is quasiregular if the 1-dimensional foliation F 1 induced by the Reeb field on M has compact fibers. Quasiregular Sasakian manifolds are always obtained from the construction described in Example 1.9 (see [BG00]). Therefore, to prove Theorem 1.11, we need to show that a given CR-manifold M admits a quasi-regular Sasakian structure, if it admits some Sasakian structure.…”

Section: Proofmentioning

confidence: 99%

Sasakian structures on CR-manifolds

Verbitsky

2007

Geom Dedicata

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A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of R >0 , with the symplectic form homogeneous of degree 2. If X is, in addition, Kähler, and its metric is also homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S 1 -bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding from M to an algebraic cone C. We show that this embedding is uniquely defined by the CRstructure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.

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“…15) that were classified in the years 1982-1985 [13,21,24] when Kaluza Klein supergravity was very topical. The Sasakian structure [30,31,32,33] of G/H reflects its holonomy and is the property that guarantees N = 2 supersymmetry both in the bulk AdS 4 and on the boundary M 3 . Kaluza Klein spectra for D = 11 supergravity compactified on the manifolds (1.15) have already been constructed [29] or are under construction [34] and, once the corresponding superconformal theory has been identified, it can provide a very important tool for comparison and discussion of the AdS/CFT correspondence.…”

Section: The Conceptual Environment and Our Goalsmentioning

confidence: 99%

Osp (calN|4) supermultiplets as conformal superfields on partial AdS ₄ and the generic form of calN = 2, d = 3 gauge theories

Fabbri¹,

Fré²,

Gualtieri³

et al. 1999

Class. Quantum Grav.

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In this paper we fill a necessary gap in order to realize the explicit comparison between the Kaluza Klein spectra of supergravity compactified on AdS 4 × X 7 and superconformal field theories living on the world volume of M2-branes. On the algebraic side we consider the superalgebra Osp(N |4) and we study the double intepretation of its unitary irreducible representations either as supermultiplets of particle states in the bulk or as conformal superfield on the boundary. On the lagrangian field theory side we construct, using rheonomy rather than superfield techniques, the generic form of an N = 2, d = 3 gauge theory. Indeed the superconformal multiplets are supposed to be composite operators in a suitable gauge theory.

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On Sasakian–einstein Geometry

Cited by 149 publications

References 41 publications

Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons

Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons

Sasakian structures on CR-manifolds

Osp (calN|4) supermultiplets as conformal superfields on partial AdS ₄ and the generic form of calN = 2, d = 3 gauge theories

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On Sasakian–einstein Geometry

Cited by 149 publications

References 41 publications

Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons

Bohm and Einstein-Sasaki metrics, black holes, and cosmological event horizons

Sasakian structures on CR-manifolds

Osp (calN|4) supermultiplets as conformal superfields on partial AdS 4 and the generic form of calN = 2, d = 3 gauge theories

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Osp (calN|4) supermultiplets as conformal superfields on partial AdS ₄ and the generic form of calN = 2, d = 3 gauge theories