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32
Toric Geometry, Sasaki–Einstein Manifolds and a new Infinite Class of AdS/CFT duals
"... Recently an infinite family of explicit Sasaki–Einstein metrics Y p,q on S 2 × S 3 has been discovered, where p and q are two coprime positive integers, with q < p. These give rise to a corresponding family of Calabi–Yau cones, which moreover are toric. Aided by several recent results in toric geome ..."
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Cited by 114 (13 self)
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Recently an infinite family of explicit Sasaki–Einstein metrics Y p,q on S 2 × S 3 has been discovered, where p and q are two coprime positive integers, with q < p. These give rise to a corresponding family of Calabi–Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kähler quotients C 4 //U(1), namely the vacua of gauged linear sigma models with charges (p,p, −p + q, −p − q), thereby generalising the conifold, which is p = 1,q = 0. We present the corresponding toric diagrams and show that these may be embedded in the toric diagram for the orbifold C 3 /Zp+1 × Zp+1 for all q < p with fixed p. We hence find that the Y p,q manifolds are AdS/CFT dual to an infinite class of N = 1 superconformal field theories arising as infra–red (IR) fixed points of toric quiver gauge theories with gauge group SU(N) 2p. As a non–trivial example, we show that Y 2,1 is an explicit irregular Sasaki–Einstein metric on the horizon of the complex cone over the first del Pezzo surface. The dual quiver gauge theory has already been
Parallel spinors and connections with skewsymmetric torsion in string theory
, 2008
"... We describe all almost contact metric, almost hermitian and G2structures admitting a connection with totally skewsymmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇parallel spinors. In p ..."
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Cited by 89 (5 self)
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We describe all almost contact metric, almost hermitian and G2structures admitting a connection with totally skewsymmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5, 6 and 7.
On EtaEinstein Sasakian Geometry
 Comm. Math. Phys
"... The purpose of this paper is to study a special kind of Riemannian metrics on Sasakian manifolds. A Sasakian manifold M of dimension 2n + 1 with a Sasakian structure S = (ξ, η,Φ, g) is said to be ηEinstein if the Ricci curvature tensor of the metric g satisfies the equation Ricg = λg + νη ⊗ η for s ..."
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Cited by 33 (9 self)
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The purpose of this paper is to study a special kind of Riemannian metrics on Sasakian manifolds. A Sasakian manifold M of dimension 2n + 1 with a Sasakian structure S = (ξ, η,Φ, g) is said to be ηEinstein if the Ricci curvature tensor of the metric g satisfies the equation Ricg = λg + νη ⊗ η for some constants λ, ν ∈ R.
New Einstein Metrics in Dimension Five
, 2001
"... The purpose of this note is to prove the existence of new SasakianEinstein met rics on S 2 x S a and on (S 2 x Sa)#(S = x Sa). These give the first known examples of nonregular SasakianEinstein 5manifolds. Our method involves describing the SasakianEinstein structures as links of certain is ..."
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Cited by 20 (10 self)
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The purpose of this note is to prove the existence of new SasakianEinstein met rics on S 2 x S a and on (S 2 x Sa)#(S = x Sa). These give the first known examples of nonregular SasakianEinstein 5manifolds. Our method involves describing the SasakianEinstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Kolltr who obtained new examples of KiihlerEinstein del Pezzo surfaces with quotient singularities.
Einstein metrics on spheres
 Ann. of Math
, 2005
"... Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3, m> 1 are known to have another Sp(m + 1)homogeneous Einstein metric discovered by Jensen [Jen73 ..."
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Cited by 18 (13 self)
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Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3, m> 1 are known to have another Sp(m + 1)homogeneous Einstein metric discovered by Jensen [Jen73]. In addition,
Cohomogeneity one EinsteinSasaki 5manifolds
, 2006
"... Abstract. We consider hypersurfaces in EinsteinSasaki 5manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5dimensional metric from such a hypersurface, analogous to the (nearly) hypo and halfflat evolution equation ..."
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Cited by 11 (1 self)
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Abstract. We consider hypersurfaces in EinsteinSasaki 5manifolds which are tangent to the characteristic vector field. We introduce evolution equations that can be used to reconstruct the 5dimensional metric from such a hypersurface, analogous to the (nearly) hypo and halfflat evolution equations in higher dimensions. We use these equations to classify EinsteinSasaki 5manifolds of cohomogeneity one. From a Riemannian point of view, an EinsteinSasaki manifold is a Riemannian manifold (M, g) such that the conical metric on M × R + is Kähler and Ricciflat. In particular, this implies that (M, g) is odddimensional, contact and Einstein with positive scalar curvature. The EinsteinSasaki manifolds that are simplest to describe are the regular ones, which arise as circle bundles over KählerEinstein manifolds. In five dimensions, there is a classification of regular EinsteinSasaki manifolds [11], in which precisely two homogeneous examples appear, namely the sphere S 5 and the Stiefel manifold (1) V2,4 = SO(4)/SO(2) ∼ = S 2 × S 3.
Sasakian geometry, homotopy spheres and positive Ricci curvature
, 2002
"... We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such hom ..."
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Cited by 10 (4 self)
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We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres Σ 2n+1 the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the 2 2m distinct diffeomorphism types of homotopy real projective spaces RP 4m+1.
Dibaryon spectroscopy
 JHEP
, 2003
"... The AdS/CFT correspondence relates dibaryons in superconformal gauge theories to holomorphic curves in KählerEinstein surfaces. The degree of the holomorphic curves is proportional to the gauge theory conformal dimension of the dibaryons. Moreover, the number of holomorphic curves should match, in ..."
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Cited by 10 (1 self)
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The AdS/CFT correspondence relates dibaryons in superconformal gauge theories to holomorphic curves in KählerEinstein surfaces. The degree of the holomorphic curves is proportional to the gauge theory conformal dimension of the dibaryons. Moreover, the number of holomorphic curves should match, in an appropriately defined sense, the number of dibaryons. Using AdS/CFT backgrounds built from the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999), we show that the gauge theory prediction for the dimension of dibaryonic operators does indeed match the degree of the corresponding holomorphic curves. For AdS/CFT backgrounds built from cones over del Pezzo surfaces, we are able to match the degree of the curves to the conformal dimension of dibaryons for the nth del Pezzo surface, 1 ≤ n ≤ 6. Also, for the del Pezzos and the Ak type generalized conifolds, for the dibaryons of smallest conformal dimension, we are able to match the number of holomorphic curves with the number of possible dibaryon operators from gauge theory. AdS/CFT correspondence [1, 2, 3] asserts that type IIB string theory on AdS5 × X is
On Positive Sasakian Geometry
 Comm. Math. Phys
"... : A Sasakian structure S=(;j;\Phi;g) on a manifold M is called positive if its basic first Chern class c1 (F ) can be represented by a positive (1;1)form with respect to its transverse holomorphic CRstructure. We prove a theorem that says that every positive Sasakian structure can be deformed to ..."
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Cited by 9 (6 self)
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: A Sasakian structure S=(;j;\Phi;g) on a manifold M is called positive if its basic first Chern class c1 (F ) can be represented by a positive (1;1)form with respect to its transverse holomorphic CRstructure. We prove a theorem that says that every positive Sasakian structure can be deformed to a Sasakian structure whose metric has positive Ricci curvature. This allows us by example to give a completely independent proof of a result of Sha and Yang [SY] that for every nonnegative integer k the 5manifolds k#(S 2 \ThetaS 3 ) admits metrics of positive Ricci curvature.
FANO VARIETIES WITH MANY SELFMAPS
, 2006
"... Abstract. We study log canonical thresholds of effective divisors on weighted threedimensional Fano hypersurfaces to construct examples of Fano varieties of dimension six and higher having infinite, explicitly described, discrete groups of birational selfmaps. 1. Introduction. Birational automorphis ..."
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Cited by 8 (0 self)
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Abstract. We study log canonical thresholds of effective divisors on weighted threedimensional Fano hypersurfaces to construct examples of Fano varieties of dimension six and higher having infinite, explicitly described, discrete groups of birational selfmaps. 1. Introduction. Birational automorphisms of higherdimensional Fano varieties play an important role in algebraic geometry. For instance, the following result is proved in [7]. Theorem 1.1. Let X be a smooth quartic threefold in P 4. Then Bir(X) = Aut(X). It follows from Theorem 1.1 that every smooth quartic threefold is nonrational, because