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173
An infinite family of superconformal quiver gauge theories with SasakiEinstein duals
 JHEP
, 2005
"... Abstract: We describe an infinite family of quiver gauge theories that are AdS/CFT dual to a corresponding class of explicit horizon Sasaki–Einstein manifolds. The quivers may be obtained from a family of orbifold theories by a simple iterative procedure. A key aspect in their construction relies on ..."
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Cited by 158 (33 self)
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Abstract: We describe an infinite family of quiver gauge theories that are AdS/CFT dual to a corresponding class of explicit horizon Sasaki–Einstein manifolds. The quivers may be obtained from a family of orbifold theories by a simple iterative procedure. A key aspect in their construction relies on the global symmetry which is dual to the isometry of the manifolds. For an arbitrary such quiver we compute the exact R–charges of the fields in the IR by applying a–maximization. The values we obtain are generically quadratic irrational numbers and agree perfectly with the central charges and baryon charges computed from the family of metrics using the AdS/CFT correspondence. These results open the way for a systematic study of the quiver gauge theories and their dual geometries. Contents
Gauge theories from toric geometry and brane tilings
, 2005
"... We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi–Yau singularity. Our method combines information from the geometry and topology of Sasaki–Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quan ..."
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Cited by 147 (25 self)
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We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi–Yau singularity. Our method combines information from the geometry and topology of Sasaki–Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3–branes probing a toric Calabi–Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki–Einstein manifolds La,b,c is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform amaximisation as well as Zminimisation to compute the exact Rcharges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily La,b,a, whose smallest member is the
The geometric dual of amaximisation for toric SasakiEinstein manifolds
, 2005
"... We show that the Reeb vector, and hence in particular the volume, of a Sasaki– Einstein metric on the base of a toric Calabi–Yau cone of complex dimension n may be computed by minimising a function Z on R n which depends only on the toric data that defines the singularity. In this way one can extrac ..."
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Cited by 130 (15 self)
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We show that the Reeb vector, and hence in particular the volume, of a Sasaki– Einstein metric on the base of a toric Calabi–Yau cone of complex dimension n may be computed by minimising a function Z on R n which depends only on the toric data that defines the singularity. In this way one can extract certain geometric information for a toric Sasaki–Einstein manifold without finding the metric explicitly. For complex dimension n = 3 the Reeb vector and the volume correspond to the R–symmetry and the a central charge of the AdS/CFT dual superconformal field theory, respectively. We therefore interpret this extremal problem as the geometric dual of a–maximisation. We illustrate our results with some examples, including the Y p,q singularities and the complex cone over the
SasakiEinstein manifolds and volume minimisation
, 2006
"... We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian ..."
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Cited by 112 (7 self)
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We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone M, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat– Heckman formula and also to a limit of a certain equivariant index on M that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of any Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki
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 91 (10 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.
Transverse Kähler geometry of Sasaki manifolds and toric SasakiEinstein manifolds
, 2007
"... In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invarian ..."
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Cited by 85 (8 self)
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In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant f1 for the first Chern class case becomes an obstruction to the existence of transverse Kähler metric of constant scalar curvature. We prove the existence of transverse KählerRicci solitons (or SasakiRicci soliton) on compact toric Sasaki manifolds whose basic first Chern form of the normal bundle of the Reeb foliation is positive and the first Chern class of the contact bundle is trivial. We will further show that if S is a compact toric Sasaki manifold with the above assumption then by deforming the Reeb field we get a SasakiEinstein structure on S. As an application we obtain irregular toric SasakiEinstein metrics on the unit circle bundles of the powers of the canonical bundle of the twopoint blowup of the complex projective plane.
Supersymmetry breaking from a CalabiYau singularity
, 2005
"... We conjecture a geometric criterion for determining whether supersymmetry is spontaneously broken in certain string backgrounds. These backgrounds contain wrapped branes at CalabiYau singularites with obstructions to deformation of the complex structure. We motivate our conjecture with a particular ..."
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Cited by 70 (4 self)
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We conjecture a geometric criterion for determining whether supersymmetry is spontaneously broken in certain string backgrounds. These backgrounds contain wrapped branes at CalabiYau singularites with obstructions to deformation of the complex structure. We motivate our conjecture with a particular example: the Y 2,1 quiver gauge theory corresponding to a cone over the first del Pezzo surface, dP1. This setup can be analyzed using ordinary supersymmetric field theory methods, where we find that gaugino condensation drives a deformation of the chiral ring which has no solutions. We expect this breaking to be a general feature of any theory of branes at a singularity with a smaller number of possible deformations than independent anomalyfree fractional branes.
Obstructions to the Existence of SasakiEinstein Metrics
, 2008
"... We describe two simple obstructions to the existence of Ricci–flat Kähler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki–Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kähler–Einstein metric ..."
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Cited by 58 (12 self)
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We describe two simple obstructions to the existence of Ricci–flat Kähler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki–Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kähler–Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3–fold and 4–fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R–charge of a gauge invariant chiral primary operator