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27
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
Fractional Branes and Dynamical Supersymmetry Breaking
, 2005
"... We study the dynamics of fractional branes at toric singularities, including cones over del Pezzo surfaces and the recently constructed Y p,q theories. We find that generically the field theories on such fractional branes show dynamical supersymmetry breaking, due to the appearance of nonperturba ..."
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Cited by 43 (6 self)
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We study the dynamics of fractional branes at toric singularities, including cones over del Pezzo surfaces and the recently constructed Y p,q theories. We find that generically the field theories on such fractional branes show dynamical supersymmetry breaking, due to the appearance of nonperturbative superpotentials. In special cases, one recovers the known cases of supersymmetric infrared behaviors, associated to SYM confinement (mapped to complex deformations of the dual geometries, in the gauge/string correspondence sense) or N = 2 fractional branes. In the supersymmetry breaking cases, when the dynamics of closed string moduli at the singularity is included, the theories show a runaway behavior (involving moduli such as FI terms or equivalently dibaryonic operators), rather than stable nonsupersymmetric minima. We comment on the implications of this gauge theory behavior for the infrared smoothing of the dual warped throat solutions with 3form fluxes, describing duality cascades ending in such field theories. We finally provide a description of the different fractional branes in the recently introduced brane tiling configurations.
Toric SasakiEinstein manifolds and Heun Equations
, 2006
"... Symplectic potentials are presented for a wide class of five dimensional toric SasakiEinstein manifolds, including L a,b,c which was recently constructed by Cvetič et al. The spectrum of the scalar Laplacian on L a,b,c is also studied. The eigenvalue problem leads to two Heun’s differential equatio ..."
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Cited by 21 (0 self)
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Symplectic potentials are presented for a wide class of five dimensional toric SasakiEinstein manifolds, including L a,b,c which was recently constructed by Cvetič et al. The spectrum of the scalar Laplacian on L a,b,c is also studied. The eigenvalue problem leads to two Heun’s differential equations and the exponents at regular singularities are directly related to the toric data. By combining knowledge of the explicit symplectic potential and the exponents, we show that the ground states, or equivalently holomorphic functions, have onetoone correspondence with the integral lattice points in the convex polyhedral cone. The scaling dimensions of the holomorphic functions are simply given by the scalar products of the Reeb vector and the integral vectors, which are consistent with Rcharges of the BPS states in the dual quiver gauge theories.
UNIQUENESS AND EXAMPLES OF COMPACT TORIC SASAKIEINSTEIN METRICS
, 2007
"... In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class, one can find a deformed Sasaki structure on which a SasakiEinstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds mod ..."
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Cited by 15 (3 self)
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In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class, one can find a deformed Sasaki structure on which a SasakiEinstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric SasakiEinstein metrics on S 5 ♯k(S 2 × S 3) for each positive integer k.
Deformations of toric singularities and fractional branes
 JHEP 0610 (2006) 080 [arXiv:hepth/0603253
"... Fractional branes added to a large stack of D3branes at the singularity of a CalabiYau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to ..."
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Cited by 12 (1 self)
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Fractional branes added to a large stack of D3branes at the singularity of a CalabiYau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann’s rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional branes can be classified by looking at suitable weights associated with the external legs of the (p,q) web. We check the proposal on many examples and match in some interesting cases the moduli space of the gauge theory with the deformed geometry.
On Classification of Brane Tilings
, 2009
"... We present a computationally efficient algorithm that can be used to generate all possible brane tilings. Brane tilings represent the largest class of superconformal theories with known AdS duals in 3+1 and also 2+1 dimensions and have proved useful for describing the physics of both D3 branes and ..."
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Cited by 8 (4 self)
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We present a computationally efficient algorithm that can be used to generate all possible brane tilings. Brane tilings represent the largest class of superconformal theories with known AdS duals in 3+1 and also 2+1 dimensions and have proved useful for describing the physics of both D3 branes and also M2 branes probing CalabiYau singularities. This algorithm has been implemented and is used to generate all possible brane tilings with at most 6 superpotential terms, including consistent and inconsistent brane tilings. The collection of inconsistent tilings found in this work form the most
Mutationperiodic quivers, integrable maps and associated Poisson algebras
 Phil. Trans. R. Soc. A
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