Toric Geometry, Sasaki–Einstein Manifolds and a New Infinite Class of AdS/CFT Duals

By D. Martelli, J. Sparks. 2004

378 citations recorded by the source corpus.

Recently an infinite family of explicit Sasaki–Einstein metrics Yp,q on S2×S3 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 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 for all q<p with fixed p. We hence find that the Yp,q manifolds are AdS/CFT dual to an infinite class of superconformal field theories arising as IR fixed points of toric quiver gauge theories with gauge group SU(N)2p. As a non–trivial example, we show that Y2,1 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 constructed for this case and hence we can predict the exact central charge of this theory at its IR fixed point using the AdS/CFT correspondence. The value we obtain is a quadratic irrational number and, remarkably, agrees with a recent purely field theoretic calculation using a-maximisation.

Subfields: Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/f541666922a8993b50b4dbec9dbc527315380657

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Citations in the corpus

Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities (2009, methodology): “Martelli and Sparks [26] constructed explicitly a Sasaki-Einstein metric on the sphere bundle of the canonical line bundle of the blown up CP 2 at one point which is a irregular Sasaki manifold.”
Fractional branes and dynamical supersymmetry breaking (2005, background): “The Y p,q family Recently an infinite class of Sasaki-Einstein 5d metrics on S(2) × S(3) has been constructed, denoted Y , which can be used to build an infinite class of 6d conical Calabi-Yau geometries [45, 46, 47, 48, 49].”
New checks and subtleties for AdS/CFT and a-maximization (2004, methodology): “perfect agreement with the results of [ 6 ], Eqs.”
On Eta-Einstein Sasakian Geometry (2004, background): “In dimension 5 the new examples produce infinitely many toric Sasakian-Einstein metric on S 2 × S 3 . The homogeneous Sasakian-Einstein metric on S 2 × S 3 examined in the context of the AdS/CFT duality by Klebanov and Witten [KW99] emerges as a special case of this cons truc tion [MS05, MSY05].”
Toric Sasaki-Einstein metrics on S2 × S3 (2004, methodology): “In particular, following [ 3 ], we first introduce complex coordinates on the Calabi–Yau cone.”
The Geometric Dual of a–Maximisation for Toric Sasaki–Einstein Manifolds (2005, methodology): “Remarkable agreement was found for these two computations in the case of the Y 2,1 metric [9], and the a–maximisation calculation [21] for the quiver gauge theory corresponding to the first del Pezzo surface [18].”
The Dual superconformal theory for L**pqr manifolds (2005, methodology): “Now notice that the four vectors orthogonal to the faces of the polyhedral cone are just the killing vectors that vanish on such faces [ 23 ]; the coordinates of such vectors in the basis ei are just the integers vectors in R3 that describe the toric diagram”
From Sasaki-Einstein spaces to quivers via BPS geodesics: Lp,q|r (2005, neutral): “In [14] the toric description was found.”
Sasaki–Einstein Manifolds and Volume Minimisation (2006, methodology): “However, the canonical circle bundle over this del Pezzo surface does admit a Sasaki–Einstein metric [10,23] – the Kähler–Einstein metric is only a transverse metric13.”
Brane tilings and their applications (2008, background): “Some reviews on this topic (mainly from Sasaki-Einstein side) include [149], which summarizes early developments and mostly concentrates on Y .”
Counting gauge invariants: The Plethystic program (2007, neutral): “e.g. [7, 8, 39 , 40, 41]) was studied in detail in [4].”
Dirichlet Branes and Mirror Symmetry (2009, background): “For example, a large family of new explicit Sasaki-Einstein metrics has been obtained and studied in this way [350, 351].”
Counting BPS operators in gauge theories: quivers, syzygies and plethystics (2006, neutral): “,[48, 49, 50]).”
Brane dimers and quiver gauge theories (2005, background): “Gauntlett, Martelli, Sparks, and Waldram [21] found an infinite class of Sasaki-Einstein (SE) metrics; previous to their work, only two explicit SE metrics were known.”
Gauge theories from toric geometry and brane tilings (2005, background): “p + q, c = p these reduce to the Y p,q family of metrics, which have an enhanced SU(2) isometry [10, 11, 12].”
Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds (2006, background): “In particular there exists an irregular toric Sasaki-Einstein metric on the circle bundle associated with the powers of the canonical line bundle of the two-point blow-up of the complex projective plane.”