James Sparks

Gauge theories from toric geometry and brane tilings

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 a-maximisation as well as Z-minimisation to compute the exact R-charges 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 Suspended Pinch Point.

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

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.

An infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals.

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.

Toric Sasaki–Einstein metrics on S2×S3

Abstract We show that by taking a certain scaling limit of a Euclideanised form of the Plebanski–Demianski metrics one obtains a family of local toric Kahler–Einstein metrics. These can be used to construct local Sasaki–Einstein metrics in five dimensions which are generalisations of the Y p , q manifolds. In fact, we find that these metrics are diffeomorphic to those recently found by Cvetic, Lu, Page and Pope. We argue that the corresponding family of smooth Sasaki–Einstein manifolds all have topology S 2 × S 3 . We conclude by setting up the equations describing the warped version of the Calabi–Yau cones, supporting ( 2 , 1 ) three-form flux.

Supersymmetric AdS5 solutions of type IIB supergravity

We analyse the most general bosonic supersymmetric solutions of type IIB supergravity whose metrics are warped products of five-dimensional anti-de Sitter space (AdS5) with a five-dimensional Riemannian manifold M5. All fluxes are allowed to be non-vanishing consistent with SO(4,2) symmetry. We show that the necessary and sufficient conditions can be phrased in terms of a local identity structure on M5. For a special class, with constant dilaton and vanishing axion, we reduce the problem to solving a second order non-linear ODE. We find an exact solution of the ODE which reproduces a solution first found by Pilch and Warner. A numerical analysis of the ODE reveals an additional class of local solutions.

Moduli spaces of Chern-Simons quiver gauge theories and AdS(4)/CFT(3)

We analyze the classical moduli spaces of supersymmetric vacua of 3D N=2 Chern-Simons quiver gauge theories. We show quite generally that the moduli space of the 3D theory always contains a baryonic branch of a parent 4D N=1 quiver gauge theory, where the 4D baryonic branch is determined by the vector of 3D Chern-Simons levels. In particular, starting with a 4D quiver theory dual to a 3-fold singularity, for certain general choices of Chern-Simons levels this branch of the moduli space of the corresponding 3D theory is a 4-fold singularity. Our results lead to a simple general method, using existing 4D techniques, for constructing candidate 3D N=2 superconformal Chern-Simons quivers with AdS{sub 4} gravity duals. As simple, but nontrivial, examples, we identify a family of Chern-Simons quiver gauge theories which are candidate AdS{sub 4}/CFT{sub 3} duals to an infinite class of toric Sasaki-Einstein seven-manifolds with explicit metrics.

Obstructions to the existence of Sasaki-Einstein metrics

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 violates the unitarity bound.

The large N limit of quiver matrix models and Sasaki-Einstein manifolds

We study the matrix models that result from localization of the partition functions of N=2 Chern-Simons-matter theories on the three-sphere. A large class of such theories are conjectured to be holographically dual to M-theory on Sasaki-Einstein seven-manifolds. We study the M-theory limit (large N and fixed Chern-Simons levels) of these matrix models for various examples, and show that in this limit the free energy reproduces the expected AdS/CFT result of N^{3/2}/Vol(Y)^{1/2}, where Vol(Y) is the volume of the corresponding Sasaki-Einstein metric. More generally we conjecture a relation between the large N limit of the partition function, interpreted as a function of trial R-charges, and the volumes of Sasakian metrics on links of Calabi-Yau four-fold singularities. We verify this conjecture for a family of U(N)^2 Chern-Simons quivers based on M2 branes at hypersurface singularities, and for a U(N)^3 theory based on M2 branes at a toric singularity.

The gravity dual of supersymmetric gauge theories on a squashed three-sphere

We present the gravity dual to a class of three-dimensional N = 2 supersymmetric gauge theories on a U(1) × U(1)-invariant squashed three-sphere, with a non-trivial background gauge field. This is described by a supersymmetric solution of four-dimensional N = 2 gauged supergravity with a non-trivial instanton for the graviphoton field. The particular gauge theory in turn determines the lift to a solution of eleven-dimensional supergravity. We compute the partition function for a class of Chern-Simons quiver gauge theories on both sides of the duality, in the large N limit, finding precise agreement for the functional dependence on the squashing parameter. This constitutes an exact check of the gauge/gravity correspondence in a non-conformally invariant setting.

Localization on Three-Manifolds

A bstractWe consider supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere. The three-manifold is always equipped with a contact structure and an associated Reeb vector field. We show that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function. In the large N limit our formula agrees with a recently discovered two-parameter family of dual supergravity solutions. We also explain how our results may be applied to prove vortex-antivortex factorization. Finally, we comment on the extension of our results to three-manifolds with non-trivial fundamental group.