Toric Sasaki-Einstein Manifolds Research Articles

Bounding toric singularities with normalized volume

AbstractWe study the normalized volume of toric singularities. As it turns out, there is a close relation to the notion of (nonsymmetric) Mahler volume from convex geometry. This observation allows us to use standard tools from convex geometry, such as the Blaschke–Santaló inequality and Radon's theorem to prove nontrivial facts about the normalized volume in the toric setting. For example, we prove that for every there are only finitely many ‐Gorenstein toric singularities with normalized volume at least . From this result it directly follows that there are also only finitely many toric Sasaki–Einstein manifolds of volume at least in each dimension. Additionally, we show that the normalized volume of every toric singularity is bounded from above by that of the rational double point of the same dimension. Finally, we discuss certain bounds of the normalized volume in terms of topological invariants of resolutions of the singularity. We establish two upper bounds in terms of the Euler characteristic and of the first Chern class, respectively. We show that a lower bound, which was conjectured earlier by He, Seong, and Yau, is closely related to the nonsymmetric Mahler conjecture in convex geometry.

Black holes with baryonic charge and mathcal{I} -extremization

Recently it was discovered that twisted superconformal index mathcal{I} can be used to understand the Bekenstein-Hawking entropy of magnetically charged black holes in AdS spacetime. In this paper we apply the so-called mathcal{I} -extremization procedure to three-dimensional gauge field theories and their geometric dual, focusing in particular on the seven-dimensional Sasaki-Einstein manifold M1,1,1. We generalize recent studies on relations among toric geometry, variational principles, and black hole entropy to the case of AdS2× Y9, where Y9 is a fibration of toric Sasaki-Einstein manifold M1,1,1 over a two-dimensional Riemann surface Σg. The nine-dimensional variational problem is given in terms of an entropy functional. In order to illustrate the computations as explicitly as possible, we consider cases where either only mesonic or baryonic fluxes are turned on. By employing the operator counting method, we calculate the S3 free energy and the topologically twisted index mathcal{I} at large-N. The result for mathcal{I} , it turns out, can be also obtained from the variational principle of the entropy functional with mesonic fluxes. We also study asymptotically AdS4 black holes which are magnetically charged with respect to the vector field in the Betti multiplet. By extremizing the entropy functional with baryonic flux, we compute the entropy and find that it agrees with the entropy of an explicit solution in a four-dimensional gauged supergravity which is a consistent truncation of eleven-dimensional supergravity in AdS4× M1,1,1.

7D supersymmetric Yang-Mills on a 3-Sasakian manifold

In this paper we study 7D maximally supersymmetric Yang-Mills on a specific 3-Sasakian manifold that is the total space of an SO(3)-bundle over ℂP2. The novelty of this example is that the manifold is not a toric Sasaki-Einstein manifold. The hyperkähler cone of this manifold is a Swann bundle with hypertoric symmetry and this allows us to calculate the perturbative part of the partition function of the theory. The result is also verified by an index calculation. We also discuss a factorisation of this result and compare it with analogous results for S7.

7D supersymmetric Yang-Mills on curved manifolds

We study 7D maximally supersymmetric Yang-Mills theory on curved manifolds that admit Killing spinors. If the manifold admits at least two Killing spinors (Sasaki-Einstein manifolds) we are able to rewrite the supersymmetric theory in terms of a cohomological complex. In principle this cohomological complex makes sense for any K-contact manifold. For the case of toric Sasaki-Einstein manifolds we derive explicitly the perturbative part of the partition function and speculate about the non-perturbative part. We also briefly discuss the case of 3-Sasaki manifolds and suggest a plausible form for the full non-perturbative answer.

Mathcal{N}=2 supersymmetric gauge theory on connected sums of S2 \xd7 S2

We construct 4D mathcal{N}=2 theories on an infinite family of 4D toric manifolds with the topology of connected sums of S2× S2. These theories are constructed through the dimensional reduction along a non-trivial U(1)-fiber of 5D theories on toric Sasaki-Einstein manifolds. We discuss the conditions under which such reductions can be carried out and give a partial classification result of the resulting 4D manifolds. We calculate the partition functions of these 4D theories and they involve both instanton and anti-instanton contributions, thus generalizing Pestun’s famous result on S4.

Integrability of geodesics and action-angle variables in Sasaki–Einstein space $$T^{1,1}$$ T 1 , 1

We briefly describe the construction of Stäkel–Killing and Killing–Yano tensors on toric Sasaki–Einstein manifolds without working out intricate generalized Killing equations. The integrals of geodesic motions are expressed in terms of Killing vectors and Killing–Yano tensors of the homogeneous Sasaki–Einstein space $$T^{1,1}$$ . We discuss the integrability of geodesics and construct explicitly the action-angle variables. Two pairs of frequencies of the geodesic motions are resonant giving way to chaotic behavior when the system is perturbed.

Modular properties of full 5D SYM partition function

We study properties of the full partition function for the $U(1)$ 5D $\mathcal{N}=2^*$ gauge theory with adjoint hypermultiplet of mass $M$. This theory is ultimately related to abelian 6D (2,0) theory. We construct the full non-perturbative partition function on toric Sasaki-Einstein manifolds by gluing flat copies of the Nekrasov partition function and we express the full partition function in terms of the generalized double elliptic gamma function $G_2^C$ associated with a certain moment map cone $C$. The answer exhibits a curious $SL(4,\mathbb{Z})$ modular property. Finally, we propose a set of rules to construct the partition function that resembles the calculation of 5D supersymmetric partition function with the insertion of defects of various co-dimensions.

Complete integrability of geodesics in toric Sasaki-Einstein spaces

We describe a method for constructing Killing-Yano tensors on toric Sasaki- Einstein manifolds using their geometrical properties. We take advantage of the fact that the metric cones of these spaces are Calabi-Yau manifolds. The complete list of special Killing forms can be extracted making use of the description of the Calabi-Yau manifolds in terms of toric data. This general procedure for toric Sasaki-Einstein manifolds is exemplified in the case of the 5-dimensional spaces Yp,q and T1,1. Finally we discuss the integrability of geodesic motion in these spaces.

Toric data, Killing forms and complete integrability of geodesics in Sasaki–Einstein spaces [formula omitted

In the present paper we show that the complete list of special Killing forms on the 5-dimensional Sasaki–Einstein spaces Yp,q can be extracted using the symplectic potential and the classical Delzant construction. The results achieved here agree with previous ones obtained by direct computation, proving the reliability of the method which stands in fact as a general algorithm for toric Sasaki–Einstein manifolds. Finally, we discuss the integrability of geodesic motion in Yp,q spaces.

Hidden symmetries on toric Sasaki-Einstein spaces

We describe the construction of Killing-Yano tensors on toric Sasaki-Einstein manifolds. We use the fact that the metric cones of these spaces are Calabi-Yau manifolds. The description of the Calabi-Yau manifolds in terms of toric data, using the Delzant approach to toric geometries, allows us to find explicitly the complex coordinates and write down the Killing-Yano tensors. As a concrete example we present the complete set of special Killing forms on the five-dimensional homogeneous Sasaki-Einstein manifold T1,1.

Gluing Nekrasov Partition Functions

In this paper we summarise the localisation calculation of 5D super Yang-Mills on simply connected toric Sasaki-Einstein (SE) manifolds. We show how various aspects of the computation, including the equivariant index, the asymptotic behaviour and the factorisation property are governed by the combinatorial data of the toric geometry. We prove that the full perturbative partition function on a simply connected SE manifold corresponding to an n-gon toric diagram factorises to n copies of perturbative Nekrasov partition function. This leads us to conjecture the full partition function as gluing n copies of full Nekrasov partition function. This work is a generalisation of some earlier computation carried out on $Y^{p,q}$ manifolds, whose moment map cone has a quadrangle and our result is valid for manifolds whose moment map cones have pentagon base, hexagon base, etc. The algorithm we used for dealing with general cones may also be of independent interest.

Killing forms and toric Sasaki-Einstein spaces

The construction of the special Killing forms on toric Sasaki-Einstein manifolds is presented. This goal is achieved using the interplay between complex coordinates of the Calabi-Yau metric cone and the special Killing forms on the toric Sasaki-Einstein space. As a concrete example, we present the complete set of special Killing forms on the five-dimensional Einstein-Sasaki Yp,q spaces. It is pointed out the existence of two additional special Killing forms associated with the complex holomorphic volume form of Calabi-Yau cone manifold.

Special Killing forms on toric Sasaki–Einstein manifolds

In this paper we study the interplay between complex coordinates on the Calabi–Yau metric cone and the special Killing forms on the toric Sasaki–Einstein manifold. First, we give a procedure to locally construct the special Killing forms. Finally, we exemplify the general scheme in the case of the five-dimensional spaces, identifying the additional special Killing 2-forms which were previously obtained using a different method by Visinescu (2012 Mod. Phys. Lett. A 27 1250217).

5D Super Yang–Mills on Y p,q Sasaki–Einstein Manifolds

On any simply connected Sasaki-Einstein five dimensional manifold one can construct a super Yang-Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki-Einstein manifolds known as $Y^{p,q}$ manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large $N$ behaviour for the case of single hypermultiplet in adjoint representation and we derive the $N^3$-behaviour in this case.

Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds

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 Kähler-Ricci solitons (or Sasaki-Ricci 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 Sasaki-Einstein structure on S. As an application we obtain Sasaki-Einstein metrics on the $U(1)$-bundles associated with the canonical line bundles of toric Fano manifolds, including as a special case an irregular toric Sasaki-Einstein metrics on the unit circle bundle associated with the canonical bundle of the two-point blow-up of the complex projective plane.

Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories

We propose a new method to find gravity duals to a large class of three-dimensional Chern-Simons-matter theories, using techniques from dimer models. The gravity dual is given by M-theory on AdS_4\times Y_7, where Y_7 is an arbitrary seven-dimensional toric Sasaki-Einstein manifold. The cone of Y_7 is a toric Calabi-Yau 4-fold, which coincides with a branch of the vacuum moduli space of Chern-Simons-matter theories.

Free Yang-Mills theory versus toric Sasaki-Einstein manifolds

It has been known that the Bekenstein-Hawking entropy of the black hole in ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ agrees with the free $\mathcal{N}=4$ super Yang-Mills entropy up to the famous factor $\frac{4}{3}$. This factor can be interpreted as the ratio of the entropy of the free Yang-Mills theory to the entropy of the strongly coupled Yang-Mills theory. In this paper we compute an analogous factor for infinitely many $\mathcal{N}=1$ superconformal field theories (SCFTs) which are dual to toric Sasaki-Einstein manifolds. We observed that this ratio always takes values within a narrow range around $\frac{4}{3}$. We also present explicit values of volumes and central charges for new classes of toric Sasaki-Einstein manifolds.

New example of infinite family of quiver gauge theories

We construct a new infinite family of quiver gauge theories which blow down to the X p , q quiver gauge theories found by Hanany, Kazakopoulos and Wecht. This family includes a quiver gauge theory for the third del Pezzo surface. We show, using Z-minimization, that these theories generically have irrational R-charges. The AdS/CFT correspondence implies that the dual geometries are irregular toric Sasaki–Einstein manifolds, although we do not know the explicit metrics.

The Geometric Dual of a–Maximisation for Toric Sasaki–Einstein Manifolds

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 $$\mathbb {R}^{n}$$ 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 second del Pezzo surface.

Toric Sasaki–Einstein manifolds and Heun equations

Symplectic potentials are presented for a wide class of five-dimensional toric Sasaki–Einstein 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 one-to-one 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 R-charges of the BPS states in the dual quiver gauge theories.