We construct an explicit diffeomorphism between the Sasaki-Einstein spaces Y^{p,q} and the product space S^3 \times S^2 in the cases q \le 2. When q=1 we express the Kähler quotient coordinates as an SU(2) bundle over S^2 which we trivialize. When q=2 the quotient coordinates yield a non-trivial SO(3) bundle over S^2 with characteristic class p, which is rotated to a bundle with characteristic class 1 and re-expressed as Y^{2,1}, reducing the problem to the case q=1. When q>2 the fiber is a lens space which is not a Lie group, and this construction fails. We relate the S^2 \times S^3 coordinates to those for which the Sasaki-Einstein metric is known. We check that the RR flux on the S^3 is normalized in accordance with Gauss' law and use this normalization to determine the homology classes represented by the calibrated cycles. As a by-product of our discussion we find a diffeomorphism between T^{p,q} and Y^{p,q} spaces, which means that T^{p,q} manifolds are also topologically S^3 \times S^2.
We analyze the transverse Kähler-Ricci flow equation on Sasaki-Ein\-stein space $Y^{p,q}$. Explicit solutions are produced representing new five-dimensional Sasaki structures. Solutions which do not modify the transverse metric preserve the Sasaki-Einstein feature of the contact structure. If the transverse metric is altered, the deformed metrics remain Sasaki, but not Einstein.
We find point-like and classical string solutions on the AdS_5 x X^5, where X^5 are the 5-dimensional Sasaki-Einstein manifolds Ypq and Lpqr. The number of acceptable solutions is limited drastically in order to satisfy the constraints on the parameters and coordinates of the manifolds. The energy of the solutions depends on the parameters of the Sasaki-Einstein manifolds and on the conserved momenta transcendentally. A discussion on BPS solutions is presented as well.
Holographic spectral functions for Sasaki-Einstein 5-manifolds
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；School of Physics and Shing-Tung Yau Center, Southeast University（东南大学物理学院及施一公中心）
The superconformal index of the quiver gauge theory dual to type IIB string theory on the product of an arbitrary smooth Sasaki-Einstein manifold with five-dimensional AdS space is calculated both from the gauge theory and gravity viewpoints. We find complete agreement. Along the way, we find that the index on the gravity side can be expressed in terms of the Kohn-Rossi cohomology of the Sasaki-Einstein manifold and that the index of a quiver gauge theory equals the Euler characteristic of the cyclic homology of the Ginzburg dg algebra associated to the quiver.
Methods of Hamiltonian dynamics are applied to study the geodesic flow on the resolved conifolds over Sasaki-Einstein space $T^{1,1}$. We construct explicitly the constants of motion and prove complete integrability of geodesics in the five-dimensional Sasaki-Einstein space $T^{1,1}$ and its Calabi-Yau metric cone. The singularity at the apex of the metric cone can be smoothed out in two different ways. Using the small resolution the geodesic motion on the resolved conifold remains completely integrable. Instead, in the case of the deformation of the conifold the complete integrability is lost…
We use the action-angle variables to describe the geodesic motions in the $5$-dimensional Sasaki-Einstein spaces $Y^{p,q}$. This formulation allows us to study thoroughly the complete integrability of the system. We find that the Hamiltonian involves a reduced number of action variables. Therefore one of the fundamental frequency is zero indicating a chaotic behavior when the system is perturbed.
Given a Sasaki-Einstein manifold, M_7, there is the N=2 supersymmetric AdS_4 x M_7 Freund-Rubin solution of eleven-dimensional supergravity and the corresponding non-supersymmetric solutions: the perturbatively stable skew-whiffed solution, the perturbatively unstable Englert solution, and the Pope-Warner solution, which is known to be perturbatively unstable when M_7 is the seven-sphere or, more generally, a tri-Sasakian manifold. We show that similar perturbative instability of the Pope-Warner solution will arise for any Sasaki-Einstein manifold, M_7, admitting a basic, primitive, transverse…
Studying nonsupersymmetric yet imaginary self-dual three-form fluxes in type IIB supergravity backgrounds on Sasaki-Einstein manifolds we find a new analytic solution that restores supersymmetry in the IR, breaks it at higher energies, yet suffers from curvature singularities in the UV, when a certain SUSY-breaking parameter becomes large. Consequently we investigate a variety of possibilities to cure the singularity by either introducing additional sources, changing the fluxes or deforming the geometry. Since it is not possible to cure the singularities making physically reasonably assumption…
We study generalised calibrations described by Exceptional Sasaki-Einstein structures on AdS backgrounds in type IIB and M-theory, placing the focus on the generalised Reeb vectors. The inequalities for the energy bound are derived by decomposing a $κ$-symmetry condition and equivalently, bispinors in calibration conditions from existing literature. We explain how the closure of the calibration forms is related to the integrability conditions of the Exceptional Sasaki-Einstein structure, in particular for AdS space-filling or point-like branes. Doing so, we show that the form parts of the twis…
We present a consistent truncation of IIB supergravity on manifolds admitting a Sasaki-Einstein structure, which keeps the metric and five real scalar fields. This theory can be further truncated to a constrained one-parameter family that depends on only the metric and one scalar, as well as to a theory with a metric and three scalars. The reduced theory admits supersymmetric and non-supersymmetric AdS_5 and AdS_4 x R solutions. We analyze the spectrum around the AdS critical points and identify the dual operators.
We revisit non-rotating, dyonically charged, and supersymmetric AdS$_4$ black holes, which are solutions of $\mathcal{N}=2$ gauged supergravity with vector- and hyper-multplets. Uplifting the near horizon solutions to $D=11$ supergravity on seven-dimensional Sasaki-Einstein manifolds, we show that dyonic AdS$_4$ black holes correspond to AdS$_2$ solutions with electric and magnetic baryonic fluxes in $D=11$ supergravity. We identify the off-shell AdS$_4$ black hole solutions with parameters of $D=11$ AdS$_2$ solutions without imposing the equations of motion. We calculate the entropy of dyonic…
We study super Yang-Mills theories on five-dimensional Sasaki-Einstein manifolds. Using localisation techniques, we find that the contribution from the vector multiplet to the perturbative partition function can be calculated by counting holomorphic functions on the associated Calabi-Yau cone. This observation allows us to use standard techniques developed in the context of quiver gauge theories to obtain explicit results for a number of examples; namely $S^5$, $T^{1,1}$, $Y^{7,3}$, $Y^{2,1}$, $Y^{2,0}$, and $Y^{4,0}$. We find complete agreement with previous results obtained by Qiu and Zabzin…
This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
We decompose the de Rham Laplacian on Sasaki-Einstein manifolds as a sum over mostly positive definite terms. An immediate consequence are lower bounds on its spectrum. These bounds constitute a supergravity equivalent of the unitarity bounds in dual superconformal field theories. The proof uses a generalization of Kahler identities to the Sasaki-Einstein case.
We provide a consistent N=4 Kaluza-Klein truncation of type IIB supergravity on general 5-dimensional squashed Sasaki-Einstein manifolds. Our reduction ansatz keeps all and only the supergravity modes dual to the universal gauge sector of the associated conformal theories, via the gauge/gravity correspondence. The reduced 5-dimensional model displays remarkable features: it includes both zero-modes as well as massive iterations of the Kaluza-Klein operators on the internal manifold; it contains tensor fields dual to vectors charged under a non-abelian gauge group; it has a scalar potential wit…
Motivated by recent interest in applications of the AdS/CFT correspondence to condensed matter applications involving fermions, we present the supersymmetric completion of the recent massive truncations of IIB supergravity on Sasaki-Einstein manifolds. In particular, we reduce the fermionic sector of IIB supergravity to obtain five dimensional N=2 supergravity coupled to one hypermultiplet and one massive vector multiplet. The supersymmetry transformations and equations of motion are presented and analyzed. Finally, a particularly interesting truncation to N=2 supergravity coupled to a single …
We present an extensive analysis on several string solutions in AdS_5 x Ypq and find some interesting properties of their energy-spin relations. Their energy depends always on the parameter a(p,q) which characterizes these manifolds. The range of this parameter for the string solutions is constrained by the Sasaki-Einstein constraints that the solutions should satisfy. Hence some string solutions we find are not valid for the whole class of Ypq manifolds. For some of our solutions, when the maximum allowed value of a(p,q) corresponds to the string approaching the poles of the squashed sphere i…
The most general class of warped AdS5 x M6 supersymmetric solutions in D=11 supergravity permit a consistent truncation to D=5 N=2 minimal gauged supergravity. Here we extend this truncation, for a particular subclass of M6 manifolds, to D=5 N=4 gauged supergravity coupled to two vector multiplets. We obtain the reduction ansatz by T-duality of a recently discussed type IIB truncation on a generic Sasaki-Einstein five-fold, which becomes non-trivial in D=11 and displays non-standard features due to the G-structure of the internal M6. Using this truncation, we construct two new classes of warpe…
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 betw…
We discuss the dimensional reduction of fermionic modes in a recently found class of consistent truncations of D=11 supergravity compactified on squashed seven-dimensional Sasaki-Einstein manifolds. Such reductions are of interest, for example, in that they have $(2+1)$-dimensional holographic duals, and the fermionic content and their interactions with charged scalars are an important aspect of their applications. We derive the lower-dimensional equations of motion for the fermions and exhibit their couplings to the various bosonic modes present in the truncations under consideration, which m…
Recent work on holographic superconductivity and gravitational duals of systems with non-relativistic conformal symmetry have made use of consistent truncations of D=10 and D=11 supergravity retaining some massive modes in the Kaluza-Klein tower. In this paper we focus on reductions of IIB supergravity to five dimensions on a Sasaki-Einstein manifold, and extend these previous truncations to encompass the entire bosonic sector of gauged D=5, N=2 supergravity coupled to massive multiplets up to the second Kaluza-Klein level. We conjecture that a necessary condition for the consistency of massiv…
We show that for every Sasaki-Einstein manifold, $M_5$, the AdS$_5\times M_5$ background of type IIB supergravity admits two universal deformations leading to supersymmetric AdS$_4$ solutions. One class of solutions describes an AdS$_4$ domain wall in AdS$_5$ and is dual to a Janus configuration with $\mathcal{N}=1$ supersymmetry. The other class of backgrounds is of the form AdS$_4\times S^1\times M_5$ with a non-trivial SL$(2,\mathbb{Z})$ monodromy for the IIB axio-dilaton along the $S^1$. These AdS$_4$ solutions are dual to three-dimensional $\mathcal{N}=1$ SCFTs. Using holography we expres…
In this note we give an explicit construction of Sasaki-Einstein metrics on a class of simply connected 7-manifolds with the rational cohomology of the 2-fold connected sum of $S^2\times S^5$. The homotopy types are distinguished by torsion in $H^4$.
Seven-dimensional inhomogeneous Sasaki-Einstein manifolds $Y^{p,k}(KE_4)$ present a challenging example of AdS/CFT correspondence. At present, their field theory duals for $KE_4=\mathbb{CP}^2$ base are proposed only within a restricted range $3p/2\le k \le 2p$ as ${\cal N}=2$ quiver Chern-Simons-matter theories with $SU(N)\times SU(N)\times SU(N)$ gauge group, nine bifundamental chiral multiplets interacting through a cubic superpotential. To further elucidate this correspondence, we use particle approximation both at classical and quantum level. We setup a concrete AdS/CFT mapping of conserve…
In this paper, we calculate the topological free energy for a number of ${\mathcal N} \geq 2$ Yang-Mills-Chern-Simons-matter theories at large $N$ and fixed Chern-Simons levels. The topological free energy is defined as the logarithm of the partition function of the theory on $S^2 \times S^1$ with a topological A-twist along $S^2$ and can be reduced to a matrix integral by exploiting the localization technique. The theories of our interest are dual to a variety of Calabi-Yau four-fold singularities, including a product of two asymptotically locally Euclidean singularities and the cone over var…
We construct exact static inhomogeneous solutions to Einstein's equations with counter flow of particle fluid and a positive cosmological constant by using the Sasaki metrics on three-dimensional spaces. The solutions, which admit an arbitrary function that denotes inhomogeneous number density of particles, are a generalization of Einstein's static universe. On some examples of explicit solutions, we discuss non-linear density contrast and deviation of the metric functions.
Localization methods reduce the path integrals in {\cal N} >= 2 supersymmetric Chern-Simons gauge theories on S^3 to multi-matrix integrals. A recent evaluation of such a two-matrix integral for the {\cal N}=6 superconformal U(N) x U(N) ABJM theory produced detailed agreement with the AdS/CFT correspondence, explaining, in particular the N^{3/2} scaling of the free energy. We study a class of p-matrix integrals describing {\cal N}=3 superconformal U(N)^p Chern-Simons gauge theories. We present a simple method that allows us to evaluate the eigenvalue densities and the free energies in the larg…
我们表明，通过采取一定的标度限制的欧几里德形式的Plebanski-Demianski度量得到一个家庭的本地环面Kahler-Einstein度量。这些可以用来构造5维的局部Sasaki-Einstein度量，它们是Y^{p，q}流形的推广。事实上，我们发现，这些指标是同构的Cvetic，Lu，Page和教皇最近发现的。我们证明了相应的光滑Sasaki-Einstein流形族都具有拓扑S^2 x S^3。最后，我们建立了描述翘曲形式的卡-丘锥的方程，支持（2，1）三形式通量。
We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form $M^d \times X_{1,1}$ with d-dimensional Riemannian manifold $M^d$ and the Aloff-Wallach space $X_{1,1}$= SU(3)/U(1) endowed with its Sasaki-Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff-Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly an…
We discuss the dimensional reduction of fermionic modes in a recently found class of consistent truncations of type IIB supergravity compactified on squashed five-dimensional Sasaki-Einstein manifolds. We derive the lower dimensional equations of motion and effective action, and comment on the supersymmetry of the resulting theory, which is consistent with N=4 gauged supergravity in $d=5$, coupled to two vector multiplets. We compute fermion masses by linearizing around two $AdS_{5}$ vacua of the theory: one that breaks N=4 down to N=2 spontaneously, and a second one which preserves no supersy…
We present the complete set of Killing-Yano tensors on the five-dimensional Einstein-Sasaki Y(p,q) spaces. Two new Killing-Yano tensors are identified, associated with the complex volume form of the Calabi-Yau metric cone. The corresponding hidden symmetries are not anomalous and the geodesic equations are superintegrable.
We show that for every positive curvature Kahler-Einstein manifold in dimension 2n there is a countably infinite class of associated Sasaki-Einstein manifolds X_{2n+3} in dimension 2n+3. When n=1 we recover a recently discovered family of supersymmetric AdS_5 x X_5 solutions of type IIB string theory, while when n=2 we obtain new supersymmetric AdS_4 x X_7 solutions of D=11 supergravity. Both are expected to provide new supergravity duals of superconformal field theories.
We consider supergravity in five-dimensional Anti-De Sitter space $AdS_{5}$ with minimal supersymmetry, encoded by a Sasaki-Einstein metric on a five-dimensional compact manifold $M$. Our main result reveals how the Sasaki-Einstein metric emerges from a canonical state in the dual CFT, defined by a superconformal gauge theory in four dimensional Minkowski space $\mathbb{R}^{3,1}$in the t'Hooft limit where the rank $N$ tends to infinity. We obtain explicit finite $N-$approximations to the Sasaki-Einstein metric, expressed in terms of a canonical (i.e. background free) BPS-state on the gauge the…
We briefly describe the construction of Stä\-kel-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 Kill\-ing-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.
We study the class of pulsating strings in AdS_5 x Y^{p,q} and AdS_5 x L^{p,q,r}. Using a generalized ansatz for pulsating string configurations, we find new solutions for this class in terms of Heun functions, and derive the particular case of AdS_5 x T^{1,1}, which was analyzed in arXiv:1006.1539 [hep-th]. Unfortunately, Heun functions are still little studied, and we are not able to quantize the theory quasi-classically and obtain the first corrections to the energy. The latter, due to AdS/CFT correspondence, is supposed to give the anomalous dimensions of operators of the gauge theory dual…
In the present paper we show that the complete list of special Killing forms on the 5-dimensional Sasaki-Einstein spaces $Y^{p,q}$ 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 $Y^{p,q}$ spaces.
