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We investigate the deformations of the Sasaki–Einstein structures of the five-dimensional spaces and by exploiting the transverse structure of the Sasaki manifolds. We consider local deformations of the Sasaki structures preserving the Reeb vector fields but modify the contact forms. In this class of deformations, we analyze the transverse Kähler–Ricci flow equations. We produce some particular explicit solutions representing families of new Sasakian structures.
Sasakian geometry is often referred to as an odd-dimensional cousin of Kähler geometry and so is of independent interest. A Sasakian structure sits beween two Kähler structures, namely the one on its metric cone and the one on the normal bundle of its Reeb foliation. In physics, a prominent role is played by Sasaki–Einstein manifolds due to their applications in the so-called AdS/CFT correspondence.
During the last years, there is a lot of work done on the original AdS/CFT correspondence in maximally supersymmetric theories. Similar ideas have been applied to theories with less supersymmetry. Some of them are the gauge/gravity dualities that preserve supersymmetry. They are of the form , where is a five-dimensional Sasaki–Einstein manifold [1,2].
The first nontrivial example in the AdS/CFT correspondence with the use of these Sasaki–Einstein manifolds was made in the case of homogeneous manifold . Significant progress has been made in Sasaki–Einstein backgrounds and their dual field theories when an infinite family of inhomogeneous metrics on was found . All these manifolds have a Reeb vector field, which is a constant norm Killing vector field and, under the AdS/CFT correspondence, is isomorphic to the R-symmetry of dual field theory. The construction of Reference  was immediately generalized to higher dimensions . For example, is a supersymmetric solution of eleven-dimensional supergravity that is expected to be dual to a three-dimensional superconformal field theory . However, dimension five is the most interesting physically and the purpose of this work is to investigate deformations of Sasaki–Einstein structures in the frame of Sasaki–Ricci flow.
In the context of the relationship between the Sasakian structure and the two Kähler structures, we mention that, in the case of a Sasaki–Einstein manifold, the Riemannian metric cone is Ricci-flat and the transverse Kähler structure is Kähler–Einstein.
A well-known method for generating Einstein metrics on manifolds is the Ricci flow introduced by Hamilton in Reference . Recently, the method was applied to Sasaki manifolds in Reference . When one considers the problem of finding a Sasaki–Einstein metric, which is one of the main interests in physics, it is reduced to the problem of finding a transverse Kähler–Einstein metric. In Reference , the existence of transverse Kähler–Ricci solitons on compact toric Sasaki manifolds, of which the 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, is proven.
In this paper, we investigate the Sasaki–Ricci flow equations on five-dimensional Sasaki–Einstein spaces and . For this purpose, we perform deformations of Sasakian structures exploiting the transverse Kähler structure of Sasaki manifolds. These deformations of Sasaki–Einstein structures have important implications in holography and string theory . We introduce local complex coordinates to parametrize the transverse holomorphic structure of the Sasaki–Kähler potential. In spite of the complexity of the Sasaki–Ricci flow equations, we are able to produce some explicit particular solutions. The perturbations that do not modify the transverse metric preserve the Sasaki–Einstein structures, whereas if the transverse metric is changed, the Sasaki structures are preserved but are not Einstein anymore. Preliminary results concerning the Sasaki–Ricci flow on these spaces have been reported in References [10,11,12]. Now, we give some explicit analytical solutions of the Sasaki–Ricci flow equations and discuss their relevance to the symmetries of the deformed Sasaki structures.
The paper is organized as follows. In the next section, we review fundamentals of Sasaki geometry, deformations of Sasaki structures, and Sasaki–Ricci flow. In Section 3, we investigate the Sasaki–Ricci flow equations on the Sasaki–Einstein spaces and . In the last section, we provide some closing remarks.
In this section, we recall definitions and some basic facts of Sasakian manifolds and their deformations. We refer to the monograph of Reference  for details.
2.1. Sasakian Manifolds
Let be a -dimensional Riemannian manifold, ∇ be the Levi–Civita connection of the Riemannian metric g, and Ric denote the Ricci tensor of ∇.
A Riemannian manifoldis Sasakian if its metric conewith metricis Kähler with r as the coordinate on.
M is a contact manifold with the contact 1-form such that . There is a canonical vector field , called Reeb vector field, defined by
for any vector field X on M.
defines a -dimensional vector bundle over M, and the Sasakian metric g gives an orthogonal splitting of the tangential bundle :
where is the trivial bundle generated by the Reeb vector . The restriction of the Sasaki metric g to gives a well-defined Hermitian metric , which is in fact Kähler. The metric is related to the Sasakian metric g by
We define a tensor a tensor of type satisfying
for any vector fields on M.
Concerning the Einstein condition of a -dimensional Sasaki manifold , the following three conditions are equivalent:
g is Einstein with ;
the metric cone is a Ricci-flat Kähler manifold (i.e., Calabi–Yau manifold); and
the transverse Kähler metric satisfies .
Note that we often refer to as the Kähler form of the transverse Kähler metric . The transverse Ricci form represents the first Chern class , and let us denote by the de Rham cohomology class of .
We also recall the Sasakian structure and its transverse structure on local coordinates. Let be an open covering of M and submersions such that
is biholomorphic when is not empty. One can choose local coordinates charts on and local coordinates charts on such that . We shall use the following notations:
For the study of the deformations of the Sasaki structures, it is necessary to introduce the basic forms. A differential r-form is said to be basic if
where denotes the inner product. In the system of local coordinates , a basic r-form of type , has the form
where does not depend on x. In particular a function, is basic if and only if .
Note that, in the chart , we may write
where is a local basic function called Sasaki potential  and , .
Finally, we introduce the notion of toric Sasaki manifold.
A Sasaki manifoldis said to be toric if the Kähler cone manifoldis toric, namely a-dimensional torus G acts oneffectively as holomorphic isometries.
2.2. Deformations of Sasaki Structures
Let us assume that defines a Sasaki structure on M. We deform the Sasakian structure keeping the Reeb vector field fixed and perturbing the contact form with a basic function :
Here, we introduced the canonical basic Dolbeault operators
and . Accordingly, the tensor and the metric are modified as follows:
A simple proof can be done using the local frame and by observing that the Sasaki potential K is replaced by .
A complex vector field X on a Sasaki manifold is called a Hamiltonian vector field if
is a holomorphic vector field onand
the complex vector field
Such a functionis called a Hamiltonian function .
In the foliation chart on , X is written as
If the contact form η is modified according to Equation (14) with a basic function φ, the Hamilton functionis deformed to.
2.3. Sasaki–Ricci Flow
In what follows, we assume and . We consider the flow with an initial condition
This flow is called Sasaki-Ricci flow.
Let with as a family of basic functions similar to the deformations considered in Equation (14). The flow can be written as
where h is a basic function. It was proved in Reference  that this flow is well posed, preserving the Sasakian structure of M.
A Sasaki structure with a Hamiltonian holomorphic vector field X is called a transverse Kähler–Ricci soliton or Sasaki–Ricci soliton if
where stands for the Lie derivative by X. In Reference , it is proved that, on any toric Sasaki manifold, there exists a Sasaki–Ricci soliton.
On a Sasaki–Einstein manifold, choosing the the vector field X proportional to the Reeb vector field, i.e.,with c a constant, the Hamilton functionis c.
3. Sasaki–Ricci Flow on Spaces and
In what follows, we consider the Sasaki–Ricci flow on five-dimensional Sasaki–Einstein spaces and , looking for some explicit solutions of the flow equations.
3.1. Sasaki–Ricci flow on Sasaki–Einstein Space
The Sasaki–Einstein space is one the most renowned examples of homogeneous Sasaki–Einstein space in five dimensions.
The standard metric on this manifold is as follows [15,16]:
where , , , and . The contact 1-form is
and the Reeb vector field has the form
The isometries of the metric in Equation (24) form the group , with the Reeb vector field in Equation (26) being one of the Killing vectors.
Writing the metric in Equation (24) as in Equation (3) with the contact form of Equation (25), we get for the transverse metric
As on , the transverse structure is locally isomorphic to a product ; for each sphere, the complex coordinate is related to the spherical coordinates as
The Sasaki potential of the transverse metric is
Assuming a deformation of the contact form with a basic function as in Equation (14), the Ricci flow equation has the following form :
This equation is quite involved, and it is not expected that the general solution can be found. Instead, we search for particular solutions imposing a factorization of the dependence on the variable t and angle coordinates as follows:
where the functions are to be determined. Note also that the dependence on the angles is separated.
At last, we look for solutions of the form in Equation (31), imposing the following additional constraints:
where is some arbitrary real constants.
Owing to these assumptions, the Ricci flow equation in Equation (30) reduces to an ordinary differential equation for :
Concerning the functions , we get the following explicit expressions:
and are other arbitrary real constants of integration.
As long as is 0, we have the following:
Any metric of the form
with arbitrary real constants, defined on the local chart considered above represents a deformation of the canonical metric on. The deformed contact structure remains Sasaki–Einstein with the contact form
Moreover, if the constants are 0, Equation (34) becomes
having the elementary solution with the initial condition :
The presence of the constantsin Equation (38) entails that the anglesinterfere in the deformed metric and that the Reeb vector field in Equation (26) remains the only Killing vector. Therefore, the initial toric symmetry ofis broken in the deformed Sasaki–Einstein spaces.
If the constants , the transverse metric is also modified but the contact structure remains Sasaki:
The deformed contact structure with the contact form
remains Sasaki with the metric
Regarding other tensors, they can be evaluated using Equations (11)–(13) with the Sasaki potential . Their explicit expressions are omitted here.
3.2. Sasaki–Ricci flow on Sasaki–Einstein Space
The Einstein–Sasaki geometries are the subject of much attention in connection with the supersymmetric backgrounds relevant to the AdS/CFT correspondence. An interesting class of inhomogeneous Sasaki–Einstein metrics is represented by the toric structures on denoted , where are positive integers.
The metric of the Sasaki–Einstein space is as follows :
The cubic equation
has three roots . For any value of the constant , .
The range of the angular coordinates is . The variable is connected with another variable
which has the range
The Sasaki–Einstein space has the contact form
and the Reeb vector field is
To describe the transverse structure of the space, we introduce the following set of local complex coordinates :
In terms of the above complex coordinates, the Sasaki–Kähler potential is
In the case of space, the Ricci flow equation is as follows :
which is more involved than in the case of the Sasaki space .
Taking into account the complexity of this equation, we look for particular solutions. Using the same procedure as in the case of the space, we assume that the dependence of on and separates
where the functions are to be determined.
Compared to Equation (31), now, it is more convenient to write the separation of variables using the complex coordinatesin Equations (52) and (53). In fact the Sasaki–Ricci flow equation in Equation (22) is written in terms of complex coordinates of the transverse Kähler space. In the case of thespace, the correspondence between the complex coordinates and angle coordinates is bijective according to Equation (28). A separation of variables written in spherical coordinates or complex coordinates is the same. Concerning the complex coordinates in Equations (52) and (53) describing the transverse structure of thespace, the angle variable θ intervenes in both complex coordinates. Therefore, to analyze the Ricci flow equation in Equation (58), the separation assumption that is required is Equation (59).
Some particular exact solutions of the Sasaki–Ricci flow equation can be obtained assuming
where are arbitrary constants.
With these assumptions, we get the following for :
involving the arbitrary constants .
Similarly, for , we obtain the following solution:
where are other arbitrary constants and
As in the case of space, for , the contact structure remains Sasaki–Einstein and we can state the following:
The families of basic functions
witharbitrary constants, stand as solutions of the transverse Kähler–Ricci flow equation on the manifold.
The corresponding deformed contact structures remain Sasaki–Einstein with the contact forms
We observe that the explicit presence of the coordinatedin the functionsof Equation (62) andof Equation (63) makes the Reeb vector field in Equation (51) the only Killing vector of the deformed metric. As it is noted in Remark 3, the inceptive toric symmetry is broken during the Ricci flow deformation.
In the case , we have the following:
The deformed contact structures with the contact forms
witharbitrary constants remain Sasaki with the deformed metrics
In this case, the function obeys a differential equation without a simple, explicit solution.
Again, making use of Equations (11)–(13), we can evaluate other tensors which describe the deformed contact structures.
In this paper, we examine the Kähler structures of the Sasaki manifolds. We perform deformations of the contact structures fixing the Reeb vector field but vary the contact form by means of smooth basic functions.
The Sasaki–Ricci flow equations are quite involved, but eventually, we are able to produce some particular explicit analytical solutions representing deformations of the Sasaki–Einstein spaces and .
The deformations considered in this paper can be compared with the so-called or deformations of Sasaki–Einstein manifolds considered in Reference .
In Reference , the constants of geodesic motion in spaces and were explicitly constructed. The angles and are cyclic variables for geodesic motion in spaces and , respectively. Consequently, the geodesic motion in these spaces are completely integrable. We note that, in the deformed Sasaki metrics considered in the present paper, some of the angles are no longer cyclic variables and the complete integrability is lost. It would be interesting to investigate the Hamiltonian holomorphic vector fields in connection with the integrability and action-angle variables on the perturbed Sasaki–Einstein spaces.
It is worth extending the study of deformations of contact structures and the Sasaki–Ricci flow on higher dimensional Sasaki–Einstein spaces.
Finally, we note the relevance of contact Hamiltonian systems in irreversible thermodynamics, statistical physics, systems with dissipation, etc. (see, e.g., Reference  for a recent review of applications of contact Hamiltonian dynamics in various fields).
This research has been partially supported by the project NUCLEU PN 19 06 01 01/2019.
The author would like to thank Branko Dragovich for an invitation to this very successful meeting.
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