m. visinescu: hidden symmetries of the five-dimensional sasaki - einstein spaces

Hidden symmetries of thefive-dimensional Sasaki-Einstein metrics

Mihai Visinescu

Department of Theoretical PhysicsNational Institute for Physics and Nuclear Engineering ”Horia Hulubei”

Bucharest, Romania

BALKAN WORKSHOP 2013– Beyond the Standard Models –

Vrnjacka Banja, Serbia, 25 – 29 April, 2013

Outline

1. Symmetries and conserved quantities2. Killing forms3. Kahler, Sasaki manifolds4. Killing forms on Kerr-NUT-(A)dS spaces5. Y (p,q) spaces6. Killing forms on mixed 3-Sasakian manifolds7. Outlook

Symmetries and conserved quantities (1)

Let (M,g) be a n-dimensional manifold equipped with a(pseudo-)Riemmanian metric g and denote by

H =12

g ijpipj ,

the Hamilton function describing the motion in a curved space.In terms of the phase-space variables(x i ,pi) the Poissonbracket of two observables P,Q is

P,Q =∂P∂x i

∂Q∂pi− ∂P∂pi

∂Q∂x i .

Symmetries and conserved quantities (2)

A conserved quantity of motions expanded as a power series inmomenta:

K = K0 +

p∑k=1

1k !

K i1···ik (x)pi1 · · · pik .

Vanishing Poisson bracket with the Hamiltonian, K ,H = 0,implies

K (i1···ik ;i) = 0 ,

Such symmetric tensor K i1···ikk is called a Stackel-Killing (SK)

tensor of rank k

Killing forms (1)

A vector field X on a (pseudo-)Riemannian manifold (M,g) issaid to be a Killing vector field if the Levi-Civita connection ∇of g satisfies

g(∇Y X ,Z ) + g(Y ,∇Z X ) = 0,

for all vector fields Y , Z on M.

Killing forms (2)

A natural generalization of Killing vector fields is given by theconformal Killing vector fields , i.e. vector fields with a flowpreserving a given conformal class of metrics.

More general, a conformal Killing-Yano tensor( also called conformal Yano tensor or conformal Killingform or twistor form ) of rank p on a (pseudo-) Riemannianmanifold (M,g) is a p -form ω which satisfies:

∇Xω =1

p + 1X−| dω −

1n − p + 1

X ∗ ∧ d∗ω,

for any vector field X on M, where n is the dimension of M , X ∗

is the 1-form dual to the vector field X with respect to the metricg , −| is the operator dual to the wedge product and d∗ is theadjoint of the exterior derivative d .

Killing forms (3)If ω is co-closed then we obtain the definition of a Killing-Yanotensor, also called Yano tensor or Killing form :

ωi1...ik−1(ik ;j) = 0 .

Moreover, a Killing form ω is said to be a special Killing form ifit satisfies for some constant c the additional equation

∇X (dω) = cX ∗ ∧ ω ,

for any vector field X on M .

These two generalizations of the Killing vectors could berelated. Given two Killing-Yano tensors ωi1,...,ir and σi1,...,ir it ispossible to associate with them a Stackel-Killing tensor of rank2 :

K (ω,σ)ij = ωii2...irσ

i2...irj + σii2...irω

i2...irj .

Kahler, Sasaki manifolds (1)Even dimensions (1)

An almost Hermitian structure on a smooth manifold M is apair (g, J), where g is a Riemannian metric on M and J is analmost complex structure on M, which is compatible with g, i.e.

g(JX , JY ) = g(X ,Y ),

for all vector fields X ,Y on M. In this case, the triple (M, J,g) iscalled an almost Hermitian manifold.If J is parallel with respect to the Levi-Civita connection ∇ of g,then (M, J,g) is said to be a Kahler manifold.On a Kahler manifold, the associated Kahler form, i.e thealternating 2-form Ω defined by

Ω(X ,Y ) = g(JX ,Y )

is closed.


In local holomorphic coordinates (z1, ..., zm), the associatedKahler form Ω can be written as

Ω = igj kdz j ∧ dzk =∑

X ∗j ∧ Y ∗j =i2

∑Z ∗j ∧ Z ∗j ,

where (X1,Y1, ...,Xm,Ym) is an adapted local orthonormal field(i.e. such that Yj = JXj ), and (Zj , Zj) is the associated complexframe given by

Zj =12

(Xj − iYj), Zj =12

(Xj + iYj) .


Volume form isdV =

1m!

Ωm ,

where dV denotes the volume form of M, Ωm is the wedgeproduct of Ω with itself m times, m being the complex dimensionof M . Hence the volume form is a real (m,m)-form on M.If the volume of a Kahler manifold is written as

dV = dV ∧ dV

then dV is the complex volume holomorphic (m,0) form of M.The holomorphic form of a Kahler manifold can be written in asimple way with respect to any (pseudo-)orthonormal basis,using complex vierbeins ei + Jei . Up to a power factor of theimaginary unit i , the complex volume is the exterior product ofthese complex vierbeins.

Kahler, Sasaki manifolds (4)Odd dimensions (1)

Let M be a smooth manifold equipped with a triple (ϕ, ξ, η),where ϕ is a field of endomorphisms of the tangent spaces, ξ isa vector field and η is a 1-form on M.If we have:

ϕ2 = −I + η ⊗ ξ , η(ξ) = 1

then we say that (ϕ, ξ, η) is an almost contact structure on M.

Kahler, Sasaki manifolds (5)Odd dimensions (2)

A Riemannian metric g on M is said to be compatible with thealmost contact structure (ϕ, ξ, η) iff the relation

g(ϕX , ϕY ) = g(X ,Y )− η(X )η(Y )

holds for all pair of vector fields X ,Y on M.An almost contact metric structure (ϕ, ξ, η,g) is a Sasakianstructure iff the Levi-Civita connection ∇ of the metric gsatisfies

(∇Xϕ)Y = g(X ,Y )ξ − η(Y )X

for all vector fields X ,Y on M

Kahler, Sasaki manifolds (6)Metric cone (1)

A Sasakian structure may also be reinterpreted andcharacterized in terms of the metric cone as follows. The metriccone of a Riemannian manifold (M,g) is the manifold

C(M) = (0,∞)×M

with the metric given by

g = dr2 + r2g,

where r is a coordinate on (0,∞).M is a Sasaki manifold iff its metric cone C(M) is Kahler.The cone C(M) is equipped with an integrable complexstructure J and a Kahler 2-form Ω, both of which are parallelwith respect to the Levi-Civita connection ∇ of g.

Kahler, Sasaki manifolds (7)Metric cone (2)

M has odd dimension 2n + 1, where n + 1 is the complexdimension of the Kahler cone.The Sasakian manifold (M,g) is naturally isometricallyembedded into the cone via the inclusion

r = 1 ×M ⊂ C(M)

and the Kahler structure of the cone (C(M), g) induces analmost contact metric structure (φ, ξ, η, g) on M.

Kahler, Sasaki manifolds (8)Einstein manifolds

An Einstein-Sasaki manifold is a Riemannian manifold (M,g)that is both Sasaki and Einstein, i.e.

Ricg = λg

for some real constant λ, where Ricg denotes the Ricci tensorof g. Einstein manifolds with λ = 0 are called Ricci-flatmanifolds.An Einstein-Kahler manifold is a Riemannian manifold (M,g)that is both Kahler and Einstein. The most important subclassof Kahler-Einstein manifolds are the Calabi-Yau manifolds(i. e. Kahler and Ricci-flat).A Sasaki manifold M is Einstein iff the cone metric C(M) isKahler Ricci-flat. In particular the Kahler cone of aSasaki-Einstein manifold has trivial canonical bundle and therestricted holonomy group of the cone is contained in SU(m),where m denotes the complex dimension of the Kahler cone.

Kahler, Sasaki manifolds (9)Progression from Einstein-Kahler to Einstein-Sasaki to Calabi-Yau manifolds (1)

Suppose we have an Einstein-Sasaki metric gES on a manifoldM2n+1 of odd dimension 2n + 1. An Einstein-Sasaki manifoldcan always be written as a fibration over an Einstein-Kahlermanifold M2n with the metric gEK twisted by the overall U(1)part of the connection

ds2ES = (dψn + 2A)2 + ds2

EK ,

where dA is given as the Kahler form of the Einstein-Kahlerbase. The metric of the cone manifold M2n+2 = C(M2n+1) is

ds2cone = dr2 + r2ds2

ES = dr2 + r2(

(dψn + 2A)2 + ds2EK

).

Kahler, Sasaki manifolds (10)Progression from Einstein-Kahler to Einstein-Sasaki to Calabi-Yau manifolds (2)

The cone manifold is Calabi-Yau and its Kahler form is

Ωcone = rdr ∧ (dψn + 2A) + r2ΩEK ,

and the Kahler condition dΩcone = 0 implies

dA = ΩEK ,

where ΩEK is Kahler form of the Einstein-Kahler base manifoldM2n.The Sasakian 1-form of the Einstein-Sasaki metric is

η = 2A + dψn ,

which is a special unit-norm Killing 1-form obeying for all vectorfields X

∇Xη =12

X−| dη , ∇X (dη) = −2X ∗ ∧ η .

Kahler, Sasaki manifolds (11)Hidden symmetries (1)

The hidden symmetries of the Sasaki manifold M2n+1 aredescribed by the special Killing (2k + 1)−forms

Ψk = η ∧ (dη)k , k = 0,1, · · · ,n.

Besides these Killing forms, there are n closed conformalKilling forms ( also called ∗-Killing forms)

Φk = (dη)k , k = 1, · · · ,n.


Moreover, in the case of holonomy SU(n + 1) , i.e. the coneM2n+2 = C(M2n+1) is Kahler and Ricci-flat, or equivalentlyM2n+1 is Einstein-Sasaki, it follows that we have two additionalKilling forms of degree n + 1 on the manifold M2n+1. Theseadditional Killing forms are connected with the additionalparallel forms of the Calabi-Yau cone manifold M2n+2 given bythe complex volume form and its conjugate.


In order to extract the corresponding additional Killing forms onthe Sasaki-Einstein space we make use of the fact that for anyp -form ω on the space M2n+1 we can define an associated(p + 1) -form ωC on the cone C(M2n+1)

ωC := rp dr ∧ ω +rp+1

p + 1dω .

The 1-1-correspondence between special Killing p -forms onM2n+1 and parallel (p + 1) -forms on the metric cone C(M2n+1)allows us to describe the additional Killing forms onSasaki-Einstein spaces.

Killing forms on Kerr-NUT-(A)dS spaces (1)

In the case of Kerr-NUT-(A)dS spacetime the Einstein-Kahlermetric gEK and the Kahler potential A are

gEK =∆µdx2

µ

Xµ+

Xµ∆µ

n−1∑j=0

σ(j)µ dψj

2

,

Xµ = −4n+1∏i=1

(αi − xµ)− 2bµ ,

A =n−1∑k=0

σ(k+1)dψk ,


with

∆µ =∏ν 6=µ

(xν − xµ) ,

σ(k)µ =

∑ν1<···<νkνi 6=µ

xν1 . . . xνk , σ(k) =∑

ν1<···<νk

xν1 . . . xνk .

Here, coordinates xµ (µ = 1, . . . ,n) stands for the Wick rotatedradial coordinate and longitudinal angles and the Killingcoordinates ψk (k + 0, . . . ,n − 1) denote time and azimuthalangles with Killing vectors ξ(k) = ∂ψk .αi (i = 1, . . . ,n + 1) and bµ are constants related to thecosmological constant, angular momenta, mass and NUTparameters.


Write the metric gEK on the Einstein-Kahler manifold M2n in theform

gEK = oµoµ + oµoµ ,

and the Kahler 2-form

Ω = dA = oµ ∧ oµ .

where

oµ =

√∆µ

Xµ(xµ)dxµ ,

oµ =

√Xµ(xµ)

∆µ

n−1∑j=0

σ(j)µ dψj .


We introduce the following complex vierbeins on Einstein-Kahler manifold M2n

ζµ = oµ + i oµ , µ = 1, · · · ,n .

On the Calabi-Yau cone manifold M2n+2 the complex vierbeinsare Λµ = rζµ for µ = 1, · · · ,n and

Λn+1 =drr

+ iη .

The standard complex volume form of the Calabi-Yau conemanifold M2n+2 is

dV = Λ1 ∧ Λ2 ∧ · · · ∧ Λn+1 .


As real forms we obtain the real part respectively the imaginarypart of the complex volume form. For example, writing

Λj = λ2j−1 + iλ2j , j = 1, ...,n + 1,

the real part of the complex volume is given by

Re dV =

[ n+12 ]∑

p=0

∑1≤i1<i2<...<in+1≤2n+2 (C)

(−1)pλi1∧λi2∧...∧λin+1 (1)

where the condition (C) means that in the second sum aretaken only the indices i1, ..., in+1 such thati1 + ...+ in+1 = (n + 1)2 + 2p and (ik , ik+1) 6= (2j − 1,2j), for allk ∈ 1, ...,n and j ∈ 1, ...,n + 1.


The imaginary part of the complex volume is given by

Im dV =

[ n2 ]∑

p=0

∑1≤i1<i2<...<in+1≤2n+2 (C′)

(−1)pλi1∧λi2∧...∧λin+1 (2)

where the condition (C′) in means that in the second sum areconsidered only the indices i1, i2, ..., in+1 such thati1 + ...+ in+1 = (n + 1)2 + 2p + 1 and (ik , ik+1) 6= (2j − 1,2j), forall k ∈ 1, ...,n and j ∈ 1, ...,n + 1.


Finally, the Einstein-Sasaki manifold M2n+1 is identified with thesubmanifold r = 1 of the Calabi-Yau cone manifoldM2n+2 = C(M2n+1) and the additional (n + 1)-Killing forms areaccordingly acquired.

The 1-1-correspondence between special Killingp -forms on M2n+1 and parallel (p + 1) -forms on the metriccone C(M2n+1) allows us to describe the additional Killingforms on Einstein-Sasaki spaces.

In order to find the additional Killing forms on the Einstein -Kahler manifold M2n+1 we must identify the ωM form in thecomplex volume form of the Calabi-Yau cone.

Y(p,q) spaces (1)

Infinite family Y (p,q) of Einstein-Sasaki metrics on S2 × S3

provides supersymmetric backgrounds relevant to the AdS/CFTcorrespondence. The total space Y (p,q) of an S1-fibration overS2 × S2 with relative prime winding numbers p and q istopologically S2 × S3.Explicit local metric of the 5-dimensional Y (p,q) manifold givenby the line element

ds2ES =

1− c y6

(dθ2 + sin2 θ dφ2) +1

w(y)q(y)dy2

+q(y)

9(dψ − cos θ dφ)2

+ w(y)

[dα +

ac − 2y + c y2

6(a− y2)[dψ − cos θ dφ]

]2

,

where

Y(p,q) spaces (2)

w(y) =2(a− y2)

1− cy, q(y) =

a− 3y2 + 2cy3

a− y2

and a, c are constants. The constant c can be rescaled by adiffeomorphism and in what follows we assume c = 1.The coordinate change α = −1

6β −16c ψ′ , ψ = ψ′ takes the line

element to the following form ( with p(y) = w(y) q(y) )

ds2ES =

1− y6

(dθ2 + sin2 θ dφ2) +1

p(y)dy2

+p(y)

36(dβ + cos θ dφ)2

+19

[dψ′ − cos θ dφ+ y(dβ + cos θ dφ)]2 ,

Y(p,q) spaces (3)

One can write

ds2ES = dS2

EK + (13

dψ′ + σ)2

The Sasakian 1-form of the Y (p,q) space is

η =13

dψ′ + σ ,

withσ =

13

[− cos θ dφ+ y(dβ + cos θ dφ)] .

connected with local Kahler form ΩEK .This form of the metric with the 1-form η is the standard one fora locally Einstein-Sasaki metric with ∂

∂ψ′ the Reeb vector field.

Y(p,q) spaces (4)

The local Kahler and holomorphic (2,0) form for ds2EK are

ΩEK =1− y

6sin θdθ ∧ dφ+

16

dy ∧ (dβ + cos θdφ)

dVEK =

√1− y6p(y)

(dθ + i sin θdφ) ∧[dy + i

p(y)

6(dβ + cos θdφ)

]

Y(p,q) spaces (5)

From the isometries SU(2)× U(1)× U(1) the momentaPφ,Pψ,Pα and the Hamiltonian describing the geodesicmotions are conserved. Pφ is the third component of the SU(2)angular momentum, while Pψ and Pα are associated with theU(1) factors. Additionally, the total SU(2) angular momentumgiven by

J2 = P2θ +

1sin2 θ

(Pφ + cos θPψ)2 + P2Ψ ,

is also conserved.

Y(p,q) spaces (6)Specific conserved quantities for Einstein-Sasaki spaces (1)

First of all from the 1-form η

Ψ = η ∧ dη

=19

[(1− y) sin θ dθ ∧ dφ ∧ dψ′ + dy ∧ dβ ∧ dψ′

+ cos θ dy ∧ dφ ∧ dψ′ − cos θ dy ∧ dβ ∧ dφ+ (1− y)y sin θ dβ ∧ dθ ∧ dφ] .

is a special Killing form. Let us note also that

Ψk = (dη)k , k = 1,2 ,

are closed conformal Killing forms (?-Killing forms).


On the Calabi-Yau manifold the Kahler form is

Ωcone = rdr ∧ η + r2ΩEK .

and the holomorphic (3,0) form is

dVcone = eψ′r2dVEK ∧ [dr + ir ∧ η]

= eψ′r2

√1− y6p(y)

(dθ + i sin θdφ

)∧[dy + i

p(y)

6(dβ + cos θdφ)

]∧[dr + i

r3

[ydβ + dψ′ − (1− y) cos θdφ]


The additional Killing 3-forms of the Y (p,q) spaces areextracted from the volume form dVcone.

Using the the 1-1-correspondence between special Killing p-forms on M2n+1 and parallel (p + 1) -forms on the metric coneC(M2n+1) for p = 2 we get the following additional Killing2-forms of the Y (p,q) spaces written as real forms:

Ξ = ReωM =

√1− y6 p(y)

×(

cosψ′[−dy ∧ dθ +

p(y)

6sin θ dβ ∧ dφ

]− sinψ′

[− sin θ dy ∧ dφ− p(y)

6dβ ∧ dθ

+p(y)

6cos θ dθ ∧ dφ

])


Υ = ImωM =

√1− y6 p(y)

×(

sinψ′[−dy ∧ dθ +

p(y)

6sin θ dβ ∧ dφ

]+ cosψ′

[− sin θ dy ∧ dφ− p(y)

6dβ ∧ dθ

+p(y)

6cos θ dθ ∧ dφ

])


The Stackel-Killing tensors associated with the Killing formsΨ ,Ξ ,Υ are constructed as usual. Together with the Killingvectors Pφ,Pψ,Pα and the total angular momentum J2 theseStackel-Killing tensors provide the superintegrability of theY (p,q) geometries.

Killing forms on mixed 3-Sasakian manifolds (1)3-Sasakian manifolds (1)

A Riemannian manifold (M,g) of real dimension m is3-Sasakian if the holonomy group of the metric cone

(C(M) , g) = (R+ ×M , dr2 + r2g)

reduces to a subgroup of Sp(m+1

4

). In particular,

m = 4n + 3 ,n ≥ 1 and (C(M) , g) is hyperKahler.

A 3-Sasakian manifold admits three characteristic vector fields(ξ1 , ξ2 , ξ3), satisfying any of the corresponding conditions ofthe Sasakian structure, such that

g(ξα , ξβ) = δαβ ,

and[ξα , ξβ] = 2εαβγξγ .

Killing forms on mixed 3-Sasakian manifolds (2)3-Sasakian manifolds (2)

Let (M,g) be a 3-Sasakian manifold and (ϕα, ξα, ηα),α ∈ 1,2,3 , be its 3-Sasakian structure. then

ηα(ξβ) = δαβ ,

ϕα(ξβ) = −εαβγξγ ,

ϕαϕβ − ξα ⊗ ηβ = −εαβγϕγ − δαβI .

TheoremEvery 3-Sasakian manifold (M ,g) of dimension 4n + 3 isEinstein with Einstein constant λ = 4n + 2.

Killing forms on mixed 3-Sasakian manifolds (3)

A mixed 3-structure on a smooth manifold M is a triple ofstructures (ϕα, ξα, ηα), α ∈ 1,2,3 , which are almostparacontact structures for α = 1,2 and almost contact structurefor α = 3 , satisfying the following compatibility conditions

ηα(ξβ) = 0,

ϕα(ξβ) = τβξγ , ϕβ(ξα) = −ταξγ ,

ηα ϕβ = −ηβ ϕα = τγηγ ,

ϕαϕβ − ταηβ ⊗ ξα = −ϕβϕα + τβηα ⊗ ξβ = τγϕγ ,

where (α, β, γ) is an even permutation of (1,2,3) andτ1 = τ2 = −τ3 = −1.


If a manifold M with a mixed 3-structure (ϕα, ξα, ηα)α=1,3 admitsa semi-Riemannian metric g such that:

g(ϕαX , ϕαY ) = τα[g(X ,Y )− εαηα(X )ηα(Y )],

for all X ,Y ∈ Γ(TM) and α = 1,2,3 , where

εα = g(ξα, ξα) = ±1

then we say that M has a metric mixed 3-structure and g iscalled a compatible metric.


A manifold M endowed with a (positive/negative) metric mixed3-structure (ϕα, ξα, ηα,g) is said to be a (positive/negative)mixed 3-Sasakian structure if (ϕ3, ξ3, η3,g) is a Sasakianstructure, while both structures (ϕ1, ξ1, η1,g) and (ϕ2, ξ2, η2,g)are para-Sasakian, i.e.

(∇Xϕα)Y = τα[g(X ,Y )ξα − εαηα(Y )X ]

for all vector fields X ,Y on M and α = 1,2,3.

TheoremAny (4n + 3)−dimensional manifold endowed with a mixed3-Sasakian structure is an Einstein space with Einsteinconstant λ = (4n + 2)θ, with θ = ∓1, according as the metricmixed 3-structure is positive or negative, respectively.


In (mixed) 3-Sasakian case any linear combination of the formsΨk1,k2,k3 defined by

Ψk1,k2,k3 =k1

k1 + k2 + k3[η1 ∧ (dη1)k1−1] ∧ (dη2)k2 ∧ (dη3)k3

+k2

k1 + k2 + k3(dη1)k1 ∧ [η2 ∧ (dη2)k2−1] ∧ (dη3)k3

+k3

k1 + k2 + k3(dη1)k1 ∧ (dη2)k2 ∧ [η3 ∧ (dη3)k3−1]

for arbitrary positive integers k1, k2, k3 , is a special Killing formon M.

Outlook

I Complete integrability of geodesic equationsI Separability of Hamilton-Jacobi, Klein-Gordon, Dirac

equationsI Hidden symmetries of other spacetime structures

References

I M. Visinescu, Mod. Phys. Lett. A 27, 1250217 (2012)I M. Visinescu, G. E. Vılcu, SIGMA 8, 058 (2012)I M. Visinescu, Mod. Phys. Lett. A 26, 2719 (2011)I M. Visinescu, SIGMA 7, 037 (2011)I M. Visinescu, J. Phys.: Conf. Series 411, 012030 (2013)

m. visinescu: hidden symmetries of the five-dimensional sasaki - einstein spaces

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