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arXiv:hep-th/0403002v3 16 Jun 2004
Imperial/TP/3-04/8
hep-th/0403002
Sasaki–Einstein Metrics on S2× S3
Jerome P. Gauntlett1∗, Dario Martelli2, James Sparks2and Daniel Waldram2
1Perimeter Institute for Theoretical Physics
Waterloo, ON, N2J 2W9, Canada
E-mail: jgauntlett@perimeterinstitute.ca
2Blackett Laboratory, Imperial College
London, SW7 2BZ, U.K.
E-mail: d.martelli, j.sparks, d.waldram@imperial.ac.uk
Abstract
We present a countably infinite number of new explicit co-homogeneity
one Sasaki–Einstein metrics on S2× S3, in both the quasi-regular and
irregular classes. These give rise to new solutions of type IIB supergravity
which are expected to be dual to N = 1 superconformal field theories in
four dimensions with compact or non-compact R-symmetry and rational
or irrational central charges, respectively.
∗On leave from: Blackett Laboratory, Imperial College, London, SW7 2BZ, U.K.
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1 Introduction
A Sasaki–Einstein five-manifold X5may be defined as an Einstein manifold whose
metric cone is Ricci-flat and K¨ ahler – that is, a Calabi–Yau threefold. Such manifolds
provide interesting examples of the AdS/CFT correspondence [1].
AdS5×X5, with suitably chosen self-dual five-form field strength, is a supersymmetric
solution of type IIB supergravity that is conjectured to be dual to an N = 1 four-
dimensional superconformal field theory arising from a stack of D3-branes sitting at
the tip of the corresponding Calabi–Yau cone [2, 3, 4, 5]. It is striking that the only
Sasaki–Einstein five-metrics that are explicitly known are the round metric on S5
(or S5/Z3) and the homogeneous metric T1,1on S2× S3(or T1,1/Z2). For S5the
Calabi–Yau cone is simply C3while for T1,1it is the conifold. Here we will present a
countably infinite number of explicit co-homogeneity one Sasaki–Einstein metrics on
S2× S3.
The new metrics were found rather indirectly. In [6] we analysed general super-
symmetric solutions of D = 11 supergravity consisting of a metric and four-form field
strength, whose geometries are the warped product of five-dimensional anti-de-Sitter
space (AdS5) with a six-dimensional manifold. A variety of different solutions were
presented in explicit form. In one class of solutions, the six-manifold is topologically
S2×S2×T2. Dimensional reduction on one of the circle directions of the torus gives
a supersymmetric solution of type IIA supergravity in D = 10. A further T-duality
on the remaining circle of the original torus then leads to a supersymmetric solution
of type IIB supergravity in D = 10 of the form AdS5× X5, with the only non-trivial
fields being the metric and the self-dual five-form. It is known [5] that for such solu-
tions to be supersymmetric, X5should be Sasaki–Einstein, and these are the metrics
we will discuss here. The global analysis presented in this paper, requiring X5to be
a smooth compact manifold, is then equivalent to requiring that the four-form field
strength in the D = 11 supergravity solution is quantised, giving a good M-theory
background.
Sasaki–Einstein geometries always possess a Killing vector of constant norm (for
some general discussion see, for example, [7]). If the orbits are compact, then we have
a U(1) action. The quotient space is always locally K¨ ahler–Einstein with positive
curvature. If the U(1) action is free then the quotient space is a K¨ ahler–Einstein
manifold with positive curvature. Such Sasaki–Einstein manifolds are called regular,
and the five-dimensional compact variety are completely classified [8]. This follows
from the fact that the smooth four-dimensional K¨ ahler–Einstein metrics with positive
curvature on the base have been classified by Tian and Yau [9, 10]. These include the
special cases CP2and S2× S2, with corresponding Sasaki–Einstein manifolds being
the homogeneous manifolds S5(or S5/Z3) and T1,1(or T1,1/Z2), respectively. For
the remaining metrics, the base is a del Pezzo surface obtained by blowing up CP2
at k generic points with 3 ≤ k ≤ 8 and, although proven to exist, these metrics are
not known explicitly.
More generally, since the U(1) Killing vector has constant norm, the action it
generates has finite isotropy subgroups. Only if the isotropy subgroup of every point is
trivial is the action free. Thus in general the base – the space of leaves of the canonical
In particular,
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U(1) fibration – will have orbifold singularities. This class of metrics is called quasi-
regular [11]. Recently [11, 12, 13, 14], a rich set of examples of quasi-regular Sasaki–
Einstein metrics have been shown to exist on #l(S2×S3) with l = 1,...,9, but they
have not yet been given in explicit form1. In particular, there are 14 inhomogeneous
Sasaki–Einstein metrics on S2× S3.
Some of the new metrics presented here are foliated by U(1) and are thus in the
quasi-regular class. We do not yet know if they include any of the 14 discussed in
[12]. We also find Sasaki–Einstein metrics where the Killing vector has non-compact
orbits, which are hence irregular. These seem to be the first examples of such metrics.
It is worth emphasising that the isometries generated by the canonical Killing
vector on the Sasaki–Einstein manifold are dual to the R-symmetry in the four-
dimensional superconformal field theory. Thus the regular and quasi-regular examples
are dual to field theories with compact U(1) R-symmetry. In contrast, the irregular
examples are dual to field theories with a non-compact R-symmetry. In other words,
these theories would be invariant under the superconformal algebra but not the su-
perconformal group. We will also calculate the volumes of the new Sasaki–Einstein
metrics. In the dual conformal field theory these are inversely related to the central
charge of the conformal field theory. We show that the metrics associated with com-
pact U(1) R-symmetry are associated with rational central charges while those with
non-compact R-symmetry are associated with irrational central charges.
2 The metrics
Our starting point is the explicit local metric given by the line element [6]:
ds2=1 − cy
6
(dθ2+ sin2θdφ2) +
1
w(y)q(y)dy2+q(y)
9
[dψ − cosθdφ]2
?2
+ w(y)
?
dα +ac − 2y + y2c
6(a − y2)
[dψ − cosθdφ]
(2.1)
with
w(y) =
2(a − y2)
1 − cy
a − 3y2+ 2cy3
a − y2
q(y) =
. (2.2)
A direct calculation shows that this metric is Einstein with Ric = 4g, for all values
of the constants a,c. Locally the space is also Sasaki. Note that the definition of
Sasaki–Einstein can be given in several (more-or-less) equivalent ways. For example,
one can define a Sasaki–Einstein geometry in terms of the existence of a certain
contact structure, or in terms of the existence of a solution to the Killing spinor
1Very recently an infinite class of explicit inhomogeneous Einstein metrics have been constructed
in [15]. These include Einstein metrics on S2×S3, but they are not expected to be Sasaki–Einstein.
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equation. The simplest way to demonstrate that we have a local Sasaki-Einstein
metric is to write the metric in a canonical way, which we do in section 4. The
corresponding local Killing spinors are then easily obtained (see the discussion in
[17]).
The next step is to analyse when we can extend the local expression for the metric
to a global metric on a complete manifold. In this section we will demonstrate that
this can indeed be done, with the manifold being S2×S3. The point of section 4 will
be to show that the Sasaki structure also extends globally, ensuring we have globally
defined Killing spinors.
We find the global extensions in two steps. First we show that, for given range of
a, one can choose the ranges of the coordinates (θ,φ,y,ψ) so that this “base space”
B4(forgetting the α direction) is topologically the product space S2×S2. The second
step is to show that, for a countably infinite number of values of a in this range, one
can choose the period of α so that the five-dimensional space is then the total space
of an S1fibration (with S1coordinate α) over B4. Topologically this five-manifold
turns out to be S2×S3. A detailed analysis of the possible values for a is the content
of section 3.
As we see in section 5, when c = 0 the metric is the local form of the standard
homogeneous metric on T1,1(the parameter a can always be rescaled by a coordinate
transformation). Thus we will focus on the case c ?= 0. In this case we can, and will,
rescale y to set c = 1. This leaves us with a local one-parameter family of metrics,
parametrised by a.
The base B4
First, we choose 0 ≤ θ ≤ π and 0 ≤ φ ≤ 2π so that the first two terms in (2.1),
at fixed y, give the metric on a round two-sphere. Moreover, the two-dimensional
(y,ψ)-space, defined by fixing θ and φ, is clearly fibred over this two-sphere. To
analyse the fibre, we fix the range of y so that 1 − y > 0, a − y2> 0, which implies
that w(y) > 0. We also demand that q(y) ≥ 0 and that y lies between two zeroes
of q(y), i.e. y1≤ y ≤ y2with q(yi) = 0. Since the denominator of q(y) is always
positive, yiare roots of the cubic
a − 3y2+ 2y3= 0 .(2.3)
All of these conditions can be met if we fix the range of a to be
0 < a < 1 .(2.4)
In particular, for this range, the cubic has three real roots, one negative and two
positive, and q(y) ≥ 0 if y1 is taken to be the negative root and y2 the smallest
positive root. The case a = 1 is special since the two positive roots coalesce into a
single double root at y = 1. In fact we will show in section 5 that when a = 1 the
metric (2.1) is locally that of S5.
We now argue that by taking ψ to be periodic with period 2π, the (y,ψ)-fibre,
at fixed θ and φ, is topologically a two-sphere. To see this, note the space is a circle
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fibred over an interval, y1 ≤ y ≤ y2, with the circle shrinking to zero size at the
endpoints, yi. This will be diffeomorphic to a two-sphere provided that the space is
free from conical singularities at the end-points. For 0 < a < 1, near either root we
have q(y) ≈ q′(yi)(y − yi). Fixing θ and φ in (2.1) (and ignoring the α direction for
the time being), this gives the metric near the poles y = yiof the two-sphere
1
w(yi)q′(yi)(y − yi)dy2+q′(yi)(y − yi)
9
dψ2. (2.5)
Introducing the co-ordinate R = [4(y − yi)/w(yi)q′(yi)]1/2this can be written as
dR2+q′(yi)2w(yi)R2
36
dψ2.(2.6)
We now note the remarkable fact that since q′(yi) = −3/yi and w(yi) = 4y2
follows that q′(yi)2w(yi)/36 = 1 at any root of the cubic. Thus the potential conical
singularities at the poles y = y1and y = y2can be avoided by choosing the period of
ψ to be 2π.
Note that the properties of the function q(y) allow the introduction of an angle
ζ(y) defined by
i, it
cosζ= q(y)1/2=
?a − 3y2+ 2y3
a − y2
?1/2
sinζ= −
2y
w(y)1/2
(2.7)
with ζ ranging from π/2 to −π/2 between the two roots (cf. [6]). It is not simple to
explicitly change coordinates from y to ζ, so we continue to work with y.
At fixed value of y between the two roots, the (ψ,θ,φ)-space is a U(1) = S1bundle,
parametrised by ψ, over the round two-sphere parametrised by θ and φ. Such bundles
are, up to isomorphism, in one-to-one correspondence with H2(S2;Z) = Z, which is
the Chern number, or equivalently the integral of the curvature two-form over the
base space. In our case, this integral is
1
2π
?
S2d(−cosθdφ) = 2 .(2.8)
This identifies the three-space at fixed y as the Lens space S3/Z2= RP3. Equiva-
lently, it is the total space of the bundle of unit tangent vectors of S2. This establishes
that the four-dimensional base manifold is a two-sphere bundle over a two-sphere.
Moreover, the R2= C bundle over S2obtained by deleting the north pole from each
two-sphere fibre is just the tangent bundle of S2.
In general, oriented S2bundles over S2are classified up to isomorphism by an
element of π1(SO(3)) = Z2. One constructs any such bundle by taking trivial bundles,
i.e. products, over the northern and southern hemispheres and gluing them together
along the equator, with the appropriate group element. The gluing is given by a map
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from the equatorial S1into SO(3). The topology of the bundle depends only on the
homotopy type of the map and hence there are only two bundles, classified by an
element in π1(SO(3)) = Z2, one trivial and one non-trivial2. In fact in our case we
have the trivial fibration and hence, topologically, the four-dimensional base space is
simply S2× S2as claimed earlier.
To see this, consider the gluing element of π1(SO(3)) corresponding to the map
φ →
cos(Nφ) −sin(Nφ) 0
sin(Nφ)cos(Nφ)
0
0
10
(2.9)
where φ is a coordinate on the equator of the base S2, with 0 ≤ φ ≤ 2π. Here, the
upper-left 2 × 2 block of the matrix describes the twisting of the equatorial plane in
the fibre (the ψ coordinate in our metric) and the bottom-right entry refers to the
“polar” direction. If one projects out the polar direction, leaving the 2 × 2 block
in the upper-left corner, the above map then gives the element N ∈ π1(U(1)) = Z
corresponding to the Chern class of a U(1) = SO(2) bundle over S2– this bundle is
just a charge N Abelian monopole.
Now, the above SO(3) matrix is well-known to correspond (with a choice of sign)
to the SU(2) matrix
?eiNφ/2
0
0
e−iNφ/2
?
(2.10)
where we recall that SU(2) is the simply-connected double-cover of SO(3). It follows
that, for N even, the resulting curve in SU(2) = S3is a closed cycle, and therefore
corresponds to the trivial element of π1(SO(3)) = H1(SO(3);Z) = Z2. Thus all even
N give topologically product spaces S2× S2. For N odd, the curve in SU(2) is not
closed – it starts at one pole of S3and finishes at the antipodal pole. Of course,
when projected to SO(3) this curve now becomes closed and thus represents a non-
trivial cycle. In fact this is the generator of π1(SO(3)). Thus all the odd N give
the non-trivial S2bundle over S2. In particular, recall that for us the U(1) bundle
corresponding to ψ was RP3with N = 2, and thus we have topologically a product
space
B4= S2× S2.
In what follows it will be useful for us to have an explicit basis for the homology
group H2(B4;Z) = Z ⊕ Z of two-dimensional cycles on B4= S2× S2. The natural
choice is simply the two S2cycles C1, C2themselves, but since our metric on B4is
not a product metric, it is not immediately clear where these two two-spheres are. In
fact, we can take C1to be the fibre S2at some fixed value of θ and φ on the round
S2. Returning to the metric (2.1) we note that there are two other natural copies of
S2located at the south and north poles of the fibre, i.e. at the roots y = y1and y2,
or equivalently ζ = π/2 and ζ = −π/2. Call these S1and S2. Then the other cycle3
2The non-trivial bundle is obtained by adding a point to the fibres of the chiral spin bundle of
S2and is not a spin manifold. It gives the same manifold as the Page instanton [18] on CP2#CP
3One way to check this is to work out the intersection numbers of the cycles. We have C2
C2
2= 0, C1· C2= 1, S2
(2.11)
2.
1=
1= 2, S2
2= −2, S1· S2= 0, S1· C1= S2· C1= 1.
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is just C2= S2+ C1= S1− C1. Thus we have
2C1= S1− S2
2C2= S1+ S2. (2.12)
For completeness let us also give explicitly the dual elements in the cohomology
H2(S2× S2;Z). We have
1
4πcosζdζ ∧ (dψ − cosθdφ) +
1
4πsinθdθ ∧ dφ
ω1=
1
4πsinζ sinθdθ ∧ dφ
ω2=
(2.13)
with?
The circle fibration
Ciωj= δij.
Now we turn to the fibre direction, α, of the full five-dimensional space. It is conve-
nient to write the five-dimensional metric (2.1) in the form
ds2= ds2(B4) + w(y)(dα + A)2
(2.14)
where ds2(B4) is the non-trivial metric on S2× S2just described, and the local
one-form A is given by
A =a − 2y + y2
6(a − y2)
[dψ − cosθdφ] .(2.15)
Note that the norm-squared of the Killing vector ∂/∂α is w(y), which is nowhere-
vanishing.
In order to get a compact manifold, we would like the α coordinate to describe
an S1bundle over B4. We take
0 ≤ α ≤ 2πℓ (2.16)
where the period 2πℓ of α is a priori arbitrary. Thus, rescaling by ℓ−1, we have that
ℓ−1A should be a connection on a U(1) bundle over B4= S2×S2. However, this puts
constraints on A. In general, such U(1) bundles are completely specified topologically
by the gluing on the equators of the two S2cycles, C1and C2. These are measured
by the corresponding Chern numbers in H2(S2;Z) = Z which we label p and q. The
corresponding five-dimensional spaces will be denoted by Yp,q. In general, we will
find that for any p and q, such that 0 < q/p < 1, we can always choose ℓ and the
parameter 0 < a < 1 such that ℓ−1A is a bona fide U(1) connection. For p and q
relatively prime, which we can always achieve by taking an appropriate ℓ, i.e. the
maximal period, it turns out that Yp,qare all topologically S2× S3. We now fill in
the details of this argument.
The essential point is to show that ℓ−1A is a connection on a U(1) bundle. As
mentioned above, a U(1) bundle over B4= S2×S2is characterised by the two Chern
numbers p and q. These are given by the integrals of the U(1)-curvature two-form
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ℓ−1dA/2π over the two-cycles C1and C2which form the basis of H2(S2× S2;Z) =
Z ⊕ Z. Let us define the two periods P1and P2as
1
2π
Pi=
?
Ci
dA(2.17)
which, in general, will be functions of a. The corresponding integrals of ℓ−1dA/2π
must give the Chern numbers p and q. That is, we require P1= ℓp and P2= ℓq.
Since we are free to choose ℓ, the only constraint is that we must find a such that
P1/P2= p/q . (2.18)
In particular, we will choose ℓ so that p and q are coprime. Then with
ℓ = P1/p = P2/q (2.19)
we get a five-dimensional manifold which is an S1bundle over B4= S2× S2with
winding numbers p and q, denoted by Yp,q. We will find that (2.18) can be satisfied
for a countably infinite number of values of a.
Let us first check that dA is properly globally defined. Recall that, in general,
a connection one-form is not a globally well-defined one-form – if it is globally well-
defined, the curvature is exact and the bundle is topologically trivial. Rather, a
connection one-form is defined only locally in patches, with gauge transformations
between the patches. However, the curvature is a globally well-defined smooth two-
form. Let us now check this is true for our metric. At fixed value of y between the two
roots, y1< y < y2, we see that A is proportional to the “global angular form” on the
U(1) bundle with fibre ψ and base parametrised by (θ,φ). This is actually globally
well-defined since a gauge transformation on ψ is cancelled by the corresponding
gauge transformation of the connection −cosθdφ. Thus in fact dA is exact on a slice
of the four-manifold B4at fixed y. This is also clearly true on the whole of the total
space of the “cylinder bundle” obtained by deleting the north and south poles of the
fibres of the S2bundle. One must now check that dA is smooth as one approaches
the poles of the fibres. There are two terms, and the only term of concern smoothly
approaches a form proportional to dy ∧ dψ near the poles, at fixed value of θ and
φ. We must now recall that the proper radial coordinate is (e.g. at the south pole)
R ∝ (y − yi)1/2. Thus dy ∝ RdR, and this piece of dA is a smooth function times
the canonical volume form RdR ∧ dψ on the open subset of R2in the fibre near to
the poles. Thus dA is a globally-defined smooth two-form on B4= S2×S2, and thus
represents an element of H2
To calculate the periods Piit is easiest to first calculate the integrals of dA over
the cycles Siat the north and south poles of the (y,ψ) fibre. We find
de Rham(B4).
1
2π
?
Si
dA =
1
2π
?
Si
a − 2yi+ y2
6(a − y2
i
i)
sinθdθdφ =yi− 1
3yi
(2.20)
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and hence, given the relations (2.12), we have
P1 =
y1− y2
6y1y2
2y1y2− y1− y2
6y1y2
P2 =
= −(y1− y2)2
9y1y2
(2.21)
and so
P1
P2
=
3
2(y2− y1). (2.22)
Thus our requirement (2.18) is that
y2− y1is rational. (2.23)
In the next section we will show that there are infinitely many values of a for
which this is true, in the range 0 < a < 1 and with 0 < q/p < 1. Furthermore,
as shown in Appendix A, for any p and q coprime, the space Yp,qis topologically
S2× S3. This then completes our argument about the regularity of the metrics on
S2× S3.
Our new metrics admit a number of Killing vectors which generate isometries.
Clearly there is a U(1) generated by ∂/∂α. In addition there is an SU(2) × U(1)
action. To see this, we write the metric in terms of left-invariant one-forms σi,
i = 1,2,3 on SU(2)
ds2=1 − y
6
(σ2
1+ σ2
2) +
1
w(y)q(y)dy2+q(y)
dα +a − 2y + y2
9
σ2
3
+ w(y)
?
6(a − y2)σ3
?2
.
(2.24)
This displays the fact that the metrics admit an SU(2) left-action and a U(1) right-
action. Together with the U(1) isometry generated by ∂/∂α, this gives an isometry
group SU(2) ×Z2U(1)2, where we have noted that the element (−12,−1,−1) acts
trivally.
We end by noting that the volume of these spaces is given by
vol =4π3
9
ℓ (y1− y2)(y1+ y2− 2) . (2.25)
3 All solutions for the parameter a
We have shown that it is necessary and sufficient that P1/P2 = p/q is rational in
order to get metrics on a complete manifold. Clearly it is sufficient that the roots
y1 and y2 of the cubic (2.3) (and hence all three of the roots) are rational. This
leads to a number theoretic analysis which is presented below, and we find an infinite
number of values of a for which the roots are rational. In these cases the volume of
the manifolds are rationally related to the volume of the round five-sphere. We will
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argue later that these values of a give rise to quasi-regular Sasaki–Einstein manifolds.
However, it is also possible to achieve rational P1/P2for an infinite number of values
of a when the roots are irrational as the following general analysis reveals. We will
see later that these cases give rise to irregular Sasaki–Einstein metrics.
General case
Assume P1/P2= p/q is rational. As discussed above, this implies that
y2− y1=3q
2p≡ λ . (3.1)
We first observe that the three roots of the cubic satisfy:
y1+ y2+ y3 = 3/2
y1y2+ y1y3+ y2y3 = 0
2y1y2y3 = −a . (3.2)
Next note that y1+ y2= 0 if and only if a = 0, which is excluded from our consider-
ations. Then we deduce that
y1+ y2=2
3(y2
1+ y1y2+ y2
2) (3.3)
with the third root given by y3=3
λ. Since y1is required to be the smallest root of the cubic, one takes the smaller of
the two roots to obtain
2− y1− y2. One may now solve for y1in terms of
y1=1
2
?
1 − λ −
?
1 − λ2/3
?
. (3.4)
Since y2> y1this means that 0 < λ ≤√3, where the upper bound ensures that y1is
real. Notice that y1< 0, as it should be.
We now require that y1be a root of the cubic. To ensure this, we simply define
a = a(λ) = 3y2
1(λ) − 2y3
1(λ) . (3.5)
One requires that 0 < a < 1 which is in fact automatic for the range of λ already
chosen. However, some values of a are covered twice since the function a(λ) has a
maximum of 1 at the value λ = 3/2. Specifically, this range is easily computed to be
1/2 ≤ a ≤ 1. The range 0 < a < 1/2 is covered only once. Finally, we must ensure
that y2= y1+ λ is the smallest positive root. Comparing with y3=3
see that this implies that λ <3
To summarise, any rational value of λ = 3q/2p is allowed within the range 0 <
λ < 3/2, and this is achieved by choosing a given by (3.4), (3.5). Moreover, the range
of a is 0 < a < 1 and is covered in a monotonic increasing fashion. The period of the
coordinate α is given by 2πℓ where
2− y1− y2, we
2.
ℓ =
q
3q2− 2p2+ p(4p2− 3q2)1/2.(3.6)
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Note that we can also recast our formula for the volume in terms of p and q to give
vol =
q2[2p + (4p2− 3q2)1/2]
3p2[3q2− 2p2+ p(4p2− 3q2)1/2]π3
(3.7)
which is generically an irrational fraction of the volume π3of a unit round S5. This
implies that we have irrational central charges in the dual superconformal field theory.
In fact the result is stronger: the central charge can be written only in terms of square-
roots of rational numbers. Note that by setting q = 1 and letting p become large, we
see that the volume can be arbitrarily small. The largest volume given by our metrics
on Yp,qoccurs for p = 2,q = 1 with vol ≈ 0.29π3, and corresponds to an irrational
case.
Case of rational roots
We now show that for a countably infinite number of values of a the roots y1and y2
are rational. First note from (3.4) that y1is rational if and only if 1 − λ2/3 is the
square of a rational. Then y2(and also y3) is rational since λ is necessarily rational.
Substituting λ = 3q/2p, the problem reduces to finding all solutions to the quadratic
diophantine
4p2− 3q2= n2
(3.8)
where p,q ∈ N, n ∈ Z, (p,q) = 1, q < p.
To find the general solution, we proceed as follows. First define r = 2p, so that
(3.8) may be written
(r − n)(r + n) = 3q2. (3.9)
We first argue that a prime factor t > 3 of r + n must appear with an even power.
Suppose it did not. We have t divides q2and so t divides q. Then, since the power
of t in q2is clearly even, t must divide r −n. This is now a contradiction since t now
divides both p and q which are, by assumption, coprime. Using a similar argument
for t = 3 we conclude that
r + n = 3A22k1
r − n = B22k2
(3.10)
where A,B,ki∈ N and we have used the freedom of switching the sign of n in fixing
the factor of 3. Moreover, A,B satisfy
(A,B) = 1,B ?= 0 mod 3, A,B ?= 0mod 2 .(3.11)
We deal with the factors of 2 in two steps. First, if n is odd then k1= k2= 0 and we
have the following solutions
p =1
4(3A2+ B2),q = AB, (3.12)
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with n =1
2(3A2− B2). To ensure that p > q we need to also impose
A > B,or B > 3A . (3.13)
Alternatively, for solutions with n even, one can show that q is always even and p is
then always odd. A little effort reveals two families of solutions. The first is:
p = 3A2+ B222k,q = AB2k+2
(3.14)
with n = 2(3A2− B222k), k ≥ 1 and A,B satisfying (3.11) and also
A > B2k, orB2k> 3A . (3.15)
The second is:
p = 3A222k+ B2,q = AB2k+2
(3.16)
with n = 2(3A222k− B2), k ≥ 1 and A,B satisfying (3.11) and also
A2k> B, or B > 3A2k. (3.17)
Finally, returning to the volume formula (3.7), we note that when the roots are ra-
tional, since 4p2−3q2= n2, clearly the volume is a rational fraction of that of the unit
round S5. This corresponds to rational central charges in the dual superconformal
field theory.
4 The metrics in canonical form
At this stage we have shown that we have a countably infinite number of new Einstein
metrics on S2× S3. We now establish that the geometries admit a Sasaki structure
which extends globally and that the metrics admit globally defined Killing spinors.
We do this by first showing that the metric can be written in a canonical form
implying it has a local Sasaki structure. We then show that this Sasaki-Einstein
structure, defined in terms of contact structures, is globally well-defined. Finally,
given we have a simply-connected five-dimensional manifold with a spin structure,
using theorem 3 of [8], we see that this implies we have global Killing spinors.
Employing the change of coordinates α = −β/6 − cψ′/6, ψ = ψ′the local metric
(2.1) becomes
ds2=1 − cy
6
(dθ2+ sin2θdφ2) +
dy2
w(y)q(y)+136w(y)q(y)(dβ + ccosθdφ)2
+1
9[dψ′− cosθdφ + y(dβ + ccosθdφ)]2
(4.1)
where we have temporarily reinstated the constant c. This has the standard form
ds2= ds2
4+?1
3dψ′+ σ?2
(4.2)
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where ds2
have the local K¨ ahler form
4is a local K¨ ahler–Einstein metric and the form σ satisfies dσ = 2J4. We
J4=1
6(1 − cy)sinθdθ ∧ dφ +1
6dy ∧ (dβ + ccosθdφ) . (4.3)
In the quasi-regular case, J4extends globally to the K¨ ahler form on the four-dimensional
base orbifold.
One can check explicitly that the four-dimensional metric is K¨ ahler–Einstein and
has Ricci-form equal to six times the K¨ ahler form. This form of the metric (4.2) is
the standard one for a locally Sasaki–Einstein metric, with ∂/∂ψ′the constant norm
Killing vector. As noted earlier, the normalisation is canonical with the Ricci tensor
being four times the metric.
Note that the SU(2) × U(1)2isometry group of our metrics (2.24) implies that
we can introduce a set of SU(2) left-invariant one-forms, ˜ σi, and write ds2
locally as a bi-axially squashed SU(2)-invariant K¨ ahler–Einstein metric. The most
general metric of this type was found in [19], where it was shown to depend on two
parameters (one being the overall scale). Global properties of such metrics were then
discussed, to some extent, in [20]. To recast our metric in the form given in [19], we
introduce ρ2= 2(1 − y)/3 and ˜ σito get, with c = 1,
1
∆dρ2+ρ2
4
∆ = 1 +4(a − 1)
27
4in (4.2)
ds2
4
=
?˜ σ2
ρ4− ρ2.
1+ ˜ σ2
2+ ∆˜ σ2
3
?
1
(4.4)
We would now like to confirm that the Sasaki structure extends globally. To do
so, we will work with the definition of a Sasaki–Einstein manifold given in terms
of a contact structure. Given the metric extends globally, this is equivalent to the
condition that the Killing vector ∂/∂ψ′and hence the dual one-form
dσ = 2J4in (4.2) are globally defined. To show this we simply consider these objects
in the original coordinates (2.1). We first observe that the Killing vector field of the
Sasaki structure is given by
∂
∂ψ′=
which is globally well defined, since both of the vectors on the right hand side are
globally well defined. Since this vector has constant norm, the dual one-form
1
3dψ′+ σ with
∂
∂ψ−1
6
∂
∂α
(4.5)
[dψ′− cosθdφ + y(dβ + ccosθdφ)](4.6)
must also be globally well defined. It is interesting to see this in the original coordi-
nates. This one-form can be written as the sum of two one-forms:
− 6y
?
dα +ac − 2y + y2c
6(a − y2)
+a − 3y2+ 2cy3
a − y2
(dψ − cosθdφ)
?
(dψ − cosθdφ) .(4.7)
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From the analysis of section 2, we conclude that the first one-form in this expression is
globally well-defined since it is, up to a smooth function, just the dual of the globally
defined vector ∂/∂α. This form is also the so-called global angular form on the total
space of the U(1) bundle U(1) ֒→ Yp,q→ S2×S2. Now, the one-form (dψ − cosθdφ)
is a global angular form on the U(1) bundle U(1) ֒→ RP3→ S2at fixed y between the
two roots/poles. Although this one-form is not well defined at the poles y = yi, the
second one-form in (4.7) is globally defined, since the pre-factor vanishes smoothly
at the poles. Finally, the exterior derivative of (4.6) is also clearly well defined and
is equal to 6J4. Thus we conclude that the Sasaki structure is globally well defined
and that we have a countably infinite number of Sasaki–Einstein manifolds.
Recall that in the original co-ordinates we argued that ψ was periodic with period
2π and that α was periodic with period 2πℓ = 2πP1/p = 2πP2/q. The form of (4.5)
shows that, for general values of a discussed in section 3, the orbits of the vector
∂/∂ψ′are not closed. In this case the Sasaki–Einstein metric is irregular. There is
no K¨ ahler–Einstein “base manifold” or even “base orbifold” for this case. Since the
orbits of ∂/∂ψ′are dense in the torus defined by the Killing vectors ∂/∂ψ and ∂/∂α,
these Sasaki–Einstein metrics are, more precisely, irregular of rank two.
However, in the special case that
6P1
p
=6P2
q
=s
r
(4.8)
for suitably chosen integers r,s, we can choose ψ′to be periodic with period 2πs.
This case occurs only when the roots of the cubic are rational. The Sasaki–Einstein
metrics are then quasi-regular. Indeed, if the base were a manifold, it would have to
be in Tian and Yau’s list. Notice also that our metrics admit an SU(2) action. As
discussed in [21], these two facts are mutually exclusive, except in the cases CP2and
CP1×CP1with canonical metrics. Thus we must be in the quasi-regular class. The
base space is then an orbifold – it would be interesting to better understand their
geometry.
5 Special cases
We discuss here the two special cases that were mentioned earlier.
Case 1: T1,1
First consider c = 0, and then rescale to set a = 3. Starting with (2.1) and introducing
the coordinates cosω = y, ν = 6α we obtain
ds2=1
6(dθ2+ sin2θ + dω2+ sin2ωdν2) +1
9[dψ − cosθdφ − cosωdν]2. (5.1)
If the period of ν is 2π we see that the four-dimensional base metric orthogonal to
∂ψis now the canonical metric on S2× S2, and if we choose the period of ψ to be
4π we recover the metric on T1,1. If the period is taken to be 2π, as in the rest of
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the family with c ?= 0, we get T1,1/Z2. Both of these metrics are well known to be
Sasaki–Einstein.
Case 2: S5
Next, returning to c = 1 and setting a = 1, we introduce the new coordinates
1 − y =3
ds2= dσ2+1
2sin2σ, ψ = ψ′′− β and α = −ψ′′/6 in (2.1) to get
4sin2σ(dθ2+ sin2θdφ2) +1
+1
9
4cos2σsin2σ(dβ + cosθdφ)2
?2
?
dψ′′−3
2sin2σ(dβ + cosθdφ)
.
(5.2)
If the period of β is taken to be 4π the base metric is the Fubini-Study metric on
CP2. If the period of ψ′′is taken to be 6π we get the round metric on S5. If the
period of ψ′′is taken to be 2π we obtain the Lens space S5/Z3. Both of these metrics
are also well known to be Sasaki–Einstein.
We can view this case as a limit a = 1 of our family of solutions. In this case ψ has
period 2π implying β has period 2π. Since y1= −1/2, y2= 1 we take P1= P2= 1/2
and p = q = 1 and hence the period α to be π. This implies that the period of ψ′′
is 6π and thus the five-dimensional space is the orbifold S5/Z2. In other words, the
mildly singular D = 11 spaces that we constructed in section 5.1 of [6] with a = c = 1
are in fact related, after dimensional reduction and T-duality, to S5/Z2.
6 Discussion
We have presented an infinite number of new Sasaki–Einstein metrics of co-homo-
geneity one on S2× S3, both in the quasi-regular and irregular classes. As far as
we know these are the first examples of irregular metrics. As type IIB backgrounds
both classes should provide supergravity duals of a family of N = 1 superconformal
field theories. Let us make some general comments about the field theories. First we
recall that the geometries generically admit an SU(2) ×U(1)2isometry. We also get
additional baryonic U(1)Bflavour symmetry factors from reducing the RR four-form
gauge potential on independent three-cycles in X5. Since here X5 = S2× S3this
gives a single U(1)Bfactor. Thus the continuous global symmetry group of the dual
field theory for all our examples is, modulo discrete identifications,
SU(2) × U(1)2× U(1)B.(6.1)
From the geometry we can also identify the R-symmetry. It is generated by the Sasaki
Killing vector4∂/∂ψ′which is a linear combination (4.5) of ℓ∂/∂α and ∂/∂ψ which
generate the U(1)2isometry group.
4This follows from the fact that the supercharges of the CFT are identified with the Killing
spinors on X5, and these indeed are charged with respect to the canonical Sasaki direction [17].
14
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