Geometries with Killing Spinors and Supersymmetric AdS Solutions

Abstract

The seven and nine dimensional geometries associated with certain classes of supersymmetric $AdS_3$ and $AdS_2$ solutions of type IIB and D=11 supergravity, respectively, have many similarities with Sasaki-Einstein geometry. We further elucidate their properties and also generalise them to higher odd dimensions by introducing a new class of complex geometries in $2n+2$ dimensions, specified by a Riemannian metric, a scalar field and a closed three-form, which admit a particular kind of Killing spinor. In particular, for $n\ge 3$, we show that when the geometry in $2n+2$ dimensions is a cone we obtain a class of geometries in $2n+1$ dimensions, specified by a Riemannian metric, a scalar field and a closed two-form, which includes the seven and nine-dimensional geometries mentioned above when $n=3,4$, respectively. We also consider various ansatz for the geometries and construct infinite classes of explicit examples for all $n$.

Full text

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arXiv:0710.2590v1 [hep-th] 13 Oct 2007
Imperial/TP/2007/JG/03
Geometries with Killing Spinors and
Supersymmetric AdS Solutions
Jerome P. Gauntlett1and Nakwoo Kim2
1Theoretical Physics Group, Blackett Laboratory,
Imperial College, London SW7 2AZ, U.K.
1The Institute for Mathematical Sciences,
Imperial College, London SW7 2PE, U.K.
2Department of Physics and Research Institute of Basic Science,
Kyung Hee University, Seoul 130-701, Korea
Abstract
The seven and nine dimensional geometries associated with certain classes
of supersymmetric AdS3and AdS2solutions of type IIB and D = 11 su-
pergravity, respectively, have many similarities with Sasaki-Einstein ge-
ometry. We further elucidate their properties and also generalise them to
higher odd dimensions by introducing a new class of complex geometries
in 2n+2 dimensions, specified by a Riemannian metric, a scalar field and a
closed three-form, which admit a particular kind of Killing spinor. In par-
ticular, for n ≥ 3, we show that when the geometry in 2n+ 2 dimensions
is a cone we obtain a class of geometries in 2n + 1 dimensions, specified
by a Riemannian metric, a scalar field and a closed two-form, which in-
cludes the seven and nine-dimensional geometries mentioned above when
n = 3,4, respectively. We also consider various ansatz for the geometries
and construct infinite classes of explicit examples for all n.

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1Introduction
An interesting class of geometries in seven and nine dimensions was recently discov-
ered in [1, 2] and further explored in [3]. The geometries are specified by a Riemannian
metric, a scalar field B and a closed two-form F and they admit Killing spinors of a
certain type. The seven dimensional geometries give rise to supersymmetric solutions
of type IIB supergravity with a three dimensional anti-de-Sitter space (AdS3) factor
and these are dual to supersymmetric conformal field theories (SCFTs) with (0,2)
supersymmetry in two-dimensions. Similarly, the nine-dimensional geometries give
rise to supersymmetric solutions of D = 11 supergravity with AdS2factors and these
are dual to superconformal quantum mechanics with two supercharges.
This geometry in 2n+1 dimensions, with n = 3,4, is strikingly similar to Sasaki-
Einstein geometry. In particular, they both have a Killing Reeb vector of constant
norm and define a U(n) or metric contact structure. The Killing vector defines a
natural foliation and in the Sasaki-Einstein case the metric transverse to these orbits
is K¨ ahler and Einstein, while for the geometry considered in [1, 2] it is K¨ ahler and in
addition satisfies
?R −1
2R2+ RijRij= 0,(1.1)
where R and Rijare the Ricci-scalar and Ricci-tensor, respectively, of the transverse
metric and we also demand1that R > 0. Moreover, locally, the whole geometry can
be reconstructed from a local K¨ ahler metric in 2n-dimensions satisfying (1.1).
Recall that a succinct definition of a Sasaki-Einstein metric in 2n+1 dimensions
is that the corresponding cone metric in 2n + 2 dimensions, with base given by
the Sasaki-Einstein metric, is Ricci-flat and K¨ ahler, or equivalently has SU(n + 1)
holonomy. A metric with SU(n+1) holonomy has an SU(n+1) structure, specified by
a fundamental two-form J and an (n+1,0)-form Ω (which together define a metric),
with vanishing intrinsic torsion,
dJ = dΩ = 0.(1.2)
Equivalently it can be characterised as admitting covariantly constant spinors.
In this paper we will determine the analogous statements for the geometries stud-
ied in [1, 2] and furthermore generalise the geometry from n = 3,4 to all n ≥ 3. We
find that the analogue of a metric with SU(n + 1) holonomy is a geometry specified
1Note that when R < 0 we can construct Lorentzian geometries which for n = 3,4 give rise to
solutions of type IIB supergravity and D = 11 supergravity with an S3and S2factor, respectively,
as described in [3].
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by a metric, a scalar field φ and a closed three-form f with an SU(n + 1) structure
(J,Ω) satisfying
d[enφΩ] = 0
d[e2(n−1)φJn] = 0
d[e2φJ] = f(1.3)
and in addition
d?e2(n−3)φ∗2n+2f?= 0, (1.4)
where ∗2n+2is the Hodge dual using the (2n + 2)-dimensional metric defined by the
SU(n + 1) structure. Note that (1.3) imply that the almost complex structure as-
sociated with the SU(n + 1) structure is integrable. We will show that complex
geometries satisfying (1.3) are equivalent to geometries that admit a certain type of
Killing spinor. Furthermore, by analysing the integrability conditions for the Killing
spinor equations, and in addition imposing (1.4), we will also determine the equa-
tions of motion satisfied by the metric, the scalar and the three-form, which are the
analogue of the property of Ricci-flatness in the case of SU(n + 1) holonomy.
If we demand that the geometry in 2n + 2 dimensions satisfying (1.3), (1.4) is a
metric cone and with the scalar and the three-form having a specific scaling:
ds2
2n+2
e−2φ
= dr2+ r2ds2
2n+1
= r
2(n−1)
n−2eB
f= r
n
2−ndr ∧ F,(1.5)
we will show that we obtain a geometry in 2n + 1 dimensions specified by a metric,
with line element ds2
of the co-ordinate r, which for n = 3,4 is precisely equivalent to the geometry of
[1, 2]. We show that for all n the 2n + 1 dimensional metric has a Killing vector
2n+1, a scalar field, B, and a closed two-form, F, all independent
of constant norm and that the transverse metric is K¨ ahler and satisfies (1.1). For
all n, locally, the whole geometry can be reconstructed from a local K¨ ahler metric
in 2n-dimensions satisfying (1.1). We determine the kind of Killing spinors that the
geometry in 2n + 1 dimensions admit and also the kind of equations of motion that
are satisfied by the metric, the scalar field B and the two-form F, which are the
analogue of the Einstein condition in the case of Sasaki-Einstein geometry.
Note that in 2n + 2 dimensions we are generalising the notion of SU(n + 1)
holonomy in the sense that if we set f = φ = 0 in (1.3), (1.4) we clearly return to the
case of SU(n+1) holonomy. However, on the base of the cone in 2n+1 dimensions,
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defined by (1.5), we are not generalising the notion of Sasaki-Einstein geometry: for
example (1.1) is not satisfied for Einstein metrics for n ≥ 3.
When n = 3 an eight dimensional geometry satisfying (1.3), (1.4) gives rise to a
supersymmetric solution of type IIB supergravity which is a warped product of R1,1
with the eight-dimensional geometry. Assuming that the eight dimensional geometry
is a cone as in (1.5) we recover the type IIB AdS3 solutions of [1]. Similarly, as
discussed in [4], when n = 4, a ten-dimensional geometry satisfying (1.3), (1.4) gives
rise to a supersymmetric solution of D = 11 supergravity which is a warped product
of R with the ten-dimensional geometry. If the ten dimensional geometry is a cone
as in (1.5) we recover the D = 11 AdS2solutions of [2].
We do not know of any physical application for the geometry when n ≥ 5. How-
ever, it is possible that the geometries in 2n+1 dimensions, with n ≥ 5, inherit some
properties dictated by physics for the seven and/or nine dimensional cases. This is by
analogy with the Sasaki-Einstein case. Recall that five-dimensional Sasaki-Einstein
geometries, SE5, give rise to AdS5× SE5solutions of type IIB supergravity. These
solutions are dual to N = 1 supersymmetric conformal field theories (SCFTs) in four
spacetime dimensions and such SCFTs exhibit the phenomenon of a-maximisation
[5]. Motivated by this observation, it was proven in [6, 7] that the volume of Sasaki-
Einstein manifolds in any dimension satisfies a variational principle.
Sections 2 and 3 of this paper will be devoted to expanding on the above discussion.
In the subsequent sections we will then consider various ansatz in order to find explicit
examples. In section 4 we will construct explicit examples of the geometries in 2n+1
dimensions. This is a direct analogue of the explicit construction of Sasaki-Einstein
metrics that was carried out in [8, 9] and generalises the analysis of [3] from n = 3,4
to all n ≥ 3. More specifically, we construct explicit local K¨ ahler metrics in 2n-
dimensions satisfying (1.1), by considering local metrics on line bundles over positively
curved K¨ ahler-Einstein manifolds in 2n−2 dimensions. We then argue that for each
choice of K¨ ahler-Einstein manifold these lead to countably infinite classes of smooth,
compact and simply connected globally defined geometries in 2n + 1 dimensions.
Section 5 will present an ansatz for geometries in 2n + 2 dimensions that depend
on a number of functions of one variable. We show that the ansatz includes the
simple case of a cone over a 2n + 1 dimensional geometry with the corresponding
2n-dimensional K¨ ahler manifold satisfying (1.1) being a product of K¨ ahler-Einstein
spaces. Such 2n+1-dimensional geometries, for n = 3,4 were studied in [3]. We also
show that the ansatz includes singular non-compact Calabi-Yau geometries, some of
which were discussed in [10]. For n = 3 the ansatz also incorporates a known solution
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in type IIB supergravity that describes an interpolation between a solution with an
AdS5factor and a solution with an AdS3× H2/Γ factor, where H2is the hyperbolic
plane and Γ is a discrete group of isometries [11, 12]. Similarly, for n = 4 the ansatz
covers a known solution in D = 11 supergravity that describes an interpolation
between a solution with an AdS4factor and a solution with an AdS2× H2/Γ factor
[13, 14].
In section 6 we will consider an ansatz for the geometries in 2n + 1 dimensions
which is inspired by the work of [15]. The resulting system boils down to solving
a differential equation for a function D of three variables, x1,x2,z. For n = 3 the
equation is linear, as in [15], and can be explicitly solved. For n ≥ 4 the equation is
∆D + z
n−4
n−3∂2
zeD= 0,(1.6)
where ∆ = ∂2
in [15]. We don’t know whether the equation is an integrable system for n ≥ 5.
Section 7 briefly concludes.
1+ ∂2
2. For n = 4 this is equivalent to the continuous Toda equation as
2 Geometry in 2n + 2 Dimensions
The geometry in 2n + 2 dimensions that we will be interested in is specified by a
Riemannian metric, g, a scalar field, φ, and a closed three-form, f:
df = 0.(2.1)
We are interested in such geometries that admit a solution to the Killing spinor
equations:
?
γα∇αφ +
?
i
12e−2φfσ1σ2σ3γσ1σ2σ3
i
24e−2φfσ1σ2σ3γασ1σ2σ3
?
?
ǫ = 0
∇α−
ǫ = 0, (2.2)
where ǫ is a Spin(2n + 2) spinor, the gamma-matrices, γα, generate the Clifford
algebra Cliff(2n+2): {γα,γβ} = 2gαβand the indices α,σ,... run from 1 to 2n+2.
We will be particularly interested in geometries admitting such Killing spinors that
in addition satisfy the following equation of motion for f:
d?e2(n−3)φ∗2n+2f?= 0.(2.3)
4