Geometry of four-dimensional Killing spinors
arXiv:0704.0247v2  [hep-th]  15 May 2007
Preprint typeset in JHEP style - HYPER VERSION IFUM-890-FT
UB-ECM-PF-07-05
Geometry of four-dimensional Killing spinors
Sergio L. Cacciatori,ad Marco M. Caldarelli,b Dietmar Klemm,cd Diego S. Mansicd
and Diederik Roeste
a Dipartimento di Scienze Fisiche e Matematiche,
Università dell’Insubria,
Via Valleggio 11, I-22100 Como.
b Departament de F́ısica Fonamental,
Universitat de Barcelona,
Diagonal, 647, 08028 Barcelona, Spain.
c Dipartimento di Fisica dell’Università di Milano,
Via Celoria 16, I-20133 Milano.
d INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano.
e Departament Estructura i Constituents de la Materia,
Facultat de F́ısica, Universitat de Barcelona,
Diagonal, 647, 08028 Barcelona, Spain.
E-mail: sergio.cacciatori@uninsubria.it, caldarelli@ub.edu,
dietmar.klemm@mi.infn.it, diego.mansi@mi.infn.it, droest@ecm.ub.es
Abstract: The supersymmetric solutions of N = 2, D = 4 minimal ungauged and
gauged supergravity are classified according to the fraction of preserved supersymmetry
using spinorial geometry techniques. Subject to a reasonable assumption in the 1/2-
supersymmetric time-like case of the gauged theory, we derive the complete form of all
supersymmetric solutions. This includes a number of new 1/4- and 1/2-supersymmetric
possibilities, like gravitational waves on bubbles of nothing in AdS4.
Keywords: Superstring Vacua, Black Holes, Supergravity Models.
http://arxiv.org/abs/0704.0247v2
mailto:sergio.cacciatori@uninsubria.it, caldarelli@ub.edu, dietmar.klemm@mi.infn.it, diego.mansi@mi.infn.it, droest@ecm.ub.es
mailto:sergio.cacciatori@uninsubria.it, caldarelli@ub.edu, dietmar.klemm@mi.infn.it, diego.mansi@mi.infn.it, droest@ecm.ub.es
http://jhep.sissa.it/stdsearch
Contents
1. Introduction 2
2. G-invariant Killing spinors in 4D 4
2.1 Orbits of Dirac spinors under the gauge group 4
2.2 The ungauged theory 7
2.3 The gauged theory 8
2.4 Generalized holonomy 10
3. Null representative 1 + ae1 11
3.1 Constant Killing spinor, da = 0 12
3.2 Killing spinor with da 6= 0 15
3.3 Half-supersymmetric backgrounds 17
4. Timelike representative 1 + be2 19
4.1 Conditions from the Killing spinor equations 19
4.2 Geometry of spacetime 20
4.3 Half-supersymmetric backgrounds 25
4.4 Time-dependence of second Killing spinor 28
5. Timelike half-supersymmetric examples 33
5.1 Static Killing spinors and b = b(z) 34
5.1.1 AdS2 ×H2 space-time (α = 0) 36
5.1.2 AdS4 space-time (β
2 = 4αγ) 39
5.1.3 The Reissner-Nordström-Taub-NUT-AdS4 family 41
5.2 Harmonic b solutions 43
5.2.1 Deformations of AdS2 ×H2 44
5.2.2 Deformations of AdS4 44
5.2.3 Deformations of Reissner-Nordström-Taub-NUT-AdS4 44
5.3 Imaginary b solutions 45
5.4 Action of the PSL(2,R) group on the imaginary b solutions 49
5.5 Gravitational Chern-Simons system and G0 = ψ− = 0 solutions 50
6. Final remarks 52
A. Spinors and forms 54
– 1 –
B. Spinor bilinears 56
C. The case P ′ = 0 57
D. Half-supersymmetric solutions with G0 = 0 58
1. Introduction
Throughout the history of string and M-theory an important part in many develop-
ments in the subject has been played by supersymmetric solutions of supergravity,
i.e. by backgrounds which admit a number of Killing spinors ǫ which are parallel with
respect to the supercovariant derivative1: Dµǫ = 0. Due to their ubiquitous role it has
long been realised that it would be advantageous to have classifications of all super-
symmetric solutions of a given theory.
For purely gravitational backgrounds the supersymmetric possibilities follow from
the Berger classification of the possible Riemannian holonomies [1] (see [2, 3] for an
extension to the Lorentzian case). However, in the presence of additional force fields
(carried by e. g. scalars, gauge potentials or a cosmological constant) it has proven very
difficult to obtain knowledge of all supersymmetric possibilities.
The reason for the complication in the presence of additional fields lies in the
holonomy of the supercurvature Rµν = D[µDν]. For purely gravitational backgrounds
the holonomy of the supercurvature is generically given by H = Spin(d − 1, 1) in d
dimensions, and hence coincides with the Lorentz group. In such cases the Lorentz
gauge freedom allows one to choose constant Killing spinors. Another simplification is
that if there is one Killing spinor with a specific stability subgroup, i.e. it is invariant
under some Lorentz subgroup, all other spinors with the same stability subgroup are
Killing as well.
For more general solutions including fields other than gravity, the holonomy is
generically extended to a larger group H ⊃ Spin(d− 1, 1). For example, in the present
paper we consider gauged minimal four-dimensional N = 2 supergravity, which has
H = GL(4,C) [4]. In such cases one cannot choose constant Killing spinors nor are all
spinors with the same stability subgroup automatically Killing. For these reasons the
1For the purpose of this discussion we will ignore possible additional Killing spinor equations coming
from the variation of dilatinos and gauginos.
– 2 –
classification of the backgrounds that allow for Killing spinors is more convoluted, or
richer, in such cases. For a long time the only classification available was in ungauged
minimal four-dimensional N = 2 supergravity [5, 6], which has H = SL(2,H).
A new impulse was given to the subject with the introduction of G-structures and
the method of spinor bilinears to solve the Killing spinor equations [7]. In this approach,
space-time forms are constructed as bilinears from a Killing spinor and one analyses
the constraints that these forms imply for the background. Using this framework, a
number of complete classifications [8–10] and many partial results (see e.g. [11–21] for
an incomplete list) have been obtained. By complete we mean that the most general so-
lutions for all possible fractions of supersymmetry have been obtained, while for partial
classifications this is only available for some fractions. Note that the complete classi-
fications mentioned above involve theories with eight supercharges and H = SL(2,H),
and allow for either half- or maximally supersymmetric solutions.
An approach which exploits the linearity of the Killing spinors has been proposed
[22] under the name of spinorial geometry. Its basic ingredients are an explicit oscillator
basis for the spinors in terms of forms and the use of the gauge symmetry to transform
them to a preferred representative of their orbit. In this way one can construct a
linear system for the background fields from any (set of) Killing spinor(s) [23]. This
method has proven fruitful in e.g. the challenging case of IIB supergravity [24–26].
In addition, it has been adjusted to impose ’near-maximal’ supersymmetry and thus
has been used to rule out certain large fractions of supersymmetry [27–30]. Finally, a
complete classification for type I supergravity in ten dimensions has been obtained [32].
In the present paper we would like to address the classification of supersymmetric
solutions in four-dimensional minimal N = 2 supergravity. As will also be reviewed in
section 2, the ungauged case has been classified completely [5,6]. For the gauged case,
the discussion of 1/4 supersymmetry splits up in a time-like and a light-like class (de-
pending on the causal nature of the Killing vector associated to the Killing spinor). The
time-like class is completely specified by a single complex function depending on three
spatial coordinates b = b(z, w, w̄), subject to a second-order differential equation which
can not be solved in general [13]. The light-like class can be given in all generality, and
in addition its restriction to 1/2-BPS solutions has been derived [16]. Furthermore,
there are no backgrounds with 3/4 supersymmetry [29] and AdS4 is the unique possi-
bility with maximal supersymmetry. Therefore the remaining open question concerns
half-supersymmetric backgrounds in the gauged theory2.
In the following, we will first re-analyse the 1/4-supersymmetric backgrounds us-
ing the method of spinorial geometry, and in fact find an additional possibility in the
2The addition of external matter was considered in [31].
– 3 –
light-like case: a half supersymmetric bubble of nothing in AdS4 and its Petrov type
II generalization, a new 1/4 BPS configuration that has the interpretation of grav-
itational waves propagating on the bubble of nothing. This completes the analysis
of the null class in all its generality. Then we will derive the constraints for half-
supersymmetric backgrounds for the timelike class. Subject to a single assumption on
the time-dependence of the second Killing spinor these will be solved in general, up to a
second order ordinary differential equation. The assumption will be justified by solving
the full set of conditions in a number of examples which illustrate the possible spatial
dependence of b. All these cases turn out to have time-dependence of the assumed
form. The different examples are:
• the b = b(z) family of solutions, comprising part of the Reissner-Nordström-Taub-
NUT-AdS4 backgrounds,
• waves on the previous backgrounds with b = b(z, w),
• solutions with b imaginary and their PSL(2,R) transformed counterparts,
• solutions of the dimensionally reduced gravitational Chern-Simons model that
can be embedded in the equations for a timelike Killing spinor [16].
We determine when these backgrounds preserve 1/2 supersymmetry and provide the
explicit Killing spinors. Moreover, in the subcases consisting of AdS4 and AdS2×H2, the
action of the isometries of these backgrounds on the Killing spinors is given explicitly.
The outline of this paper is as follows. In section 2, we discuss the orbits of Killing
spinors and review the known classification results in the theory at hand. In section
3, we go through the complete classification of the null class. In section 4, we discuss
the constraints for 1/4 and 1/2 supersymmetry in the timelike class. We derive the
time-dependence of the second Killing spinor and solve the equations for the case of
linear time-dependence (G0 = 0). A number of examples of the 1/2 BPS timelike class
are provided in section 5. Finally, in section 6 we present our conclusions and outlook.
In appendix A we review our notation and conventions for spinors, while in appendix B
the associated bilinear forms are given. Appendix C deals with the special case P ′ = 0,
to be defined in section 4.4. Finally, in appendix D, we will give the details of the
G0 = 0 case.
2. G-invariant Killing spinors in 4D
2.1 Orbits of Dirac spinors under the gauge group
In order to obtain the possible orbits of Spin(3,1) in the space of Dirac spinors ∆c,
– 4 –
we first consider the most general positive chirality spinor3 a1 + be12 (a, b ∈ C) and
determine its stability subgroup. This is done by solving the infinitesimal equation
αcdΓcd(a1 + be12) = 0 . (2.1)
First of all, notice that a1 + be12 is in the same orbit as 1, which can be seen from
eγΓ13eψΓ12eδΓ13ehΓ02 1 = ei(δ+γ)eh cosψ 1 + ei(δ−γ)eh sinψ e12 .
This means that we can set a = 1, b = 0 in (2.1), which implies then α02 = α13 = 0,
α01 = −α12, α03 = α23. The stability subgroup of 1 is thus generated by
X = Γ01 − Γ12 , Y = Γ03 + Γ23 . (2.2)
One easily verifies that X2 = Y 2 = XY = 0, and thus exp(µX + νY ) = 1 + µX + νY ,
so that X, Y generate R2.
Spinors of negative chirality are composed of odd forms, i.e. ae1 + be2. One can
show in a similar way that they are in the same orbit as e1, and the stability subgroup
is again R2, with the above generators X, Y .
For definiteness and without loss of generality we will always assume that the first
Killing spinor has a non-vanishing positive chirality component, and use (part of) the
Lorentz symmetry to bring this to the form 1. Hence we can write a general spinor as
1 + ae1 + be2. Now act with the stability subgroup of 1 to bring ae1 + be2 to a special
form:
(1 + µX + νY )(1 + ae1 + be2) = 1 + be2 + [a + 2b(ν − iµ)]e1 .
In the case b = 0 this spinor is invariant, so the representative is 1+ ae1, with isotropy
group R2. If b 6= 0, one can bring the spinor to the form 1+ be2, with isotropy group I.
The representatives4 together with the stability subgroups are summarized in table 1.
In the ungauged theory, we therefore can have the following G-invariant Killing
spinors. The R2-invariant Killing spinors are spanned by 1 and e1 and there can be up
to four of these. The I-invariant Killing spinors are spanned by all four basis elements
and there can be up to eight of these. In the first two case, the vector Va bilinear in the
spinor ǫ is lightlike, whereas in the last case it is timelike, see table 1. The existence of
a globally defined Killing spinor ǫ, with isotropy group G ∈ Spin(3,1), gives rise to a
G-structure. This means that we have an R2-structure in the null case and an identity
structure in the timelike case.
3Our conventions for spinors and their description in terms of forms can be found in appendix A.
4Note the difference in form compared to the Killing spinors of the corresponding theories in five
and six dimensions: in six dimensions these can be chosen constant [9] while in five dimensions they
are constant up to an overall function [28]. In four dimensions such a choice is generically not possible.
– 5 –
In U(1) gauged supergravity, the local Spin(3,1) invariance is actually enhanced
to Spin(3,1) × U(1). Thus, in order to obtain the stability subgroup, one determines
the Lorentz transformations that leave a spinor invariant up to an arbitrary phase
factor, which can then be gauged away using the additional U(1) symmetry. For the
representative 1, one gets in this way an isotropy group generated by X, Y and Γ13
obeying
[Γ13, X ] = −2Y , [Γ13, Y ] = 2X , [X, Y ] = 0 ,
i. e. G ∼= U(1)⋉R2. For ǫ = 1 + ae1 with a 6= 0, the stability subgroup R2 is not
enhanced, whereas the I of the representative 1+ be2 is promoted to U(1) generated by
Γ13 = iΓ•̄•. The Lorentz transformation matrix aAB corresponding to Λ = exp(iψΓ•̄•) ∈
U(1), with ΛΓBΛ
−1 = aABΓA, has nonvanishing components
a+− = a−+ = 1 , a••̄ = e
2iψ , a•̄• = e
−2iψ . (2.3)
Finally, notice that in U(1) gauged supergravity one can choose the function a in 1+ae1
real and positive: Write a = R exp(2iδ), use
eδΓ13(1 + ae1) = e
iδ1 + e−iδae1 = e
iδ(1 +Re1) ,
and gauge away the phase factor exp(iδ) using the electromagnetic U(1).
ǫ G ⊂ Spin(3,1) G ⊂ Spin(3,1) × U(1) Va = D(ǫ,Γaǫ)
1 R2 U(1)⋉R2 (1, 0,−1, 0)
1 + ae1 R
2 (a ∈ R) (1 + |a|2, 0,−1− |a|2, 0)
1 + be2 I U(1) (1 + |b|2, 0,−1 + |b|2, 0)
Table 1: The representatives ǫ of the orbits of Dirac spinors and their stability subgroups G
under the gauge groups Spin(3,1) and Spin(3,1) × U(1) in the ungauged and gauged theories,
respectively. The number of orbits is the same in both theories, the only difference lies in the
stability subgroups and the fact that a is real in the gauged theory. In the last column we
give the vectors constructed from the spinors.
In the gauged theory the classification of G-invariant spinors is therefore slightly
more complicated. There can be at most two U(1)⋉R2-invariant Killing spinors,
spanned by 1. The four R2-invariant spinors are spanned by 1 and e1. Then there
are the U(1)-invariant spinors, spanned by 1 and e2. Finally, for generic enough Killing
spinors, one does not fall in any of the above classes and the common stability subgroup
is I. Note that in the gauged theory the presence of G-invariant Killing spinors will
in general not lead to a G-structure on the manifold but to stronger conditions. The
– 6 –
structure group is in fact reduced to the intersection of G with Spin(3,1), and hence is
equal to the stability subgroup in the ungauged theory.
We will now consider the possible supersymmetric solutions to the equation Dµǫ =
0 in various sectors of N = 2, D = 4 in terms of the stability subgroup G of the Killing
spinors.
2.2 The ungauged theory
The supercovariant derivative of ungauged minimal N = 2 supergravity in four dimen-
sions reads
Dµ = ∂µ +
ωabµ Γab +
FabΓabΓµ . (2.4)
As mentioned in the introduction, a first point to notice is that there is no complex
conjugation on the Killing spinor. Therefore, the number of supersymmetries that are
preserved is always even: if ǫ is Killing, then so is iǫ.
First consider purely gravitational solutions with F = 0. In this case the superco-
variant connection truncates to the Levi-Civita connection and has Spin(3,1) holonomy.
This implies the following. If ǫ is Killing, then so are5 Γ3∗ǫ and Γ012∗ǫ (where ∗ denotes
complex conjugation). Together, the operations i, Γ3∗ and Γ012∗ generate four linearly
independent Killing spinors from any null spinor ǫ = 1 or ǫ = 1+ae1 and eight from any
time-like spinor ǫ = 1+ be2. This illustrates the general statement in the introduction:
if the gauge group equals the holonomy, as in this case, then there is only one possible
number of Killing spinors for every stability subgroup. Therefore there are only two
classes of supersymmetric solutions, which are listed in table 2, and which consist of
the gravitational wave and Minkowski space-time, respectively.
G = \ N = 4 8
Table 2: Gravitational solutions with G-invariant Killing spinors in the ungauged theory.
Now let us also allow for fluxes F . The supercovariant connection no longer equals
the Levi-Civita connection due to the flux term. In particular, this implies that Γ012∗ no
longer commutes with Dµ. However, this does still hold for the other operation: Γ3 ∗ ǫ
is Killing provided ǫ is. The combined operations of i and Γ3∗ generate four linearly
5These operations anti-commute and commute with the Γ-matrices, respectively.
– 7 –
independent spinors from any null or time-like spinor. Thus the number of supersym-
metries is always N = 4p, as illustrated in table 3. Indeed the generalised holonomy of
the supercovariant connection in the ungauged case is SL(2,H) [4], consistent with the
supersymmetries coming in quadruplets.
G = \ N = 4 8
Table 3: General solutions with G-invariant Killing spinors in the ungauged theory.
The half-supersymmetric solution have been classified by Tod [5] and consist of the
plane wave and the Israel-Wilson-Perjes metric, respectively. The maximally supersym-
metric solutions are AdS2 × S2 and its Penrose limits, the Hpp wave and Minkowski
space-time [6].
2.3 The gauged theory
The supercovariant derivative of gauged minimalN = 2 supergravity in four dimensions
reads
Dµ = ∂µ +
ωabµ Γab − iℓ−1Aµ + 12ℓ
−1Γµ +
FabΓabΓµ . (2.5)
Due to the gauging the structure of Γ-matrices is richer, but there still is no complex
conjugation on the Killing spinor. Therefore, the number of supersymmetries that are
preserved is always even: if ǫ is Killing, then so is iǫ.
Again, we first consider the purely gravitational solutions. In this case the super-
covariant derivative has SO(3,2) holonomy. The operation Γ012∗ commutes with Dµ
and therefore generates additional Killing spinors. Together, the operations i and Γ012∗
generate four linearly independent Killing spinors from generic null or time-like spinors.
The exception is the null spinor ǫ = 1+e1, in which case ǫ and Γ012∗ are linearly depen-
dent, and hence allows for two instead of four Killing spinors. The possibilities allowed
for by this analysis of the supercovariant derivative can be found in table 4.
However, although all these entries are allowed for by the spinor orbit structure
and the crude analysis of the supercurvature above, not all of them have an actual
field theoretic realisation in supergravity. In other words, there are no solutions to the
Killing spinor equations for all of the above sets of Killing spinors. The lightlike cases
were considered in [16]: The 1/4-BPS case is the Lobatchevski wave while imposing
more supersymmetries leads to the maximally supersymmetric AdS4 solution (with
– 8 –
G = \ N = 2 4 6 8
U(1)⋉ R2 × × × ×
2 √ ◦ × ×
U(1) × ◦ × ×
I × ◦ ◦
Table 4: Gravitational solutions with G-invariant Killing spinors in the gauged theory. Check
marks indicate entries with actual solutions, while circles stand for allowed entries which are
not realized.
G=1). The N = 4 and G = R2 entry is thus effectively empty. In particular, this
implies that imposing a single Killing spinor 1 + ae1 with a 6= 1 leads to AdS4. Also
note that the N = 6 and G = 1 entry must be empty since any time-like spinor plus
1+e1 leads to maximal supersymmetry, while all other Killing spinors come in groups of
four. The only remaining entries are N = 4 and G = U(1) or G = I. Using the results
of [13,16], it is straightforward to show that in these purely gravitational timelike cases
the geometry is given by
ds2 = −z
2 + n2
(dt− 2n cosh θdφ)2 + ℓ
z2 + n2
+ (z2 + n2)(dθ2 + sinh2 θdφ2) ,
where n = ±ℓ/2. But this is simply AdS4 written as a line bundle over a three-
dimensional base manifold, so both N = 4 entries are empty as well. We conclude that
there are no 1/2-supersymmetric gravitational solutions in the gauged theory, only the
1/4-supersymmetric Lobatchevski waves and maximally supersymmetric AdS4.
We now come to the general supersymmetric solutions in the gauged case. Due
to the gauging and flux terms, neither Γ012∗ nor Γ3∗ commute with Dµ. Therefore
we have the cases as listed in table 5. The supercovariant connection in the gauged
case has generalized holonomy GL(4,C) [4], again consistent with the supersymmetries
coming in doublets.
The 1/4-BPS solutions with G = R2 and G = U(1) were derived in [13], and we
will show there is no solution with G = U(1)⋉R2. In addition, it was shown in [16]
that any additional supersymmetries in the null case are always timelike, i.e. end up
in the N = 4 and G = 1 entry. Again, the N = 4 and G = R2 entry is empty. It
would be interesting to see if there is a nice explanation for this. In addition, the
maximally supersymmetric case is always AdS4. Recently, it has been shown in [29]
that the N = 6 and G = 1 entry is empty as well, because imposing three complex
Killing spinors implies that the spacetime is AdS4 and thus maximally supersymmetric.
– 9 –
G = \ N = 2 4 6 8
U(1)⋉ R2 ◦ × × ×
2 √ ◦ × ×
Table 5: General solutions with G-invariant Killing spinors in the gauged theory. Check
marks indicate entries with actual solutions, while circles stand for allowed entries which are
not realized.
The most general 1/2-BPS solution in the timelike case remains an open issue and will
be studied in this paper.
2.4 Generalized holonomy
In minimal gauged supergravity theories with eight supercharges, the generalized holon-
omy group for vacua preserving N supersymmetries, whereN = 0, 2, 4, 6, 8, is GL(8−N
2 [4]. To see this, assume that there exists a Killing spinor ǫ1. By a local
GL(4,C) transformation, ǫ1 can be brought to the form ǫ1 = (1, 0, 0, 0)
T . This is
annihilated by matrices of the form
that generate the affine group A(3,C) ∼= GL(3,C)⋉C3. Now impose a second Killing
spinor ǫ2 = (ǫ
2, ǫ2)
T . Acting with the stability subgroup of ǫ1 yields
eAǫ2 =
ǫ02 + b
, where bT = aTA−1(eA − 1) .
We can choose A ∈ gl(3,C) such that eAǫ2 = (1, 0, 0)T , and b such that ǫ02 + bT ǫ2 = 0.
This means that the stability subgroup of ǫ1 can be used to bring ǫ2 to the form
ǫ2 = (0, 1, 0, 0). The subgroup of A(3,C) that stabilizes also ǫ2 consists of the matrices
1 0 b2 b3
0 1 B12 B13
0 0 B22 B23
0 0 B32 B33
∈ GL(2,C)⋉ 2C2 .
Finally, imposing a third Killing spinor yields GL(1,C) ⋉ 3C as maximal generalized
holonomy group, which is however not realized in N = 2, D = 4 minimal gauged
– 10 –
supergravity [16, 29]. It would be interesting to better understand why such preons
actually do not exist. In section 4.3, we explicitely compute the generalized holonomy
group for N = 2, D = 4 minimal gauged supergravity in the case N = 2 and show that
it is indeed contained in A(3,C), supporting thus the classification scheme of [4].
3. Null representative 1 + ae1
In this section we will analyse the conditions coming from a single null Killing spinor.
As we saw in section 2.1, there are two orbits of such spinors, one with representative
ǫ = 1 and stability subgroup G = U(1)⋉R2 and one with ǫ = 1 + ae1 and G = R
Owing to local U(1) gauge invariance, it is always possible to choose the function a real
and positive, so in the following we set a = eχ, χ ∈ R. The Killing spinor equations
become
E •̄ − 2iF+•̄E−
= 0 ,
E •̄ + 2iF+•̄E−
= 0 ,
ω−• +
2iF−•E •̄ +
= 0 ,
ω−• +
−2iF−•E •̄ +
= 0 , (3.1)
where φ ≡ F+− + F •̄• and Ω ≡ ω+− + ω•̄•.
The conditions for the special U(1)⋉R2-orbit with ǫ = 1 can be obtained as the
singular limit χ → −∞ of the above equations. Note however that, in this limit, the
second line implies the constraint ℓ−1−iφ = 0, while the fourth line leads to ℓ−1+iφ = 0.
Clearly, for ℓ−1 6= 0 this does not allow for a solution. Hence, in the gauged theory,
there are no backgrounds with U(1)⋉R2-invariant Killing spinors.
The only null possibility is therefore given by the R2-invariant Killing spinor ǫ =
1 + eχe1. We will now analyse the above conditions for the generic case with χ finite.
In fact, we will furthermore assume it is positive. This does not constitute any loss of
generality since one can flip the sign of χ by changing chirality (a spinor 1 + eχe1 with
χ negative is gauge equivalent to a spinor e1 + e
χ̃1 with χ̃ = −χ positive), and hence
the resulting background will not depend on this sign.
From the last two equations one obtains the constraints
F−• = F−•̄ = 0 , φ = − i
tanhχ (3.2)
– 11 –
on the field strength, as well as
ω−• = ω−•̄ = − 1√
2ℓ coshχ
E− (3.3)
for the spin connection. (3.2) implies F+− = 0 and F •̄• = − i
tanhχ. The first two
equations of (3.1) yield then
ω+− = 2eχH3E
− − 1
coshχ
ω•̄• = 2i sinhχH1E
cosh 2χ
coshχ
A = −ℓ coshχH1E− − sinhχE3 ,
dχ = −2 coshχH3E− +
sinhχE1 , (3.4)
where E1 = (E• + E •̄)/
2, iE3 = (E• − E •̄)/
2, and we defined
F+• + F+•̄√
= H1 ,
F+• − F+•̄√
= iH3 .
In order to proceed, we distinguish two subcases, namely dχ = 0 and dχ 6= 0.
3.1 Constant Killing spinor, da = 0
If a and hence χ are constant, eqn. (3.4) implies χ = H3 = 0. Next we impose vanishing
torsion. The torsion two-form reads
T− = dE− +
E1 ∧ E− ,
T+ = dE+ − E1 ∧
ω+1 +
+ ω+3 ∧ E3 ,
T 1 = dE1 + E− ∧
ω+1 +
T 3 = dE3 +
E1 ∧ E3 − ω+3 ∧ E− . (3.5)
From T− = 0 one gets E−∧dE− = 0, so by Fröbenius’ theorem there exist two functions
η and u such that locally
E− = ηdu .
Plugging this into T− = 0 yields
d log η +
∧ du = 0 ,
– 12 –
so that there exists a function ξ such that
E1 = − ℓ
dη + ξdu .
The gauge field and its field strength can now be written as
A = −ℓηH1du , F =
H1dη ∧ du ,
and the Bianchi identity F = dA implies
dH1 +
H1d log η
∧ du = 0 .
This means that H1η
3/2 can depend only on u,
3/2 = −ϕ
where the prefactor and the derivative were chosen in order to conform with the notation
of [13]. Let us define a new coordinate x = −η−1/2, so that E1 = ℓ
dx+ξdu, E− = x−2du
A = −xϕ′(u)du . (3.6)
One can now use part of the residual gauge freedom, given by the stability subgroup
2 of the null spinor 1 + ae1, in order to simplify E
1. To this end, consider an R2
transformation with group element
Λ = 1 + µX + νY ,
where X and Y are given in (2.2). Defining α = µ+ iν, this can also be written as
Λ = 1 + αΓ+• + ᾱΓ+•̄ . (3.7)
Given the ordering A,B = +,−, •, •̄, the Lorentz transformation matrix aAB corre-
sponding to Λ ∈ R2 ⊆ Spin(3,1) reads
aAB =
0 1 0 0
1 −4|α|2 2ᾱ 2α
0 −2ᾱ 0 1
0 −2α 1 0
. (3.8)
The transformed vielbein αEA = aABE
B is thus given by
αE• = E• − 2αE− , αE1 = E1 −
2 (α + ᾱ)E− ,
αE •̄ = E •̄ − 2ᾱE− , αE3 = E3 +
2i (α− ᾱ)E− ,
αE− = E− , αE+ = E+ + 2ᾱE• + 2αE •̄ − 4|α|2E− . (3.9)
– 13 –
Choosing α + ᾱ = ξx2/
2, we can eliminate E1u, so one can set ξ = 0 without loss
of generality. Note that this still leaves a residual gauge freedom associated to the
imaginary part of α, which will be used below.
From dT 3 = 0 we get d(ω+3/x) ∧ du = 0, and thus there exist two functions β, β̃
such that
ω+3 = −xdβ + β̃du .
Plugging this into T 3 = 0 yields d(xE3 + βdu) = 0, which is solved by
E3 = − ℓ
dy + βdu , (3.10)
where y denotes some function that we shall use as a coordinate. Using the remaining
gauge freedom (3.8) with Imα = −βx2/2
2 allows to set also β = 0. The equation
T 1 = 0 tells us that ω+1 + E+/ℓ = γdu for some function γ. Using this together with
T+ = 0, one shows that
E− ∧ E+
E− ∧ E+
which means that the surface described by E− and E+ is integrable, so that
E+ = ℓ2
du+ hdV , (3.11)
for some functions G, h, V . The metric becomes then
ds2 = 2E−E+ +
Gdu2 + 2h
dudV + dx2 + dy2
. (3.12)
Finally, the equation T+ = 0 implies
∂xh = ∂yh = 0 , ∂V G =
∂uh , (3.13)
∂xG , β̃ = −
∂yG .
h can be eliminated by introducing a new coordinate v(u, V ) with ∂V v = h/ℓ
2 and
shifting G → G + 2∂uv, which leads to
ds2 =
Gdu2 + 2dudv + dx2 + dy2
. (3.14)
Note that, due to (3.13), G is independent of v, therefore ∂v is a Killing vector. One
easily verifies that it coincides with the Killing vector constructed from the Killing
spinor as − ℓ2
D(ǫ,Γµǫ).
– 14 –
All that remains is to impose the Maxwell and Einstein equations. One finds that
the former are automatically satisfied by the gauge potential (3.6). The same holds for
the Einstein equations, except for the uu-component, which gives the Siklos equation
with sources
∆G − 2
∂xG = −
ϕ′(u)2 . (3.15)
This family of solutions enjoys a large group of diffeomorphisms which leave the solution
invariant in form but change the function G. This is the Siklos-Virasoro invariance,
discussed in [16, 33]. In conclusion, the geometry of solutions admitting the constant
null spinor 1 + e1 is given by the Lobachevski waves with metric (3.14) and gauge field
(3.6), where G satisfies (3.15) and ϕ(u) is arbitrary. This coincides exactly with the
results of [13], where it was shown moreover that there is a second covariantly constant
spinor iff the wave profiles G and ϕ have the form
Gα(x, y, u) = −
+ 2αx3 − α2ℓ2(x2 + y2) , ϕ(u) = u , (3.16)
up to Siklos-Virasoro transformation, with α ∈ R constant. In this case, the solution
does also belong to the timelike class [13]. While the α 6= 0 solution only has the
obvious Killing vectors ∂v and ∂y, the special α = 0 case is maximally symmetric with
a five-dimensional isometry group.
3.2 Killing spinor with da 6= 0
If da and hence also dχ do not vanish, one can use the R2 stability subgroup of the
spinor 1 + eχe1 to eliminate the fluxes F+• and F+•̄. To see this, observe that under
an R2 transformation (3.8),
αF+• = F+• − 2iα
tanhχ , αF •̄• = F •̄• ,
so by choosing α = − iℓ
F+• cothχ one can achieve αF+• = 0. Note that this would not
be possible if χ = 0. With this gauge fixing, one has
sinhχE1 , A = − sinhχE3 , F = −1
tanhχE1 ∧ E3 . (3.17)
Next we impose vanishing torsion. Using (3.17), one easily shows that T− = 0 leads to
e2χ − 1
= 0 ,
and therefore one can introduce a function u with
e2χ − 1
E− = du . (3.18)
– 15 –
Before we come to the other torsion components, let us consider the Bianchi identity
and the Maxwell equations. The gauge field strength reads
F = dχ
sinh 2χ
Requiring it to be equal to dA implies that A/
tanhχ is closed, so that locally
tanhχdΨ . (3.19)
Note that the functions χ, u and Ψ must be independent, because otherwise E1, E−
and E3 would not be linearly independent. We can thus use these three functions as
coordinates.
Using
∗F = −1
tanhχE− ∧ E+ ,
the Maxwell equations d∗F = 0 imply
E− ∧ E+
sinh 2χ
E− ∧ E+
= 0 .
By Fröbenius’ theorem and (3.18), E+ can thus be written as
du+ hdV ,
where K̃, h and V are some functions, and we can use V as the remaining coordinate.
Substituing E+ into the Maxwell equations one obtains a constraint on the function h,
e2χ + 1
∧ du ∧ dV = 0 ,
and hence
h = h0(u, V )
e2χ + 1
In what follows, we define K = K̃/(e2χ + 1) and use ω+1 = (ω+• + ω+•̄)/
2, ω+3 =
(ω+• − ω+•̄)/
2i. We now come to the remaining torsion components. From T 3 = 0
and T 1 = 0 one obtains respectively
ω+3 = AE− , ω+1 = − E
ℓ coshχ
+BE− ,
where A and B are some functions to be determined. Finally, T+ = 0 yields
∂VK = 2∂uh0 , A = −
e4χ − 1
) sinhχ√
tanhχ
∂ΨK , B =
e4χ − 1
sinhχ∂χK .
– 16 –
The line element is given by
ds2 = 2E−E+ +
= cothχ
Kdu2 + 2h0dudV
ℓ2dχ2
4 sinh2 χ
sinhχ coshχ
. (3.20)
As before, one can eliminate h0 by introducing a new coordinate v(u, V ) with ∂V v = h0
and shifting K → K + 2∂uv, whereupon the metric becomes
ds2 = cothχ
Kdu2 + 2dudv
ℓ2dχ2
4 sinh2 χ
sinhχ coshχ
. (3.21)
Notice that, owing to (3.20), K is independent of v, therefore ∂v is a Killing vector.
It coincides with the Killing vector −
2D(ǫ,Γµǫ) constructed from the Killing spinor.
All that remains now is to impose Einstein’s equations. One finds that they are all
satisfied except for the uu component, which yields again a Siklos-type equation for K,
∂2ΨK + 4 tanhχ∂2χK −
cosh2 χ
∂χK = 0 . (3.22)
In conclusion, the bosonic fields for a configuration admitting a null Killing spinor
with dχ 6= 0 are given by (3.19) and (3.21), with K satisfying (3.22)6. As we will
discuss in section 5.3, the K = 0 solution is of Petrov type D and represents a bubble
of nothing in anti-De Sitter space-time. When K 6= 0, the metric becomes of Petrov
type II and the Weyl scalar signalling the presence of gravitational radiation acquires
a non-vanishing value. Hence the general solution represents a gravitational wave on a
bubble of nothing. To our knowledge these solutions have not featured in the literature
before.
3.3 Half-supersymmetric backgrounds
In the previous subsections we have addressed the conditions for preserving one null
Killing spinor of the form ǫ1 = 1 or ǫ1 = 1 + e
χe1. It is natural to enquire about the
possibility of these backgrounds admitting an additional Killing spinor with the same
2 stability subgroup, i.e. of the form ǫ2 = c01+ c1e1. Using the fact that ǫ1 is Killing,
the second Killing spinor equation Dµǫ2 = 0 can then be rewritten as
(c0 − c1)Dµ1 + ∂µc01 + ∂µc1e1 = 0 , (3.23)
6This solution escaped a majority of the present authors in [13]. The reason for this is that
equ. (4.32) of [13] is not correct; it must be R+−ij = 0, which yields no information on the constant
κ. Thus, in addition to the solutions with κ = 0 found in [13] (the Lobachevski waves), there are also
the κ = 1 solutions, which are exactly the ones found here with dχ 6= 0.
– 17 –
in the U(1)⋉R2 case and
(c0 − c1e−χ)Dµ1 + ∂µc01 + (∂µc1 − c1∂µχ)e1 = 0 , (3.24)
in the R2 case. Furthermore, we can assume that (c0 − c1) 6= 0 and (c0 − c1e−χ) 6= 0 in
the two cases, respectively, since otherwise the second Killing spinor would be linearly
dependent on the first and there would not be any additional constraints. Hence the e2
and e12 components of Dµ1 have to vanish separately. In particular, this implies that
ω−• = 0 (as can be seen from the third line of (3.1) in the singular limit χ → −∞).
However, this is clearly incompatible with (3.3). We conclude that, in the gauged
theory, there are no backgrounds with four R2-invariant Killing spinors. In other words,
there are no half-supersymmetric backgrounds with an R2-structure. This is unlike
the ungauged case, where the half-supersymmetric gravitational waves provide such
solutions.
Therefore, the only possibility to augment the supersymmetry of the null solutions
above is to add a Killing spinor which breaks the R2 invariance, i.e. with a non-vanishing
e2 and/or e12 component. From a linear combination of the first and second Killing
spinor one can then always construct a time-like Killing spinor, and hence this brings
us to the next section. For the convenience of the reader, we will already summarise
how to restrict the 1/4-supersymmetric null solutions to allow for a time-like Killing
spinor as well.
For the case with constant null Killing spinors, dχ = 0, the restriction was al-
ready discussed in [13] and is given in (3.16). For the other case, with dχ 6= 0, it is
straightforward to show that the solution (3.19), (3.21) admits a second Killing spinor
iff ∂χG = ∂ΨG = 0, so that G depends only on u. By a simple diffeomorphism one can
then set G = 0. The general solution to the Killing spinor equations reads in this case
ǫ = λ1(1 + e
χe1) +
e4χ − 1
(e2 + e
χe1 ∧ e2) , (3.25)
where λ1,2 ∈ C are constants. The invariants constructed from ǫ, as defined in appendix
B, are
2 cothχ(|λ2|2dv − |λ1|2du)−
sinh 2χ
(λ2λ̄1 − λ̄2λ1) dΨ ,
B = −
2(|λ1|2du+ |λ2|2dv) +
e4χ − 1 sinhχ
(λ̄1λ2 + λ1λ̄2) dχ ,
f = i(λ1λ̄2 − λ̄1λ2)
tanhχ , g = (λ̄1λ2 + λ1λ̄2)
cothχ .
The norm of the Killing vector V is given by
V 2 = − 2
sinh 2χ
(λ̄1λ2 + λ1λ̄2)
2 − 4|λ1λ2|2 tanhχ .
– 18 –
Since χ > 0, this is negative unless λ1 = 0 or λ2 = 0, so indeed the solution (3.19),
(3.21) with G = 0 must belong also to the timelike class. It turns out that it is identical
to the bubble of nothing of section 5.3 with imaginary b and L < 0. The coordinate
transformation
2A2(t− Ly)− z
, v = −
2A2(t− Ly)− z
Ψ = −2A2t , χ = artanhX
(3.26)
with A8 = −1/4L brings the metric (3.21) (with G = 0) to (5.60), and the field strength
of (3.19) to (5.61). Note that, in the new coordinates, the above invariants become
V = ∂t as a vector, and B = dz, in agreement with section 4.2.
4. Timelike representative 1 + be2
We will now turn to the timelike case and first recover the general 1/4-BPS solutions
[13]. Afterwards we will study the conditions for 1/2 supersymmetry. This will complete
the classification since we already know that no 3/4-supersymmetric solutions can arise
and AdS4 is the unique maximally supersymmetric possibility.
4.1 Conditions from the Killing spinor equations
Acting with the supercovariant derivative (2.5) on the representative 1 + be2 yields the
linear system
ω•̄•+ −
ω+−+ −
bA+ = 0 ,
ω•̄•+ +
ω+−+ −
F •̄• + ib√
F+− = 0 ,
ω•−+ + i
2bF•− = ω•++ = 0 , (4.1)
ω•̄•− −
ω+−− −
bA− +
F •̄• − i√
F+− = 0 ,
ω•̄•− +
ω+−− −
A− = 0 ,
b ω•+− + i
2F•+ = ω•−− = 0 , (4.2)
– 19 –
ω•̄•• −
ω+−• −
bA• − i
2F •̄− = 0 ,
ω•̄•• +
ω+−• −
A• − i
2bF •̄+ = 0 ,
ω•−• +
− ib√
F •̄• − ib√
F+− = 0 ,
b ω•+• +
F •̄• + i√
F+− = 0 , (4.3)
∂•̄b+
ω•̄••̄ −
ω+−•̄ −
bA•̄ = 0 ,
ω•̄••̄ +
ω+−•̄ −
A•̄ = 0 ,
ω•−•̄ = b ω
•̄ = 0 . (4.4)
From eqns. (4.1) - (4.4) one obtains the gauge potential and the fluxes in terms of the
spin connection and the function b,
− ∂+b
− ω•̄•+
, A− =
ω••̄− , A• =
(ω••̄• + ω
• ) ,
F+− = i√
(b ω•+• − b−1ω•−• ) , F•+ =
ω+−•̄ ,
F••̄ = i√
(b ω•+• + b
−1ω•−• ) +
, F•− = i
ω•−+ . (4.5)
Furthermore, the system (4.1) - (4.4) determines almost all components of the spin
connection (with the exception of ω••̄) in terms of the function b and its spacetime
derivatives,
ω+−+ =
, ω+−− = 0 , ω
ω+•+ = ω
•̄ = 0 , ω
− = −
, ω+•• =
ω−•+ = −b ∂•̄b̄ , ω−•− = ω−••̄ = 0 , ω−•• =
. (4.6)
In what follows, we assume b 6= 0. One easily shows that b = 0 leads to ℓ−1 = 0, so
this case appears only in ungauged supergravity.
4.2 Geometry of spacetime
In order to obtain the spacetime geometry, we consider the spinor bilinears
Vµ = D(ǫ,Γµǫ) , Bµ = D(ǫ,Γ5Γµǫ) ,
– 20 –
whose nonvanishing components are
2 b̄b , V− = −
2 , B+ =
2 b̄b , B− =
As V 2 = −4b̄b = −B2, V is timelike and B is spacelike. Using eqns. (4.1) - (4.4), it is
straightforward to show that V is Killing and B is closed, i. e. ,
∂AVB + ∂BVA − ωCB|AVC − ωCA|BVC = 0 ,
∂ABB − ∂BBA − ωCB|ABC + ωCA|BBC = 0 .
There exists thus a function z such that B = dz locally. Let us choose coordinates
(t, z, xi) such that V = ∂t and i = 1, 2. The metric will then be independent of t. Note
also that the system (4.1) - (4.4) yields
∂tb =
2 (|b|2∂− − ∂+)b = 0 ,
so b is time-independent as well. In terms of the vierbein EAµ the metric is given
ds2 = 2E+E− + 2E•E •̄ , (4.7)
where
E+µ =
Bµ + Vµ
2|b|2
, E−µ =
Bµ − Vµ
From V 2 = −4|b|2 and V = ∂t as a vector we get Vt = −4|b|2, so that V = −4|b|2(dt+σ)
as a one-form, with σt = 0. Furthermore, V
• = 0 yields E•t = 0, and thus
E• = E•zdz + E
The component E•z can be eliminated by a diffeomorphism
xi = xi(x′j , z) ,
= −EIz , I = •, •̄ .
As the matrix EIi is invertible
7, one can always solve for ∂xi/∂z. Note that the metric
is invariant under
t→ t+ χ(xi, z) , σ → σ − dχ ,
7One has det(EIi ) = − det(EAµ ), and the latter is always nonzero.
– 21 –
where χ(xi, z) denotes an arbitrary function. This second gauge freedom can be used
to eliminate σz. Hence, without loss of generality , we can take σ = σidx
i, and the
metric (4.7) becomes
ds2 = −4|b|2(dt+ σidxi)2 +
4|b|2 + 2E
iE •̄j dx
j . (4.8)
Next one has to impose vanishing torsion,
ν − ∂νEAµ + ωAµBEBν − ωAνBEBµ = 0 .
One finds that some of these equations are already identically satisfied, while the re-
maining ones yield (using the expressions (4.6) for the spin connection) the constraints
∂zσi = −
4|b|2 (E
•̄ − E•iEj•)∂j ln(b/b̄) , (4.9)
∂iσj − ∂jσi = (E•iE •̄j −E•jE •̄i )
∂z ln(b/b̄) +
, (4.10)
ω••̄t = −2|b|2∂z ln(b/b̄) +
− 2b̄
, (4.11)
j − ∂jE•i = (E•iE •̄j −E•jE •̄i )ω••̄•̄ , (4.12)
as well as
∂z + ω
∂z ln(b̄b) +
E•i = 0 . (4.13)
In (4.9), EiI denotes the inverse of E
j . In order to obtain the above equations, one has
to make use of the inverse tetrad
E+ = −
2|b|2∂z , E− =
2|b|2
2 ∂z , E• = E
•(∂i − σi∂t) .
(4.13) can be solved to give
E•i =
|b|Ê
i exp
dz ω••̄z −
, (4.14)
where Ê•i is an integration constant that depends only on the coordinates x
j . At this
point it is convenient to use the residual U(1) gauge freedom of a combined local Lorentz
and gauge transformation to eliminate ω••̄z . This is accomplished by the transformation
(2.3), with
dz ω••̄z .
– 22 –
Note that ψ is real, as it must be. Defining
Φ := − 1
, (4.15)
we have thus
E•i =
|b|Ê
i expΦ . (4.16)
Using (4.16) in (4.12), one gets for the only remaining unknown component ω••̄• of the
spin connection
ω••̄• =
ω̂••̄• − Êi•∂i
|b| exp(−Φ) ,
where ω̂••̄• denotes the spin connection following from the zweibein Ê
In what follows, we shall choose the conformal gauge for the two-metric hij =
ÊIiÊ
j , i. e. ,
hij = e
2ξ[(dx1)2 + (dx2)2] . (4.17)
with ξ depending only on the coordinates xi. Furthermore, we choose an orientation
such that
Ê•i Ê
j − Ê•j Ê •̄i = −ie2ξǫij ,
where ǫ12 = 1. To be concrete, we shall take
(ÊIi ) =
The eqns. (4.9) and (4.10) then simplify to
∂zσi = −
4|b|2 ǫij∂j ln(b/b̄) , (4.18)
∂iσj − ∂jσi = −
|b|2 e
2(Φ+ξ)ǫij
∂z ln(b/b̄) +
. (4.19)
Moreover, one has
ω••̄• = −∂• ln
|b|e−Φ−ξ
. (4.20)
In [13] it has been shown that in the case where the Killing vector constructed from
the Killing spinor is timelike, the Einstein equations follow from the Killing spinor
equations, so all that remains to do at this point is to impose the Bianchi identity and
the Maxwell equations. Using the spin connection (4.6) and (4.11) in (4.5), the gauge
– 23 –
potential and the field strength become
A = i(dt + σ)(b− b̄) + ℓ
ǫij∂j(Φ + ξ) dx
i − iℓ
d ln(b/b̄) ,
F = i(dt + σ) ∧ d (b̄− b) + 1
4|b|2dz ∧ dx
iǫij∂j(b+ b̄)
2|b|2
∂z(b+ b̄) +
e2(Φ+ξ)ǫijdx
i ∧ dxj . (4.21)
The Bianchi identity F = dA yields
∆(Φ + ξ) =
e2(Φ+ξ)
, (4.22)
with ∆ = ∂i∂i denoting the flat space Laplacian in two dimensions. As for the Maxwell
equations,
−gFµν) = 0 ,
the only nontrivial information comes from the t-component, which gives
4e2(Φ+ξ)
b2∂2z
− b̄2∂2z
+ b2∆
− b̄2∆1
= 0 , (4.23)
where we used eqns. (4.18) and (4.19).
Let us now show that the equations (4.22) and (4.23) are actually the same as the
ones in [16]. If we set
F = − 1
, eφ = 2eΦ+ξ , (4.24)
(4.22) yields exactly equation (2.3) of [16]. On the other hand, deriving (4.22) with
respect to z and using (4.15), one obtains
∆A + e2φ
3A∂zA− 3B∂zB + A3 − 3AB2 + ∂2zA
= 0 , (4.25)
where A and B denote the real and imaginary part of F respectively. This can be used
in (4.23) to get
∆B + e2φ
∂2zB + 3B∂zA+ 3A∂zB − B3 + 3A2B
= 0 ,
which, together with (4.25), yields
∆F + e2φ
F 3 + 3F∂zF + ∂
= 0 , (4.26)
i. e. , equation (2.2) of [16]. For a complete identification of the present results with
the ones in [16], one also has to set σ = ω.
– 24 –
In conclusion, the metric of the general 1/4-supersymmetric solution is given by
ds2 = −4|b|2(dt+ σ)2 + 1
4|b|2
dz2 + 4e2(Φ+ξ)dw dw̄
, (4.27)
where b and φ are determined by the system (4.22), (4.23) and w = x1 + ix2 ≡ x+ iy.
The one-form σ follows then from (4.18) and (4.19), and the gauge field strength is given
by (4.21). Note that (4.23) represents also the integrability condition for (4.18), (4.19).
As noted in [16], this system of equations is invariant under PSL(2,R) transformations8.
If we define a new coordinate z′ through the Möbius transformation
αz + β
γz + δ
, (4.28)
with α, β, γ and δ arbitrary real constants satisfying αδ − βγ = 1, then the functions
b̃(z′, xi) and Φ̃(z′, xi) defined by
(γz′ − α)2b −
γz′ − α , e
Φ̃ = (γz′ − α)2 eΦ , (4.29)
solve the system in the new coordinate system (z′, xi), with the function ξ(xi) left
invariant and z seen as a function of z′. This symmetry allows to generate new BPS
solutions from the known ones. Note however that it is only a symmetry of the equations
for 1/4 supersymmetry, and if we apply it to solutions with additional Killing spinors,
it will in general not preserve them, as we shall show explicitely in some examples.
4.3 Half-supersymmetric backgrounds
We now would like to investigate the possibility of adding a second Killing spinor. Since
the first Killing spinor ǫ1 has stability subgroup 1, one cannot use Lorentz transforma-
tions to bring the second spinor to a preferred form. Therefore we use the most general
ǫ2 = c01 + c1e1 + c2e2 + c12e1 ∧ e2 . (4.30)
The corresponding linear system simplifies significantly after inserting the results from
ǫ1. These determine all the fluxes and the spin connection in terms of the functions b,
ξ and their derivatives. First it is convenient to introduce the new basis9
b−1c2 − c0
8It might be of interest to investigate the possible relation between this ’hidden symmetry’ and the
Ehlers group for solutions of four-dimensional vacuum gravity with a Killing vector.
9Note that ǫ1 = (1, 0, 0, 0) in this basis.
– 25 –
in which the Killing spinor equations for ǫ2 read
(∂A +MA)α = 0 , (4.31)
with the connection MA given by
0 −∂+ ln b̄ 0 0
0 ∂+ ln b̄ −∂• ln b ∂• ln b
0 0 b̄−b√
∂+ ln
b̄− ∂+ ln b
0 −|b|2∂•̄ ln b̄ 0 b̄−b√2ℓ −
∂+ ln(b̄b)
0 0 |b|−2∂• ln b̄ −|b|−2∂• ln b̄
0 ∂− ln b −|b|−2∂• ln b̄ |b|−2∂• ln b̄
0 ∂•̄ ln b
b−b̄√
2ℓ|b|2 −
∂− ln(b̄b) 0
0 0 −
− ∂− ln b̄ b−b̄√2ℓ|b|2 +
∂− ln
0 −∂• ln b̄ 0 0
0 ∂• ln(b̄b) 0 0
b̄− ∂+ ln b −∂• ln
|b|e−Φ−ξ
b+ ∂+ ln b̄ 0 −∂• ln
|b|e−Φ−ξ
M•̄ =
0 0 −∂− ln b̄ ∂− ln b̄+
0 0 ∂− ln(b̄b) +
−∂− ln(b̄b)−
0 0 ∂•̄ ln
e−Φ−ξ
−∂•̄ ln b
0 0 −∂•̄ ln b̄ ∂•̄ ln
e−Φ−ξ
Let us first of all consider the simpler possibility of a second Killing spinor of the
form ǫ2 = c01 + c2e2. As discussed in section 2.1, both ǫ1 and ǫ2 are invariant under
the same U(1) symmetry, and hence this case constitutes the G = U(1) case with four
supersymmetries. As can easily be seen from the above Killing spinor equations with
α1 6= 0 and α2 = α12 = 0, this restricts the derivatives of the coefficient b to be
∂−b = −
, ∂+b = −
, ∂•b = ∂•̄b = 0 . (4.32)
Hence this corresponds to ∂zb = −1/ℓ. As will be discussed in section 5.1, this restric-
tion uniquely leads to the half-supersymmetric anti-Nariai space-time. Hence AdS2×H2
is the only possibility for backgrounds with four U(1)-invariant Killing spinors.
In the more general case with α2 and α12 non-vanishing, i.e. with trivial stability
subgroup, the Killing spinor equations do not so readily provide information about b
– 26 –
and one has to resort to their integrability conditions. The first integrability conditions
for the linear system (4.31) are
Nµνα ≡ (∂µMν − ∂νMµ + [Mµ,Mν ])α = 0 , (4.33)
where the matrices Mµ = E
µMA are given by
2(|b|2M− −M+) , Mz =
2|b|2
(M+ + |b|2M−) ,
Mw = σwMt +
eΦ+ξM• , Mw̄ = σw̄Mt +
eΦ+ξM•̄ ,
and we introduced the complex coordinates w = x + iy, w̄ = x − iy. For half-
supersymmetric solutions, the six matrices Nµν must have rank two. (As at least
one Killing spinor exists, namely ǫ1 = (1, 0, 0, 0), we already know that the Nµν can
have at most rank three. Rank one is not possible, because 3/4 BPS solutions cannot
exist [29]. Rank zero corresponds to the maximally supersymmetric case, which implies
that the spacetime geometry is AdS4 [13].) Let us define
Ñµν ≡ SNµνT ,
1 0 0 0
1 1 0 0
0 0 1 0
0 0 0 1
, T =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 1 1
The similarity transformation S corresponds to adding the first line to the second one
and T adds the last column to the third one. This does not alter the rank of Nµν . One
finds
Ñwt =
2b∂∂z b̄+
∂b̄ −2|b|
e−Φ−ξ
∂2b̄+ 1
∂b̄∂b̄
∂z b̄+
∂ ln b̄
0 −2∂(Φ + ξ)∂b̄
2b̄∂∂zb+
∂b −2|b|
e−Φ−ξ
∂2b+ 1
∂ ln b
0 −2∂(Φ + ξ)∂b)
2|b|3e−Φ−ξ∂̄∂ ln b 2b̄∂∂zb
0 −2|b|3eΦ+ξb−2
2∂zb+
∂ ln b
2|b|3e−Φ−ξ∂̄∂ ln b̄ 2b∂∂z b̄− 2ℓ∂b̄0 −2|b|3eΦ+ξ b̄−2
2∂z b̄+
∂z b̄+
∂z b̄+
∂ ln b̄
– 27 –
Ñw̄t =
2b∂̄∂z b̄ −2|b|e−Φ−ξ∂̄∂ ln b̄
∂z b̄+
∂̄ ln b̄
ℓ|b|e
2∂z b̄+
+ 2b|b|b̄e
2∂z b̄+
∂z b̄+
2b̄∂̄∂zb −2|b|e−Φ−ξ∂̄∂ ln b
∂̄ ln b
ℓ|b|e
2∂zb+
+ 2b̄|b|be
2∂zb+
2|b|b̄e−Φ−ξ
∂̄2b+ 1
∂̄b∂̄b 2b̄∂̄∂zb+
0 −2∂̄(Φ + ξ)∂̄b
∂̄ ln b
2|b|be−Φ−ξ
∂̄2b̄+ 1
∂̄b̄∂̄b̄ 2b∂̄∂z b̄− 4ℓ ∂̄b̄0 −2∂̄(Φ + ξ)∂̄b̄
∂z b̄+
∂̄ ln b̄
where ∂ = ∂w, ∂̄ = ∂w̄. The other four integrability conditions give no additional
information, because the lines of the corresponding matrices are proportional to the
lines of Ñwt and Ñw̄t
As the upper right 3× 3 determinant of Ñwt must vanish, we obtain ∂b = 0 or
e−2(Φ+ξ)b̄∂b̄
e−2(Φ+ξ)b∂b
e−2(Φ+ξ)b∂b
e−2(Φ+ξ)b̄∂b̄
= 0 . (4.34)
Let us assume that the expression in (4.34) does not vanish. One has then ∂b = 0
as well as ∂b̄ = 011. But then also (4.34) holds, which leads to a contradiction. Thus
(4.34) must be satisfied in any case.
Note that the vanishing of the first column of Ñµν implies that also the first column
of T−1NµνT is zero, and thus T
−1NµνT ∈ a(3,C), hence the generalized holonomy in
the case of one preserved complex supercharge is contained in the affine group A(3,C).
This supports the classification scheme of [4]. Of course, depending on the particular
solution, the generalized holonomy may also be a subgroup of A(3,C).
4.4 Time-dependence of second Killing spinor
In this section we will utilize the above Killing spinor equations to derive the time-
dependence of the second Killing spinor. In addition, we will show that the Killing
spinor equations can be completely solved when the second Killing spinor is time-
independent.
Let us first simplify the Killing spinor equations (4.31). In the following we set
b = reiϕ and define ψ = Φ + ξ, ψ1 = r
2α1, ψ2 = re
−ψα2, ψ12 = re
−ψα12 and ψ± =
10In order to show this, one has to make use of eqns. (4.22) and (4.26).
11This follows from the vanishing of the 3 × 3 determinant that is obtained from Ñwt by deleting
the first column and the third line.
– 28 –
ψ2 ± ψ12. First of all, use the integrability conditions (4.33), that can be rewritten as
ÑµνT
−1α = 0. Defining P = e−2ψb∂b, the second component for µ = w, ν = t gives
′ + ψ−∂P = 0 , (4.35)
with ′ = ∂z . Let us assume P
′ 6= 0 (the case P ′ = 0 is considered in appendix C and
will lead to the same conclusions). If we define g(t, z, w, w̄) = −ψ−/P ′, we get
ψ− = −gP ′ , ψ1 = g∂P .
The third component of the (w, t) integrability condition is of the form
ψ1f1 + ψ2∂b+ ψ−f− = 0 ,
for some functions f1, f− that depend on z, w, w̄ but not on t. Using the above form
of ψ1 and ψ−, this becomes
f1g∂P + ψ2∂b− f−gP ′ = 0 . (4.36)
Now, if g = 0, the latter equation implies ψ2∂b = 0, and hence (since ∂b 6= 0 due to
P ′ 6= 0) ψ2 = 0. Furthermore, ψ1 = ψ− = 0 in this case, so there exists no other Killing
spinor. Thus, g 6= 0 and we can write g = expG. Dividing (4.36) by g and deriving
with respect to t yields ∂t(ψ2/g) = 0 and hence
ψ2 = e
Gψ02(z, w, w̄) .
It is then plain that ∂tψi = ψi∂tG, i = 1, 2, 12. The Killing spinor equations are of the
form ∂µψi = Mµijψj , for some time-independent matrices Mµ. Taking the derivative
of this with respect to t, one gets ∂µ∂tG = 0, whence
G = G0t + G̃(z, w, w̄) ,
with G0 ∈ C constant. We have thus ∂tψi = G0ψi and hence also ∂tαi = G0αi. Fur-
thermore, the time-dependence of α0 can be easily deduced from the Killing spinor
equations: if G0 does not vanish it is of the same exponential form as the other com-
ponents of the second Killing spinor, i.e. ∂tα0 = G0α0, while if G0 vanishes there can
be a linear part in t, i.e. ∂tα0 = c for some constant c. Hence, in terms of the basis
elements, the time-dependence of the second Killing spinor takes the form12
G0 = 0 : ǫ2 = c01 + c1e1 + c2e2 + c12e1 ∧ e2 + ct(1 + be2) ,
G0 6= 0 : ǫ2 = eG0t(c01 + c1e1 + c2e2 + c12e1 ∧ e2) , (4.37)
12We will loosely refer to Killing spinors with G0 = 0 as time-independent, despite the possible
linear time-dependence, to distinguish from the G0 6= 0 exponential time-dependence.
– 29 –
where c0, c1, c2, c12 are time-independent functions of the spatial coordinates, and c is a
constant. This was derived assuming P ′ does not vanish, but as we show in appendix
C is in fact a completely general result. Hence, adding a second Killing spinor to
ǫ1 = 1 + be2, the Killing spinor equations imply that ǫ2 always has the above time-
dependence.
Plugging this time-dependence into the subsystem of the Killing spinor equations
not containing α0 one obtains in terms of ψi
ψ′1 −
ψ− = 0 , (4.38)
ψ′2 −
ψ12 = 0 , (4.39)
ψ′12 − e−2ψ
ψ12 = 0 , (4.40)
ψ′1 −
ψ− = 0 , (4.41)
ψ′2 + e
−2ψ ∂̄b
ψ2 = 0 , (4.42)
ψ′12 −
ψ12 = 0 , (4.43)
∂ψ1 − σwG0ψ1 = 0 , (4.44)
∂ψ2 −
σwG0 +
− 2∂ψ
ψ2 = 0 , (4.45)
∂ψ12 +
σwG0 +
− 2∂ψ
ψ12 = 0 , (4.46)
∂̄ψ1 −
σw̄G0 +
ψ1 + e
− ψ12
= 0 ,(4.47)
∂̄ψ2 −
σw̄G0 +
ψ12 = 0 ,(4.48)
∂̄ψ12 −
σw̄G0 +
ψ12 = 0 .(4.49)
ForG0 = 0, these equations simplify significantly, and allow for a complete solution.
As is shown in appendix D, under the additional assumption ψ− 6= 0, ψ1 6= 0, the metric
– 30 –
and the field strength for half-supersymmetric solutions with G0 = 0 are given in terms
of a single real function H depending only on the combination Z−w− w̄ and satisfying
the second order differential equation
1 + e−2H
Ḧ + Ḣ2
1− 3α
e2H + 1− α2
, (4.50)
where α ∈ R denotes an arbitrary constant and γ = 0, 1. The new coordinate Z is
defined by Z = z for γ = 0 and Z = ℓ ln
1 + z
for γ = 1. Furthermore, in the
remainder of this section and in appendix D, a dot denotes a derivative with respect
to Z − w − w̄. Given a solution of (4.50), one defines the functions χ, ρ by
e2H + 1− α2
− Ḣ2χ2 . (4.51)
Note that χ is imaginary and ρ is real. b and ψ are then given by
b = eγZ/ℓρ eiϕ , e2ψ = e2(H+γZ/ℓ) ,
where
tanϕ =
so that the metric reads
ds2 = −4ρ2e2γZ/ℓ(dt+ σ)2 + 1
+ e2Hdwdw̄
, (4.52)
where the shift vector satisfies
∂Zσw =
e−γZ/ℓ
, ∂σw̄ − ∂̄σw = −
e−γZ/ℓ
Finally, the gauge field strength is given by (4.21).
Equation (4.50) is actually the Euler-Lagrange equation for the following standard
action for the scalar H
d (Z − w − w̄)
M(H)Ḣ2 − V (H)
, (4.53)
where
M(H) =
e2H + 1
(e2H + 1− α2)3/2
, V (H) = − γ
e2H + 1− 2α2
(e2H + 1− α2)1/2
. (4.54)
– 31 –
Thus it is possible to use the energy conservation law of that model in order to evaluate
the “velocity” Ḣ in terms of H . Since dH = Ḣd (Z − w − w̄) one has
M(H)Ḣ2 + V (H)
= 0 , (4.55)
so that there must exist a constant E such that
[E − V (H)] =
e2H + 1− α2
e2H + 1
e2H + 1− 2α2√
e2H + 1− α2
(4.56)
The key-point is to consider now, as a new coordinate, the function H in place of
w+ w̄13 and to write down the full solution, say metric plus gauge field, in terms of H .
Using w = x+ iy, the general solution is given by
ds2 = −4ρ2e2γZ/ℓ
dt + e−γZ/ℓσ̂ydy
dy2 +
dZ2 + e2H
dZ − dH
A = ℓḢ
−2iρ2χeγZ/ℓdt+
1− e2Hχ2
− i ℓ
d log
, (4.57)
where Ḣ is given in equation (4.56), the functions χ and ρ are defined in (4.51) and
the shift vector reads
σ̂y = −
If γ = 1, a simple example of this set of solutions can be obtained by setting α = 0,
Ḣ = 1/ℓ , b =
. (4.58)
As will be shown in section 5.1.2, this corresponds to the maximally supersymmetric
AdS4 solution. More general γ = 1 solutions will be two-parameter deformations
thereof, the parameters being α and the energy E of the associated scalar system.
Setting γ = 0 the potential V (H) vanishes and the parameter E can be fixed by
a simple rescaling of the coordinates. Thus we are left with a one-parameter family
of solutions. Since the metric does no more depend explicitly on Z, it is useful to
replace the coordinate Z instead of x by H . Defining a new coordinate r such that
13This is possible by simply requiring that Ḣ 6= 0.
– 32 –
r4 ≡ 16
e2H + 1− α2
and a new parameter Q = 4
α, the complete solution reads
ds2 = −
dt− 2ℓ
r4 + ℓ2Q2
h(r)2
r4 + ℓ2Q2 − 16
dx2 + dy2
A = −Q
dy − i ℓ
d log
, (4.59)
where
h(r) =
r4 + ℓ2Q2
r4 + ℓ2Q2 − 16 . (4.60)
The parameter Q can thus be interpreted as an electric charge. The Petrov type of the
solution is D or simpler. If one sets Q = 4/ℓ the Petrov type is reduced to N , so that
there is a gravitational wave.
In order to complete the classification of G0 = 0 solutions, we need to study
separately the cases where either ψ1 or ψ− vanishes (it can easily be seen from (4.39)
and (4.40) that there is no solution if both vanish). As one can see by looking at
equations (4.38) and (4.41), the condition ψ1 = 0 leads to b = b(z), which is studied in
detail in section 5.1. The other possibility, ψ− = 0, is more involved, but as we show
in appendix D it boils down to three different cases, that can be completely solved:
the AdS2 × H2 anti-Nariai spacetime studied in section 5.1.1, the imaginary b case
solved in section 5.3, and finally the half BPS solution coming from the gravitational
Chern-Simons model, that we analyse in section 5.5.
We would like to remark that the assumptionG0 = 0 on the overall time-dependence
of the second Killing spinor seems a reasonable choice since all known 1/2-supersymmetric
solutions to be studied in the next section are contained in this class, or can be brought
to this class by a general coordinate transformation. Hence we expect the G0 = 0 class
to form an important subclass of all 1/2-supersymmetric solutions.
5. Timelike half-supersymmetric examples
The problem of finding all half BPS configurations in the timelike class involves the
solution of the integrability conditions we obtained above. To obtain explicit examples
of half BPS solutions, we shall restrict to some simple subclasses with particular b.
This will determine the fraction of preserved supersymmetry for the solutions which
are already known to be 1/4 supersymmetric, and will also lead to new solutions.
– 33 –
5.1 Static Killing spinors and b = b(z)
The timelike vector field V , constructed as a bilinear of the Killing spinor, is static
if the associated one-form V = dt + σ satisfies the Fröbenius condition V ∧ dV = 0.
Obviously, there can be static BPS solutions with V not being static itself, due to the
choice of coordinates; we shall loosely refer the Killing spinors whose vector bilinear is
static as static Killing spinors. The staticity condition, in turn, implies dσ = 0 and
puts strong constraints on the function b. Indeed, equation (4.18) implies that the
phase ϕ of b depends only on z. Then, (4.19) gives the modulus r of b in terms of its
phase,
sinϕ(z)
lϕ′(z)
. (5.1)
As a consequence, r and therefore the complete complex function b, depend on the
single variable z. The full solution is therefore determined by the single real function
ϕ, which has to satisfy the equations for supersymmetry (together with the conformal
factor ψ).
However, since the equations can be exactly solved for arbitrary b(z), we will stick
to this more general case and eventually comment on the static subcase.
If b depends only on z, the equations of motion simplify to
b2∂2z
= 0 , (5.2)
e−2ξ∆ξ =
. (5.3)
Here we have used the fact that Φ, defined in (4.15), depends only on the coordinate
z. In principle there is also an integration constant K(w, w̄) with arbitrary dependence
on the transverse coordinates, but since Φ appears only in the combination Φ+ ξ in all
the equations, we can always absorb the (w, w̄) dependence into the conformal factor
ξ. Now the left hand side of equation (5.3) depends only on the coordinates w and w̄,
while the right hand side depends only on z. This equation can be therefore satisfied
only if both sides are equal to some constant κ. The system of equations is then
e2ξ = 0 , (5.4)
e2Φ(z)
= − l
κ . (5.5)
Note that the first one is the Liouville equation, whose solution describes the transverse
two-dimensional manifold, which has therefore constant curvature κ.
– 34 –
Equations (5.2) and (5.5) can easily be solved [16]. Their solution is given by14
b̄ = −αz
2 + βz + γ
ℓ(2αz + β)
, (5.6)
with α, β, γ ∈ C. Then ξ solves the Liouville equation for a constant curvature two-
manifold with scalar curvature15
κ = 8(αγ̄ + ᾱγ)− 4ββ̄ . (5.7)
This solution generically belongs to the supersymmetric Reissner-Nordström-Taub-
NUT-AdS4 family of spacetimes. The values α = 0 and β
2 = 4αγ are special cases
and will be treated separately in the following. Note that the coefficients α, β and γ
are not three independent parameters, as they can be rescaled without changing the
function b: the solutions depend only on their ratios. For example, if α 6= 0, one can
use β/α and γ/α as independent complex parameters of the family of solutions.
The solutions with static Killing spinor form a subset of this family. For (5.6) the
staticity condition (5.1) yields the condition αβ̄− ᾱβ = 0. Recalling the expression for
the NUT charge of these solutions,
, (5.8)
this charge must vanish for non-vanishing α, as one could have guessed. On the other
hand, for α = 0 the solution is anti-Nariai, as we shall see below. We conclude that
the most general supersymmetric configuration with static Killing vector constructed
as a Killing spinor bilinear is either of the form (5.6) – i. e. in the fourth row of table
1 of [34] – with vanishing NUT charge, or it is anti-Nariai spacetime.
The supersymmetric static solutions discussed so far are generically 1/4-BPS. We
want to see what further condition ensures the presence of an additional Killing spinor.
Inserting the staticity ansatz b = b(z) into the integrability equations and requiring
these matrices to be of rank smaller or equal to two, one finds the following condition
(in particular this is obtained from the vanishing of the minor of the last row of Ñwt
and the first two rows of Ñw̄t)
2b′ +
= 0 , (5.9)
14With this definition, the constants α, β and γ coincide with a, b and c of [16] respectively.
15This scalar curvature differs from the one given in [16] for the case in which all coefficients are
real, k = 4αγ − β2 = κ/4. The factor of 4 comes from the different definition of the conformal factor
of the transverse metric, our ξ is related to the old γ by ξ = γ − ln 2.
– 35 –
As an aside, note that we have only used the ansatz b = b(z) so far and not the
staticity condition (5.1), i.e. the precise relation between r and ϕ. The static solutions
are therefore in general still a subset of the solutions under consideration.
Condition (5.9) calls for the following three different cases, corresponding to the
vanishing of its three factors.
5.1.1 AdS2 ×H2 space-time (α = 0)
Requiring the first factor of (5.9) to vanish leads to b = −z
+ ic with constant c,
corresponding to α = 0 in (5.6). We can absorb the imaginary part of c by a shift of
the coordinate z and henceforth will assume c ∈ R.
In this case κ = −4 and we have a hyperbolic transverse space. As a solution of
(5.4) we can take
e2ξ =
. (5.10)
Moreover, eΦ = l|b| and σ = 0, therefore giving the metric
ds2 = −4
dt2 +
dx2 + dy2
. (5.11)
This is the anti-Nariai AdS2 × H2 solution, with the AdS2 factor written in Poincaré
coordinates for c = 0 and in global coordinates for c 6= 0. The coordinate transforma-
tions between Poincaré coordinates (tP , zP ) (with c = 0) to global ones (tgl, zgl) (with
c 6= 0) is given by
(zgl −
z2gl + ℓ
2c2 cos(4ctgl/ℓ)) ,
tP = −
z2gl + ℓ
2c2 sin(4ctgl/ℓ)
zgl −
z2gl + ℓ
2c2 cos(4ctgl/ℓ)
. (5.12)
The electromagnetic field strength (4.21) in this case is given by
F = − 1
dx ∧ dy , (5.13)
i.e. only lives on the hyperbolic part and is independent of the coordinates of the AdS
part of space-time.
This solution preserves precisely 1/2 of the supersymmetries, as was already shown
in [35]. To obtain the form of the Killing spinors admitted by this metric we first
observe that the integrability conditions impose α2 = α12 = 0. Then the Killing spinor
equations are easily solved, but one should treat separately the cases c = 0 and c 6= 0:
– 36 –
• If c = 0, then
α0 = λ1 + λ2
, α1 =
, (5.14)
where λ1,2 ∈ C are integration constants. This yields the following Killing spinors,
spanning a two-dimensional complex space,
λ1 + λ2
1 + b
λ1 + λ2
e2 . (5.15)
Note that λ1 = 1, λ2 = 0 corresponds to the original Killing spinor. Also note
that the constant G0, corresponding to the time-dependence of the second Killing
spinor with λ2 6= 0, is zero. The form of the scalar invariant corresponding to the
general spinor ǫ is
b̃ = b
|λ1|2 + |λ2|2
λ̄1λ2 + λ1λ̄2
λ̄1λ2 − λ1λ̄2
. (5.16)
Here the first term is real, while the second is imaginary. Note that the latter
is in fact constant. Then the Killing vector Ṽ built from ǫ will have a norm
Ṽ 2 = −4|b̃|2, and will be timelike unless b̃ vanishes. This is however not possible,
because both the real and imaginary parts of b̃ should vanish, but since λ1,2 do
not depend on the coordinates, the real part cannot vanish. Therefore, every
Killing spinor of this solution belongs to the timelike class.
• If c 6= 0 we have
[λ1−iλ2+(λ1+iλ2)
−4ict/ℓ , α1 = −
|b| (λ1+iλ2)e
−4ict/ℓ , (5.17)
and the most general Killing spinor is parametrized by λ̃1,2 ∈ C as follows
(λ1 − iλ2)(1 + be2) +
2|b|(λ1 + iλ2)e
−4ict/ℓ(1 + b∗e2) . (5.18)
Note that the combination λ1− iλ2 corresponds to the first Killing spinor 1+ be2,
while the orthogonal combination λ1 + iλ2 gives rise to the second Killing spinor
proportional to 1 + b∗e2. Any combination with λ2 6= 0 has G0 = −4ic/ℓ.
In this case, the real part of the invariant b̃ is given by
Re(b̃) =
|λ1|2
(−z +
z2 + ℓ2c2 cos(4ct/ℓ)) +
|λ2|2
(−z −
z2 + ℓ2c2 cos(4ct/ℓ))+
2 + λ2λ
z2 + ℓ2c2 sin(4ct/ℓ)) , (5.19)
while the imaginary part is identical to that of (5.16).
– 37 –
It can easily be checked that the coordinate transformation (5.12) indeed relates the
complex scalar b̃, which is composed of spinor bilinears, in (5.16) and (5.19) to each
other.
Let’s now check how the isometries of AdS2 act on the Killing spinors. It is useful
to do this by embedding AdS2 with metric
ds2 = −4
dt2 +
) (5.20)
into the three-dimensional flat space Xa = (U, T,X) with metric
ds2 = −dU2 − dT 2 + dX2 . (5.21)
Then, AdS2 is obtained as the hyperboloid defined by
−U2 − T 2 +X2 = ℓ
, (5.22)
and its isometry group SO(2,1) will act as the three-dimensional Lorentz group on the
embedding coordinates Xa (here a is a three-dimensional Lorentz index).
If c = 0, the AdS2 metric (5.20) is in the Poincaré form, and can be seen to be
the induced metric on the hyperboloid by parameterizing it with the coordinates (t, z)
given by
z = U +X , t =
2(U +X)
. (5.23)
Then, if one defines the 3d Lorentz vector
|λ1|2 − |λ2|2
(λ∗1λ2 + λ1λ
2) ,−
|λ1|2 + |λ2|2
, (5.24)
one explicitly checks that the invariant b̃ can be put in the form
b̃ = XaΛ
ΛaΛa. (5.25)
Now, the real and imaginary part of b̃ are independently manifestly invariant under the
AdS2 isometries, as they should be (since they transform respectively as pseudoscalar
and scalar under diffeomorphism16).
If c 6= 0 we have AdS2 in global coordinates, and the embedding is modified to
U = − ℓ
+ c2 cos
, T = − ℓ
+ c2 sin
, X =
. (5.26)
16Note that Λ doesn’t depend on the sum of the phases of λ1,2; this is diffeomorphism invariant but
transforms under U(1) gauge transformations.
– 38 –
The invariant (5.19) takes again the manifestly invariant form (5.25), as expected, and
the isometries of AdS2 are realized linearly on the Killing spinors through their action
on Λa.
This result may be useful to study in detail quotients of AdS2 and to see whether
this operation breaks some supersymmetry.
5.1.2 AdS4 space-time (β
2 = 4αγ)
The following subcase corresponds to the vanishing of the second factor of the inte-
grability condition (5.9). The function b is then given by b = − z
+ ic, which can be
obtained as the special case β2 = 4αγ from (5.6). This corresponds to AdS4, the only
maximally supersymmetric solution of the theory. Indeed the integrability condition
matrices vanish in this case.
Let’s see in detail the form of the metric arising from different values of c. As in
the previous case we can take the constant c to be real. If c = 0, the metric is static,
σ = 0, ξ = 0 and e2Φ = |b|4, and we obtain anti-de Sitter in Poincaré coordinates,
ds2 = −z
dt2 − dx2 − dy2
dz2 . (5.27)
On the other hand, for c 6= 0, the metric appears in non-static coordinates,
σ = − ℓdy
, e2ξ =
4c2x2
, e2Φ = |b|4 , (5.28)
which give
ds2 = −
+ 4c2
dt− ℓdy
16c2x2
dx2 + dy2
+ 4c2
dz2 .
(5.29)
The field strength (4.21) vanishes in this case.
We shall now obtain the form of the Killing spinors for AdS4, and will do this in
the simpler c = 0 case. The solution of the Killing spinor equations yields
α0 = λ1 −
λ3 , α2 = −
λ2 , α12 =
1− zt
λ4 , (5.30)
where the coefficients λ1,...,4 span a four dimensional complex space, as expected in the
case of maximal supersymmetry. In the form basis of the spinors ǫ = c01+c1e1+c2e2+
– 39 –
c12e1 ∧ e2, we obtain
c0 = λ1 −
λ3 , c2 = −
λ3 + λ4 , c12 =
λ4 . (5.31)
The new Killing spinors corresponding to λ2 and λ4 both have
17 G0 = 0. To study the
action of the AdS4 isometries it is useful to embed the hyperboloid in a five-dimensional
flat space (U, V, T,X, Y ) with metric
ds2 = −dU2 + dV 2 − dT 2 + dX2 + dY 2. (5.32)
Then, AdS4 is the hypersurface −U2 + V 2 − T 2 +X2 + Y 2 = −ℓ2/4 and its isometries
are realized as the SO(3,2) isometries of the embedding space. The relation with the
Poincaré coordinates is
U − V ,
U − V ,
U − V , z = 2(U − V ) . (5.33)
If we define the vectors
ℓΛa =
|λ1|2 − |λ2|2 + |λ3|2 − |λ4|2
|λ1|2 + |λ2|2 − |λ3|2 − |λ4|2
λ3λ̄4 + λ̄3λ4 − λ̄1λ2 − λ1λ̄2
λ2λ̄4 + λ̄2λ4 − λ̄1λ3 − λ1λ̄3
λ2λ̄4 − λ̄2λ4 + λ̄1λ3 − λ1λ̄3
, Xa =
, (5.34)
where the index a = 1, . . . , 5 is an SO(3,2) index raised and lowered using the metric
(5.32), then
a = − 1
λ3λ̄4 − λ̄3λ4 + λ̄1λ2 − λ1λ̄2
]2 ≥ 0 , (5.35)
and the invariant b̃ for the Killing spinors reads
b̃ = c∗0c2 + c1c
12 = XaΛ
ΛaΛa . (5.36)
17Note that this does not hold for λ3, whose time-dependence is not of the form derived in section
4.4. There is no contradiction however, since all solutions in this class have P = 0 and hence are
treated separately in appendix C. It is interesting to find that nevertheless the time-dependence of
many Killing spinors in this class have the canonical G0 time-dependence.
– 40 –
This form of b̃ is manifestly invariant under the AdS4 isometries, and shows that under
Λa transforms in the fundamental representation of SO(3,2) under these transforma-
tions. Note that it has precisely the same form (5.25) as in the anti-Nariai case. Again,
the explicit knowledge of the AdS4 isometry group action on the Killing spinors is
important to study the supersymmetry of its quotients.
5.1.3 The Reissner-Nordström-Taub-NUT-AdS4 family
The last subcase corresponds to the vanishing of the third factor of the integrability
condition (5.9). Note that this is precisely the expression in square brackets of equation
(5.5) and the condition reads simply κ = 0. Then ξ is an harmonic function and the
transverse space is flat. In particular, the solution (5.6) admits a second Killing spinor
|β|2 = 2(αγ̄ + ᾱγ) . (5.37)
Since α 6= 0 we can define ζ = Im(β/α) and δ = Im(γ/α). Moreover, all equations
are invariant under rigid translations in the z directions, since the coordinate z never
appears explicitly in them. One can use this freedom to eliminate the real part of β/α
by performing the redefinition z 7→ z − 1
Re(β/α). Hence this complete family of 1/2
BPS solutions is determined by two real parameters ζ and δ,
b = −1
z2 − iζz + 1
ζ2 − iδ
2z − iζ . (5.38)
Then σ = −2ζ(r/ℓ)2dϑ and the resulting metric is
ds2 = −
z2 + ζ
+ (ζz + δ)
z2 + ζ
dt− 2ζ
z2 + ζ
z2 + ζ
+ (ζz + δ)
dr2 + r2 dϑ2
, (5.39)
where we used polar coordinates (r, ϑ) in the (w, w̄) plane. The charges of the solution
M = −δζ
, n =
, P = −ζ
, Q = −δ
. (5.40)
Essentially, the imaginary part of γ gives the electric charge and the imaginary part of β
determines the NUT charge. Note that the quantization condition P = −(kℓ2+4n2)/2ℓ
is also satisfied. In terms of the charges, the solution is given by
b = −1
(z − in)2 + 2n2 + iℓQ
2 (z − in) . (5.41)
– 41 –
The subfamily of static half BPS configurations is obtained by imposing the static-
ity condition ζ = 0 or equivalently vanishing NUT charge. It is parameterized by
the single parameter left, δ ∈ R and the solutions are restricted to have the following
charges
M = 0 , n = 0 , P = 0 , Q = −δ
In terms of the charges, the solution is given by
b = −1
z2 + iℓQ
. (5.42)
The metric and electromagnetic field strength for this solution read
ds2 = −
dt2 +
+ 4ℓ2z2 dwdw̄ , (5.43)
F = −Q
dt ∧ dz . (5.44)
This is simply the backreacted AdS4 filled with the electric field generated by an electric
charge Q placed in its center ζ = 0. The solution has a singularity there. Note that this
solution was already shown to be 1/2 supersymmetric in [36]. It was also shown there
that the Killing spinors are preserved if one compactifies the transverse two-dimensional
plane to a two-torus.
We will now discuss the Killing spinors for these metrics. The integrability condi-
tions impose α2 = 0 and
α4 . (5.45)
With these constraints, the Killing spinor equations simplify, and can be solved to give
α0 = λ1 + 2iζw̄λ2 , α1 = 0 , (5.46)
z2 + iζz + ζ
4z2 + ζ2
, α12 = α2 −
4z2 + ζ2 , (5.47)
where λ1,2 ∈ C parameterize the two dimensional space of Killing spinors. Then the
most general Killing spinor for these metrics is
ǫ = (λ1 + 2iζw̄λ2) 1− ℓλ2
2z + iζ
2z − iζ e1
+b (λ1 + 2iζw̄λ2) e2 −
z2 − iζz + ζ2
4z2 + ζ2
λ2 e1 ∧ e2 . (5.48)
– 42 –
Again the second Killing spinor has G0 = 0 time-dependence. Finally, the correspond-
ing orbit of the Killing spinor is determined by the invariant
b̃ = b|λ1|2 +
z2 + iζz + ζ
2z − iζ + 4ζ
2bww̄
|λ2|2 + 2iζb
w̄λ̄1λ2 − wλ1λ̄2
. (5.49)
It is easy to show now that b̃ is non vanishing for any choice of λ1,2: indeed if b̃ = 0, we
have ∂∂̄b̃ = 4ζ2b|λ2|2 = 0 and either λ2 = 0, which implies in turn λ1 = 0, or ζ = 0. In
the latter case, it is very easy to see that b̃ = 0 iff ǫ = 0. Therefore, all Killing spinors
of this family of metrics belong to the timelike class, and the solution is purely timelike.
Summary of the b = b(z) case:
1. The only supersymmetric solutions with static Killing spinor (i.e. whose timelike
Killing vector constructed as a Killing spinor bilinear is static) are AdS4, the
anti-Nariai spacetime and the Reissner-Nordström-AdS4 solutions of the fourth
row of table 1 of [34], i. e. solutions of the form (5.6) with vanishing NUT charge.
2. The only 1/2 BPS solutions with static Killing spinor are the anti-Nariai space-
time and the solution (5.43) with field strength (5.44).
3. The most general half BPS solution with b = b(z) are the anti-Nariai spacetime
and the solution (5.39) with charges (5.40) describing an electric charge in the
center of AdS4.
The natural way to continue this approach is to study half BPS solutions with b
harmonic, and this will be the subject of the next paragraph.
5.2 Harmonic b solutions
The previous class of solutions can be generalized by requiring ∆b = 0 instead of
b = b(z) [16]. This implies that ∆1/b = 0 and hence (4.23) still simplifies in exactly
the same way as in the b = b(z) case. Indeed, the solution is
b̄ = −αz
2 + βz + γ
ℓ(2αz + β)
, (5.50)
where now α, β and γ are no more constants but arbitrary functions of (w, w̄). It is
then easy to show that the ∆b = 0 condition requires these functions to be harmonic
and all (anti-)holomorphic, that is α, β and γ all depending either only on w or only
on w̄, and this is the most general solution with ∆b = 0. The b = b(z) configurations
– 43 –
are particular cases of this larger class, and are obtained for α, β and γ constant. Note
that also the ∂b = 0 and ∂b̄ = 0 subclasses fall into this family.
Let’s take for definiteness α, β, γ all anti-holomorphic, then b = b(z, w). The
requirement that the integrability conditions allow for an extra Killing spinor, i.e. that
they are of rank ≤ 2, in this case leads to several conditions. One of these is obtained
from the minor of the last three lines of Ñwt and reads
2∂z b̄+
∂z b̄+
∂b∂b − 2∂(Φ + ξ)∂b
∂b = 0. (5.51)
This gives three different cases to be analysed, corresponding to the vanishing of the
first three factors of this equation (vanishing of the fourth factor implies b = b(z) and
hence brings one back to the previous section).
5.2.1 Deformations of AdS2 ×H2
The vanishing of the first factor in (5.51) implies b = −z
+ ic(w), where c(w) is an arbi-
trary holomorphic function. These are the α(w) = 0 supersymmetric Kundt solutions
of Petrov type II, describing gravitational and electro-magnetic waves propagating on
anti-Nariai space-time [16].
The remaining integrability conditions however imply α1 = α2 = α12 = 0, in which
case there is no second Killing spinor, or ∂c = 0. Therefore there are no new half
BPS solutions with non constant c. In this class c constant is the half supersymmetric
anti-Nariai spacetime and the other preserve only 1/4 of the supersymmetries.
5.2.2 Deformations of AdS4
The vanishing of the second factor in (5.51) implies b = − z
+ ic(w). In this case we
are considering the β2 = 4αγ supersymmetric Kundt solutions, describing gravitational
and electro-magnetic waves propagating on AdS4 spacetime [16].
Again the remaining integrability equations have to solutions: α1 = α2 = α12 = 0 or
∂c = 0. Hence, as in the previous case, we find that there are no harmonic deformations
of AdS4 preserving half supersymmetry.
5.2.3 Deformations of Reissner-Nordström-Taub-NUT-AdS4
Not considering the previous two special cases, the general solution represents expand-
ing gravitational and electro-magnetic waves propagating on a Reissner-Nordström-
Taub-NUT-AdS4 spacetime [16]. When Im(β) = 0, the solution can be put in Robinson-
Trautman form and is of Petrov type II.
The vanishing of the third factor in (5.51) is given by
∂b∂b − 2∂(Φ + ξ)∂b = 0 . (5.52)
– 44 –
With b given in (5.50) this case can be solved for the derivative of Φ + ξ and implies
∂̄(Φ + ξ) =
, (5.53)
and therefore ∆(Φ + ξ) = 0. Then (5.3) fixes the transverse manifold to be flat and
κ(w) = 8(αγ̄ + ᾱγ)− 4ββ̄ = 0. (5.54)
But α,β and γ being holomorphic, this last equation can be satisfied if and only if they
are constant, and we are back to the previous case, i. e. there are no new 1/2 BPS
solutions.
Summary of the harmonic case:
There are no new half BPS solutions in the harmonic b case. The only half BPS
solutions are those with b = b(z), and as soon as one deforms these solutions by adding
some harmonic (w, w̄)-dependence, one breaks supersymmetry further to 1/4.
5.3 Imaginary b solutions
Another subcase we want to study is b̄ = −b, i. e. b purely imaginary. For notational
convenience we introduce18
b = iX ,
where X is real. From (4.15) one gets Φ = 0. All quantities in the Bianchi iden-
tity (4.22), apart from b and hence X , are then z-independent. The only consistent
possibility is to take ∂zX = 0. The remaining equations (4.22) and (4.23) read
e2ξ , ∆
e2ξ = 0 . (5.55)
Examples of 1/4 supersymmetric solutions of this class, i.e. with imaginary b, that
were discussed in [16] are X = (x/ℓ)α with α = −2 and α = 1
. These correspond to a
particular Petrov type I solution and an electrovac AdS travelling wave of Petrov type
N, respectively. It was shown that the latter actually preserves a second, null Killing
spinor. In this section we will derive the general condition for 1/2 supersymmetry in the
case of imaginary b and will find that there is a one-parameter family of such solutions.
The condition for 1/2 supersymmetry is very simple in this case. Assuming that
∂X is not equal to zero, which would clearly be incompatible with (5.55), there is
18In the following we will assume that X is positive without loss of generality.
– 45 –
only one differential constraint which needs to be satisfied for the existence of a second
Killing spinor, i. e. for the matrices of integrability conditions to have rank 2, namely
∂2X−1 − 2∂ξ∂X−1 = 0 . (5.56)
The above three differential equations can be integrated to
e2ξ = −iK̄(w̄)∂X−1 , ∂X−1 = i
, (5.57)
where K(w) is an arbitrary holomorphic function and L is a real constant. The func-
tion K(w) corresponds to the freedom to choose holomorphic coordinates on the two-
dimensional space, and hence it can be gauged away. A convenient gauge choice will
be K(w) = iℓ. Note that, for this choice, the imaginary part of the right hand side of
the last equation vanishes, and therefore that ∂yX = 0.
For L = 0, (5.57) can be integrated to give
, (5.58)
which is (up to a rescaling of the coordinate x) the example given above with α = 1
This was already found to be 1/2 supersymmetric in [16]. Here we find that this solution
is a special case of the most general possibility.
For other values of the constant L it is convenient to use X as a new coordinate
instead of solving for X(x). From (4.18) and (4.19) it follows that σ can be chosen to
. (5.59)
Then the metric reads
ds2 = −4X2
dz2 +
ℓ2dX2
X2(1 + 4LX4)
1 + 4LX4
dy2 . (5.60)
Finally, from (4.21) we obtain the gauge field strength
F = 2dt ∧ dX . (5.61)
Note that the geometry (5.60) is generically of Petrov type D, and becomes of Petrov
type N for L = 0.
Now let us turn our attention to the form of the second Killing spinor. First of all,
the integrability conditions imply that it takes the form
αT = (β1, β2, iX
3eξβ2, iX
3eξβ2) ,
– 46 –
where β1 and β2 are arbitrary space-time dependent functions. The Killing spinor
equations (4.31) yield
β1 = λ1 − 12λ2b
−2 , β2 = λ2b
where λ1 and λ2 are integration constants. This implies that the new Killing spinor
takes the form ǫ = λ1ǫ1 + λ2ǫ2, where
ǫ1 = 1 + iXe2 , ǫ2 =
X−2(1− iXe2) +
X−4 + L (e1 − iXe1 ∧ e2) . (5.62)
Note that G0 = 0 as well in this class.
One interesting aspect of the second Killing spinor ǫ2 is the norm of its associated
Killing vector Vµ = D(ǫ2,Γµǫ2). We find VµV
µ = −4X2L2, hence the second Killing
spinor is indeed null for the case L = 0, as was noticed before, while it is timelike
for L 6= 0. In the latter case, to understand whether the solution belongs also to the
null class of supersymmetric solutions, we have therefore to study the most general
linear combination of the two Killing spinors. The Killing vector Ṽ constructed from
ǫ = λ1ǫ1 + λ2ǫ2 has norm
Ṽ 2 =
λ̄1λ2 − λ1λ̄2
)2 − 4X2
L|λ1|2 + |λ2|2
which can vanish only if L ≤ 0. We have therefore three cases:
1. L > 0, pure timelike class, Petrov type D.
2. L = 0, belongs to both null and timelike classes, Petrov type N. This is the
homogeneous half BPS pp-wave in AdS. (In the terminology of [16] it has a wave
profile Gα with α = 0).
3. L < 0, belongs to both null and timelike classes, Petrov type D.
Actually the solutions (5.60) with L > 0 can be cast into a simpler form. This is
done by trading the coordinate y for a new variable ψ = Ly − t. For convenience, let
us also introduce the Schwarzschild coordinate r and rescale z,
r = − ℓ√
, ζ =
Lz . (5.63)
In the new coordinates, the metric and the gauge field strength read
ds2 = −
dt2 +
dψ2 + dζ2
, F = qe
dt ∧ dr , (5.64)
– 47 –
where we have defined qe = 2ℓ/
L. This is precisely the half BPS solution obtained
in [36], the massless limit of an electrically charged toroidal black hole, which forms a
naked singularity. It is also interesting to note that the charge qe diverges in the L→ 0
limit. This limit is naively singular in these coordinates, but it can be taken if we
perform a Penrose limit [37, 38]. The existence of this limit explains why we obtained
a one-parameter family of geometries (5.60) connecting the massless limit of toroidal
black holes and a pp-wave. Indeed, define the new coordinates (X+, X−, R, Z) and the
rescaled charge Qe by
ψ + t = 2ǫ2X+ , ψ − t = 2X− , r = 1
, ζ = ǫZ , qe =
. (5.65)
Then, the singular limit ǫ→ 0 yields is a regular solution of the theory and corresponds
to the half supersymmetric solution (5.60) with L = 0,
ds2 =
4 dX+dX− − Q
dX−2 + dR2 + dZ2
, F = Qe
dX− ∧ dR . (5.66)
In the procedure, we have blown up the metric in the neighborhood of a geodesic with
ψ + t constant near the boundary r → ∞ of AdS.
We now turn to the L < 0 case, which is both timelike and lightlike. Let us define
L = −µ2. We can perform a coordinate transformation inspired from the previous one,
ψ = Ly − t , r = − ℓ
, ζ =
z , (5.67)
under which the metric and the field strength become
ds2 =
dt2 +
− q2e
−dψ2 + dζ2
, F = qe
dt ∧ dr , (5.68)
where we have defined qe = 2ℓ/µ. We see that this is the precisely the metric for L > 0
after the double analytic continuation
t 7→ it , ψ 7→ iψ , qe 7→ −iqe . (5.69)
This solution represents therefore a bubble of nothing in AdS [39–42]. Note that the
metric is singular for r =
ℓqe. One should compactify t, in such a way to eliminate
the conical singularity on the (t, r) hypersurface. Then, if we compactify also ζ , this
S1 will have a minimal radius for r =
ℓqe (the boundary of the bubble of nothing)
and then grow with r. Note that for r → ∞ one locally recovers AdS spacetime, and
that the L = 0 solutions can again be understood as a Penrose limit of this metric.
– 48 –
5.4 Action of the PSL(2,R) group on the imaginary b solutions
We can now generate new supersymmetric solutions by acting with the PSL(2,R)
symmetry group (4.28)-(4.29) on the known ones. It is easy to check that the AdS4
and AdS2×H2 solutions are invariant under this group (although it acts non trivially on
the Killing spinors). Its action on the b = b(z) subfamily of the RNTN-AdS4 solutions
was studied in [16], where it was shown that it acts non trivially on the charges, by
mixing them. Here we want to apply it to the imaginary b solutions of the previous
paragraph.
The new solution solution of the supersymmetry equations (4.22)-(4.23) generated
by the transformation (4.28)-(4.29) is
b̃ = − γ
2γ2ℓXz + i
, e2(Φ̃+ξ) =
1 + 4LX4
, (5.70)
where, without loss of generality, we eliminated α by means of a translation of z19, and
dropped the prime of the new coordinate z′. The shift function is then determined by
solving equations (4.18) and (4.19),
σx = 0 , σy =
1 + 4LX4
4γ2X4z2
. (5.71)
Then, defining the new coordinates (T, σ, p, q) through
2ℓ2γ2
, σ =
, p = − ℓ
, q = 2ℓ2γ2z , (5.72)
the metric reads
ds2 = − Q(q)
q2 + p2
P (p)
q2 + p2
dq2 +
q2 + p2
P (p)
(q2 + p2)P (p) dσ2, (5.73)
Q(q) =
, P (p) =
p4 + 4Lℓ2
, (5.74)
and the gauge field (4.21) is
F = d
ℓ(q2 + p2)
∧ dT + d
q2 + p2
∧ dσ . (5.75)
19After this translation the limit γ → 0 is not anymore well-defined. To perform it, one has to
substitute preliminarily z with z − α/γ everywhere.
– 49 –
The form of the metric suggests some connection with the Plebanski-Demianski family
of solutions, and indeed these geometries are of Petrov type D for L 6= 0, and of Petrov
type N for L = 0, but we were not able to find the precise relation. Note also that
the parameter γ has been reabsorbed in the new variables, and we are left with a
one-parameter (L) family of solutions.
The left hand side of the necessary condition (4.34) for the existence of a second
Killing spinor reads, for this solution,
− 9iX
4 (1 + 4LX4)
ℓ2 (1 + 4γ4ℓ2X2z2)
γ2 (5.76)
which clearly vanishes only for γ = 0, i.e. if the PSL(2,R) transformation is trivial.
Therefore, the new solutions (5.73)-(5.75) preserves only 1/4 of the supersymmetries,
and we explicitly see that the PSL(2,R) transformations can break any additional
supersymmetry. Also note that if we perform the PSL(2,R) transformation adapting
the original metric to a different Killing spinor, we could in principle end up with other
supersymmetric solutions.
Surprisingly, we find that the L = 0 solution can be cast in the Lobatchevski wave
form, even though it only has a time-like Killing spinor. This can be seen by trading
the coordinates (q, p) for (x, z) defined by
, z =
, (5.77)
in the metric (5.73) with L = 0, which becomes
ds2 =
−2 dTdσ + z
x2 + z2
x2 + z2
x2 + z2
dT 2 + dz2 + dx2
. (5.78)
The field strength can be easily obtained from equation (5.75) but the result is not
particularly enlightening and therefore we do not report it. This metric represents a
1/4 BPS Lobatchevski wave, whose Killing spinor falls in the timelike class. This does
not contradict the results obtained in the null case, since the null Lobatchevski had a
field strength (3.6) of the form F = φ′(T )dT ∧dz, while this solution has a much more
complicated gauge field. It is however interesting to note that the solutions of the null
case do not exhaust all possible supersymmetric Lobatchevski waves.
5.5 Gravitational Chern-Simons system and G0 = ψ− = 0 solutions
A number of the previously studied subcases can be combined into the interesting
Ansatz
b = −1
αz2 + βz + γ
2αz + β − iη(w, w̄) , (5.79)
– 50 –
where α, β and γ are three real constants. For α = β = 0 this reduces to b imaginary,
while η = 0 leads to the real subcase of b = b(z). With this assumption, the equations
for a timelike Killing spinor reduce to
e2ξ (k − 3η) = 0 , ∆η + e2ξ
kη − η3
= 0, (5.80)
where we have defined k = 4αγ − β2 and ∆ = 4∂∂̄. Interestingly, as shown in [16],
this system of equations follows from the dimensionally reduced Chern-Simons action
[43, 44],
(2)Rη + η3
, (5.81)
if we use the conformal gauge (2)gijdx
idxj = e2ξ (dx2 + dy2) and η is the curl of a vector
potential,
(2)g ǫijη = ∂iAj−∂jAi. To obtain equations (5.80) we vary the action with
respect to Ai and ξ. When varying the dimensionally reduced Chern-Simons action
with respect to gij there is however an additional equation to (5.80).
Using the results of Grumiller and Kummer [48], one obtains the most general
solution to the dimensionally reduced Chern-Simons system [16]
e2ξ =
η4, (5.82)
where L is an integration constant and dη = e2ξdx. Trading the coordinate x for η, we
get the following configuration of the fields
ds2 = − 4
P ′22 + η
[dt + σ]
P ′22 + η
dz2 + P 22
e−2ξdη2 + e2ξdy2
A = 2
P ′22 + η
[dt + σ] +
Vdy − i ℓ
d log
, (5.83)
where P2(z) = αz
2 + βz + γ, k is defined as above and the shift function reads
αη2 +
dy . (5.84)
These solutions preserve 1/4 of the original supersymmetry. In fact, the k = 0 solutions
coincide with the imaginary b ones and their PSL(2,R) transforms of sections 5.3 and
5.4. For k non-vanishing these are different solutions.
As can be seen from the Poisson bracket (4.34), the only possibility to have 1/2
supersymmetry is α = 0 and hence k ≤ 0. In fact, starting from any solution with
k ≤ 0, one can always obtain α = 0 by an appropriate PSL(2,R) transformation. The
– 51 –
non-trivial part of the PSL(2,R) symmetry is z 7→ −1/(z + δ), whose action on the
parameters α, β and γ of the Ansatz (5.79) is given by
α 7→ αδ2 − βδ + γ , β 7→ 2αδ − β , γ 7→ α , (5.85)
which keeps k fixed. Indeed, for k ≤ 0, there is always a PSL(2,R) transformation that
sets α = 0, while this is impossible for k > 0.
The requirement α = 0 leads to the half-supersymmetric imaginary b solution of
section 5.3 for k = 0. In the case of k negative, when α = 0 one can scale β to 1 in
(5.79) without loss of generality, and γ can be put to zero by a translation in z. Hence
the function b is given by
b = −1
1− iη . (5.86)
The metric is given in (D.32) and is generically of Petrov type D. The second Killing
spinor can be found in (D.33). As shown in appendix D, the G0 = ψ− = 0 solutions
are either the imaginary b ones, anti-Nariai spacetime or the above 1/2 supersymmetric
solution with k = −1.
We would like to mention that (5.82) is the most general solution to the dimen-
sionally reduced Chern-Simons system, but not to the equations (5.80). The reason for
this is the additional constraint one obtains when varying (5.81) with respect to gij.
An example of this is provided by the Petrov type I solution with b = i(x/ℓ)2 in section
5.3 and its PSL(2,R) transform given in eq. (2.44) of [16].
6. Final remarks
In this paper, we applied spinorial geometry techniques to classify all supersymmetric
solutions of minimal N = 2 gauged supergravity in four dimensions.
In the presence of null Killing spinors, the problem can be completely solved,
and all 1/4- and 1/2-supersymmetric solutions have been written down explicitly. We
showed that there are no 1/4-BPS backgrounds with U(1)⋉R2-invariant Killing spinors
and those with R2-invariant Killing spinors have been derived in sections 3.1 and 3.2.
The backgrounds in the latter section were previously unknown and are Petrov type
II configurations describing gravitational waves propagating on a bubble of nothing
in AdS4. In addition, it turned out that there are no 1/2-BPS backgrounds with R
invariant Killing spinors and hence any additional Killing spinor is timelike. In section
3.3 we gave the backgrounds with one null and one timelike Killing spinor.
For a timelike Killing spinor we derived the conditions for the corresponding back-
grounds in section 4.1 and 4.2. We worked out the first integrability conditions nec-
essary for the existence of a second Killing spinor in section 4.3. We explicitly solved
– 52 –
these equations in a number of subcases in section 5, and thereby found several new
solutions, like the bubbles of nothing in AdS4, already obtained in the null formalism,
and their PSL(2,R)-transformed configurations. Furthermore, our results showed that
the generalized holonomy in the case of one preserved complex supercharge is contained
in A(3,C), supporting thus the classification scheme of [4].
In addition, the time-dependence of a second time-like Killing spinor was shown to
be an overall exponential factor with coefficient G0 in section 4.4. In the case G0 = 0
these equations have been solved in full generality, up to a second order ordinary
differential equation. We expect this class to comprise a large number of interesting
1/2-BPS solutions. Indeed, all the examples of section 5 either have vanishing G0 or
can be transformed to that case by a coordinate transformation.
There are several interesting points that remain to be understood. First of all, it
would be desirable to get a deeper insight into the underlying geometric structure in
the case of U(1) invariant spinors. In five dimensions, spacetime is a fibration over a
four-dimensional Hyperkähler or Kähler base for ungauged and gauged supergravity
respectively [8, 12], whereas in four-dimensional ungauged supergravity one has a fi-
bration over a three-dimensional flat space [5]. This suggests that the base for D = 4
gauged supergravity might be an odd-dimensional analogue of a Kähler manifold, i. e. ,
a Sasaki manifold. From the equations (4.22) and (4.23) this is not obvious.
Secondly, in [16], a surprising relationship between the equations (4.22), (4.23) gov-
erning 1/4 BPS solutions and the gravitational Chern-Simons theory [43] was found.
Why such a relationship should exist is not clear at all, and deserves further investiga-
tions.
The third point concerns preons, which were conjectured in [45] to be elementary
constituents of other BPS states. In type II and eleven-dimensional supergravity, it was
shown that imposing 31 supersymmetries implies that the solution is locally maximally
supersymmetric [27,30,46]. Similar results in four- and five-dimensional gauged super-
gravity were obtained in [28,29]. This implies that preonic backgrounds are necessarily
quotients of maximally supersymmetric solutions. While M-theory preons cannot arise
by quotients [47], it remains to be seen if 3/4 supersymmetric solutions to N = 2,
D = 4 or D = 5 gauged supergravities really do not exist. The only maximally super-
symmetric backgrounds in these theories are AdS4 [13] and AdS5 [12] respectively, so
the putative preonic configurations must be quotients of AdS.
Finally, it would be interesting to apply spinorial geometry techniques to classify
all supersymmetric solutions of four-dimensional N = 2 matter-coupled gauged super-
gravity. Work in this direction is in progress [49].
– 53 –
Acknowledgments
We are grateful to Alessio Celi, Marcello Ortaggio and Christoph Sieg for useful dis-
cussions. This work was partially supported by INFN, MURST and by the European
Commission program MRTN-CT-2004-005104. D.R. wishes to thank the Università
di Milano for hospitality. Part of this work was completed while he was a post-doc
at King’s College London, for which he would like to acknowledge the PPARC grant
PPA/G/O/2002/00475. In addition, he is presently supported by the European EC-
RTN project MRTN-CT-2004-005104, MCYT FPA 2004-04582-C02-01 and CIRIT GC
2005SGR-00564.
A. Spinors and forms
In this appendix, we summarize the essential information needed to realize the spinors
of Spin(3,1) in terms of forms. For more details, we refer to [50]. Let V = R3,1 be
a real vector space equipped with the Lorentzian inner product 〈·, ·〉. Introduce an
orthonormal basis e1, e2, e3, e0, where e0 is along the time direction, and consider the
subspace U spanned by the first two basis vectors e1, e2. The space of Dirac spinors is
∆c = Λ
∗(U⊗C), with basis 1, e1, e2, e12 = e1∧e2. The gamma matrices are represented
on ∆c as
Γ0η = −e2 ∧ η + e2⌋η , Γ1η = e1 ∧ η + e1⌋η ,
Γ2η = e2 ∧ η + e2⌋η , Γ3η = ie1 ∧ η − ie1⌋η , (A.1)
where
ηj1...jkej1 ∧ . . . ∧ ejk
is a k-form and
ei ∧ η =
(k − 1)!ηij1...jk−1ej1 ∧ . . . ∧ ejk−1 .
One easily checks that this representation of the gamma matrices satisfies the Clifford
algebra relations {Γa,Γb} = 2ηab. The parity matrix is defined by Γ5 = iΓ0Γ1Γ2Γ3,
and one finds that the even forms 1, e12 have positive chirality, Γ5η = η, while the odd
forms e1, e2 have negative chirality, Γ5η = −η, so that ∆c decomposes into two complex
chiral Weyl representations ∆+c = Λ
even(U ⊗ C) and ∆−c = Λodd(U ⊗ C).
Let us define the auxiliary inner product
αiei,
βjej〉 =
α∗iβi (A.2)
– 54 –
on U ⊗ C, and then extend it to ∆c. The Spin(3,1) invariant Dirac inner product is
then given by
D(η, θ) = 〈Γ0η, θ〉 . (A.3)
In many applications it is convenient to use a basis in which the gamma matrices act
like creation and annihilation operators, given by
Γ+η ≡
(Γ2 + Γ0) η =
2 e2⌋η , Γ−η ≡
(Γ2 − Γ0) η =
2 e2 ∧ η ,
Γ•η ≡
(Γ1 − iΓ3) η =
2 e1 ∧ η , Γ•̄η ≡
(Γ1 + iΓ3) η =
2 e1⌋η . (A.4)
The Clifford algebra relations in this basis are {ΓA,ΓB} = 2ηAB, where A,B, . . . =
+,−, •, •̄ and the nonvanishing components of the tangent space metric read η+− =
η−+ = η••̄ = η•̄• = 1. The spinor 1 is a Clifford vacuum, Γ+1 = Γ•̄1 = 0, and
the representation ∆c can be constructed by acting on 1 with the creation operators
Γ+ = Γ−,Γ
•̄ = Γ•, so that any spinor can be written as
φā1...ākΓ
ā1...āk1 , ā = +, •̄ .
The action of the Gamma matrices and the Lorentz generators ΓAB is summarized in
the table 6.
1 e1 e2 e1 ∧ e2
Γ+ 0 0
2e2 −
2e1 ∧ e2 0 0
2e1 0
2e1 ∧ e2 0
Γ•̄ 0
Γ+− 1 e1 −e2 −e1 ∧ e2
Γ•̄• 1 −e1 e2 −e1 ∧ e2
Γ+• 0 0 −2e1 0
Γ+•̄ 0 0 0 2
Γ−• −2e1 ∧ e2 0 0 0
Γ−•̄ 0 2e2 0 0
Table 6: The action of the Gamma matrices and the Lorentz generators ΓAB on the different
basis elements.
– 55 –
Note that ΓA = UA
aΓa, with
1 0 1 0
−1 0 1 0
0 1 0 −i
0 1 0 i
∈ U(4) ,
so that the new tetrad is given by EA = (U∗)AaE
B. Spinor bilinears
Given a Killing spinor
ǫ = c01 + c1e1 + c2e2 + c12e1 ∧ e2 , (B.1)
one can construct the bilinears
f̃ = −iD(ǫ, ǫ) = −i (c0c∗2 − c1c∗3 − c2c∗0 + c12c∗1) , (B.2)
g̃ = −iD(ǫ,Γ5ǫ) = c0c∗2 + c1c∗3 + c2c∗0 + c12c∗1 , (B.3)
Ṽ = D(ǫ,Γµǫ) dx
|c2|2 + |c12|2
− |c0|2 − |c1|2
|c2|2 + |c12|2 + |b|2
|c0|2 + |c1|2
(dt + σ)
ψ [(c2c
1 − c0c∗12) dw + (c1c∗2 − c12c∗0) dw̄] , (B.4)
B̃ = D(ǫ,Γ5Γµǫ) dx
|c2|2 − |c12|2
+ |c0|2 − |c1|2
|c2|2 − |c12|2 − |b|2
|c0|2 − |c1|2
(dt + σ)
ψ [(c2c
1 + c0c
12) dw + (c1c
2 + c12c
0) dw̄] , (B.5)
D(ǫ,Γµνǫ) dx
µ ∧ dxν = − (c0c∗2 − c1c∗12 + c2c∗0 − c12c∗1) dt ∧ dz
12 + |b|2c0c∗1
dt ∧ dw − 2e
0 + 4|b|2c1c∗0
dt ∧ dw̄
2 − c1c∗12 + c2c∗0 − c12c∗1)σw +
2|b|3 c2c
2|b|c0c
dz ∧ dw
2 − c1c∗12 + c2c∗0 − c12c∗1)σw̄ +
2|b|3 c12c
2|b|c1c
dz ∧ dw̄
– 56 –
12σw̄ − c12c∗0σw + |b|2 (c0c∗1σw̄ − c1c∗0σw)
4|b| (c0c
2 + c1c
12 − c2c∗0 − c12c∗1)
dw ∧ dw̄ . (B.6)
Given the first Killing spinor of the form ǫ1 = 1 + be2 and the second Killing spinor
ǫ2 = c01 + c1e1 + c2e2 + c12e1 ∧ e2, one can also construct mixed bilinears of the type
D(ǫ1,Γ···ǫ2), which verify the same differential equations as the bilinears built from the
original two Killing spinors:
f̂ = −i(b̄c0 − c2) , ĝ = b̄c0 + c2 , (B.7)
(c2 + bc0) (dt + σ) +
(c2 − bc0) dz +
b̄c1 − c12
dw̄ , (B.8)
(c2 − bc0) (dt+ σ) +
(c2 + bc0) dz +
b̄c1 + c12
dw̄ . (B.9)
C. The case P ′ = 0
In section 4.3, we simplified the equations for the second Killing spinor under the
assumption P ′ 6= 0, where P = e−2ψb∂b. Here we consider the case P ′ = 0. To this
end, we need the following subset of the Killing spinor equations (4.31):
∂+ψ2 −
ψ12 = 0 , (C.1)
∂+ψ12 − re−ψ∂•̄ ln b̄ ψ1 −
ψ12 = 0 , (C.2)
∂−ψ2 +
e−ψ∂•̄ ln b ψ1 −
ψ2 = 0 , (C.3)
∂−ψ12 −
ψ12 = 0 , (C.4)
re−ψ∂•
e2ψψ2
ψ1 = 0 , (C.5)
re−ψ∂•
e2ψψ12
ψ1 = 0 . (C.6)
If P ′ = 0, (4.35) implies ψ− = 0 or ∂P = 0. Let us first assume the former, i. e. ,
ψ2 = ψ12. From (C.6) – (C.5) one obtains then ψ1 = 0 or
= 0 . (C.7)
– 57 –
• If ψ1 = 0, (C.4) – (C.3) yields ψ2 = 0, and thus there exists no further Killing
spinor.
• If (C.7) holds, one can use (C.1) and (C.4) to show that ∂+ψ2 = ∂−ψ2 = 0, or
equivalently ∂tψ2 = ψ
2 = 0. Using this in (C.2) and (C.3) and deriving with
respect to t, one gets ∂̄b̄ ∂tψ1 = ∂̄b ∂tψ1 = 0. When ∂tψ1 6= 0, this means that
∂̄b = ∂b = 0, so b = b(z), which is a case analyzed in section 5.1. If instead
∂tψ1 = 0, all the ψi are independent of t, and the Killing spinor equations reduce
to the system (4.38) to (4.49) with G0 = 0.
In the case ∂P = 0, consider the integrability condition
′ + ψ−∂Q = 0 , (C.8)
where Q = e−2ψ b̄∂b̄, following from the first line of Ñwt. As long as Q
′ 6= 0, with the
same reasoning as in section 4.3, one obtains the system (4.38) to (4.49). If Q′ = 0,
(C.8) implies ψ− = 0 or ∂Q = 0. The case ψ− = 0 was already considered above, so
the only remaining case is P ′ = ∂P = Q′ = ∂Q = 0. For P = Q = 0 we get again
b = b(z), so without loss of generality we can assume P 6= 0 or Q 6= 0. Suppose that
Q = 0, P 6= 0, so b = b(w, z). Take the logarithm of e−2ψb∂b = P (w̄), derive with
respect to z, use (4.15), and apply ∂̄. This leads to ∂b = 0, which is a contradiction to
the assumption P 6= 0. In the same way one shows that P = 0, Q 6= 0 is not possible,
so that both P and Q must be nonvanishing. Now use the third row of Ñw̄t, which
leads to Q̄ψ2 = 0 and hence ψ2 = 0. Finally, the last row of Ñw̄t yields ψ− = 0, i. e. ,
the case already considered above.
Hence, the conclusion is that in the case P ′ = 0, the second Killing spinor either
has G0 time-dependence of the form (4.37), or leads to solutions with b = b(z). The
latter are treated separately in section 5.1. As can be found there, all 1/2-BPS solutions
with b = b(z) also have second Killing spinors with G0 time-dependence of the form
(4.37). Hence this time-dependence is a completely general result20 for second Killing
spinors in the time-like case.
D. Half-supersymmetric solutions with G0 = 0
From the difference of equations (4.39)−(4.43) and (4.48)−(4.49) one gets ψ− = ψ−(w).
Furthermore, [(4.42)−(4.40)+e−2ψ(4.47)] and (4.44) yield ψ1 = ψ1(z). Assuming ψ− 6=
20The only counterexample is the third Killing spinor of AdS4, see (5.30), but since this is maximally
supersymmetric it does not contradict the result.
– 58 –
0, eqns. (4.38) and (4.41) can be written in the form
= 0 ,
= 0 , (D.1)
where β = ψ1/ψ−. Deriving (D.1) with respect to w̄ gives
+ ∂∂̄
= 0 ,
+ ∂∂̄
= 0 .
Now use (D.1) in the difference between the first equation and the complex conjugate
of the second one to get
β̄β ′ − β̄ ′β
= 0 .
Observe that β̄β ′ − β̄ ′β = |ψ−|−2
1 − ψ̄′1ψ1
(z), so that for ψ̄1ψ
1 − ψ̄′1ψ1 6= 0 there
must exist a real function B(z) and a generic function h(w, w̄) such that
b = B(z)h(w, w̄) .
Plugging this into (D.1), we conclude that
= 0 ,
so that the phase of the function h is fixed, h = hR(w, w̄)e
iϕ0 , with hR real. Using
(4.18), the constancy of the phase of b implies that the shift vector σ does not depend
on z. (4.19) gives then
= 0 ,
or, using (4.15),
cosϕ0
+B′ = 0 ,
and thus
B′ = c ,
cosϕ0
= −c ,
where c denotes a real constant. Now we have to distinguish to cases:
1. c 6= 0: In this case b(z) =
B0 − 2 cosϕ03ℓ z
eiϕ0 . Plugging this into the first of
eqns. (D.1) one gets
= 0 ,
which is solved by ψ1 = ηb where η is a constant. But this yields ψ̄1ψ
1−ψ̄′1ψ1 = 0,
which contradicts our assumption.
– 59 –
2. c = 0: In this case b(w, w̄) = ihR(w, w̄). The combination (4.40)+(4.42)−(4.39)−
(4.43) leads to ψ− = 0, which again contradicts one of our assumptions.
We thus conclude that ψ̄1ψ
1− ψ̄′1ψ1 = 0, and hence ψ1 = ζ(z)eiθ0 where θ0 is a constant
and ζ(z) is a real function. Sending ψi → e−iθ0ψi we can take ψ1 real and non-negative
without loss of generality. Let us now consider the case where both ψ1 and ψ− are
non-vanishing. This allows to introduce new coordinates Z,W, W̄ such that
ψ1(z)
dz , dW =
ψ−(w)
, dW̄ =
ψ̄−(w̄)
Note that one can set ψ− = 1 using the residual gauge invariance w 7→ W (w), ψ 7→
ψ̃ = ψ − 1
ln(dW/dw)− 1
ln(dW̄/dw̄) leaving invariant the metric e2ψdwdw̄. We can
thus take W = w in the following. Equations (4.38) and (4.41) are then equivalent to
(∂Z + ∂)ϕ = 0 , ∂Z lnψ1 − (∂Z + ∂) ln r = 0 .
From the real part of the first equation we have
ϕ = ϕ(Z − w − w̄) .
Using ψ1 = ψ1(Z), the second equation implies
(∂Z + ∂)
= 0 ,
and hence
= ρ(Z − w − w̄) .
The function b must thus have the form
b(Z,w + w̄) = ψ1(Z)B(Z − w − w̄) ,
where B(Z − w − w̄) = ρ(Z − w − w̄)eiϕ(Z−w−w̄). The difference between (4.45) and
(4.46) yields
(∂Z + ∂) (lnψ1 − ψ) = 0 ,
so that lnψ1 − ψ = −H(Z − w − w̄) with H real. This gives
e2ψ = ψ1(Z)
for the conformal factor. In terms of the new coordinate Z, (4.15) reads
∂Zψ +
= 0 .
– 60 –
Using the definition of H we get
Ḣ + ∂Z lnψ1 +
= 0 , (D.2)
where a dot denotes a derivative with respect to Z−w− w̄. We can thus conclude that
∂Z lnψ1 = γ/ℓ for some constant γ, i. e. ,ψ1(Z) = ψ
γZ/ℓ. By shifting Z one can set
1 = 1. Calling χ = ψ+/ψ−, the only remaining nontrivial Killing spinor equations
∂Zχ− 2
χ+ 2iϕ̇+
= 0 ,
− Ḣ + γ
χ− 2ie−2H ϕ̇− 1
= 0 ,
∂χ + 2
χ− 2iϕ̇− 1
= 0 ,
∂̄χ+ 2
χ− 2iϕ̇ = 0 ,
χ + 2
1 + e−2H
− Ḣ + γ
= 0 .
Summing the first and the third equation yields χ = χ(Z −w− w̄), so that we are left
χ̇− 2
χ+ 2iϕ̇+
= 0 , (D.3)
− Ḣ + γ
χ− 2ie−2H ϕ̇− 1
= 0 , (D.4)
− χ̇+ 2 ρ̇
χ− 2iϕ̇ = 0 , (D.5)
χ + 2
1 + e−2H
− Ḣ + γ
= 0 . (D.6)
Adding (D.4) and (D.6) one gets
Ḣχ +
= 0 , (D.7)
which means that χ is purely imaginary. From (D.2) and (D.7) we obtain then the
function B,
+ Ḣ(1 + χ) = 0 . (D.8)
– 61 –
Using this, the remaining Killing spinor equations reduce further to
1 + e2H
= 0 , (D.9)
= 0 , (D.10)
1 + χ2
− 2 ρ̇
1 + e−2H
. (D.11)
Note that (D.9) automatically implies the integrability condition for the system (4.18),
(4.19), which reduces to
∂Zσw =
, ∂σw̄ − ∂̄σw = −
. (D.12)
Thus, also equation (4.23) is satisfied, whereas (4.22) reads
1 + e−2H
2Ḧ + Ḣ2
1 + 3χ2
. (D.13)
From (D.8) we obtain the phase ϕ and the modulus ρ of B,
tanϕ = i
− Ḣ2χ2 .
Plugging this into equation (D.10) yields
2ḦḢχ
1− χ2
− Ḣ2χ̇
1 + 3χ2
χ̇ = 0 .
Using (D.13), this can be rewritten as
1− χ2
1 + e−2H
= 0 ,
so that either Ḧ = 0 or Ḣχ (1− χ2) +
1 + e−2H
χ̇ = 0. It is straightforward to show
that the first case leads to AdS4, whereas the second one implies
e2H + 1
1− χ2 = −α
2 , (D.14)
where α is a real integration constant. Equations (D.9) and (D.11) are then identically
satisfied. Solving (D.14) for χ and plugging into (D.13) yields finally the ordinary
differential equation (4.50), which determines half-supersymmetric solutions with G0 =
0. Putting together all our results, we obtain (4.52) for the metric. Note that in the
– 62 –
case γ 6= 0 one can always set γ = 1 by rescaling the coordinates.
The second Killing spinor for these backgrounds is given by
αT = (α0, ρ
−2e−γZ/ℓ,
eH) ,
where
α0 = −
+ α̂0(Z,w, w̄) ,
and α̂0 is a solution of the system
∂Zα̂0 =
− iϕ̇ + γ
∂α̂0 = −
+ iϕ̇
, (D.15)
∂̄α̂0 = −
σw̄ +
+ iϕ̇
γχ e2H
ℓψ1ρ2
It is straightforward to verify that the integrability conditions for this system are al-
ready implied by (D.9), (D.10) and (D.12).
Consider now the case ψ− = 0. From the difference of equations (4.38) and (4.41)
it follows that b′/b is real. Then (4.38) and (4.44) imply that ψ is a real function,
depending only on z, ψ1 = ψ1(z). Moreover, since ψ12 = ψ2, the difference of equations
(4.45) and (4.46) imply that b′/b+ 1/ℓb is imaginary.
The conditions b′/b real and b′/b+ 1/ℓb imaginary can be satisfied simultaneously
in three different ways:
• b′/b = 0 hence b = b(w, w̄) is an imaginary function independent of z. This case
is solved completely in section 5.3.
• b′/b+ 1/ℓb = 0 implies b = −z/ℓ + c and corresponds to AdS2 ×H2, analyzed in
section 5.1.1. It is also a subcase of the following, general case,
• if we are not in one of the previous special cases, the function b must take the
b = −1
1− iY (w, w̄) , (D.16)
where Y (w, w̄) is some real function to be determined.
We thus have to solve just for the ansatz (D.16). Equation (4.38) implies ψ′1/ψ1 = b
than is solved by ψ1 = z, where we have reabsorbed the integrability constant in the
– 63 –
scale of z. Equation (4.39) (or equivalently (4.43)) tells us that ψ2 = ψ2(w, w̄), so that
the remaining independent equations read
iz2e−2ψ
1 + Y 2
− ψ2 = 0 ,
∂ψ2 + ∂
1 + Y 2
ψ2 − iY = 0 ,
∂̄ψ2 + ∂̄ log
1 + Y 2
ψ2 = 0 .
The first equation allows us to define a function H(w, w̄) such that
eψ = zeH(w,w̄) , (D.17)
while the last one implies that there must exist a holomorphic function C(w) such that
1 + Y 2
. (D.18)
Thus we are left with
e2HC(w) = i∂̄Y , (D.19)
e2HC(w)
= ie2HY
1 + Y 2
. (D.20)
This set of equations automatically implies the integrability condition for the system
(4.18), (4.19), which reduces to
∂zσ = i
, (D.21)
∂σ̄ − ∂̄σ = iℓ2e2HY
1 + Y 2
. (D.22)
Thus also (4.23), which reads
∂∂̄Y − e2HY
1 + Y 2
= 0 , (D.23)
is satisfied and it turns out that also the Bianchi identity (4.22), namely
∂∂̄2H − e2H
1 + 3Y 2
= 0 , (D.24)
holds. We conclude that a solution to the system (D.19), (D.20) describes a 1/2-BPS
configuration of the “gravitational Chern-Simons” system discussed in [16]. If C(w) = 0
then necessarily also Y = 0 so that we are left with AdS. If C(w) 6= 0 then we can
define new variables W and W̄ such that
∂W = C(w)∂ , ∂W̄ = C̄(w̄)∂̄ , (D.25)
– 64 –
so that we have
e2HCC̄ = i∂W̄Y ,
e2HCC̄
= ie2HCC̄Y
1 + Y 2
As what we did in the previous case, we can set C(w) = 1 using the residual gauge
invariance w 7→ W (w), ψ 7→ ψ̃ = ψ − 1
ln(dW/dw) − 1
ln(dW̄/dw̄) leaving invariant
the metric e2ψdwdw̄. We can thus take W = w without loss of generality, and get
e2H = i∂̄Y , (D.26)
2∂H = iY
1 + Y 2
(D.26) implies Y = Y [i(w − w̄)] and hence H = H [i(w− w̄)]. Denoting with a dot the
derivative with respect to the combination i(w − w̄) we have
e2H = Ẏ , (D.27)
2Ḣ = Y
1 + Y 2
. (D.28)
The equations for the shift form can now be integrated, giving
Ẏ d (w + w̄) (D.29)
Plugging (D.27) into (D.28) leads to
Ÿ = Ẏ Y (1 + Y 2) , (D.30)
which, integrated once, gives
Y 2 +
Y 4 ≡ P (Y ) , (D.31)
where L is a real constant and k = −121. We can thus use Y as a new coordinate,
instead of i(w − w̄). Call X = w + w̄, so that the solution reads
ds2 = − 4
1 + Y 2
PC(Y )dX
1 + Y 2
dz2 + z2
PC(Y )dX
PC(Y )
A = 2
1 + Y 2
dt+ ℓY
PC(Y )
1 + Y 2
1 + Y 2
1 + Y 2
. (D.32)
21The link with the notation of [48], where C and k are the Casimirs of the Poisson sigma model
equivalent to the dimensionally reduced gravitational Chern-Simons model in 2D, is given by 2C =
L/ℓ△.
– 65 –
We can thus finally compute the second Killing spinor, with the result
ǫ2 = −
1 + Y 2
PC(Y )
1 + iY
1− iY e
1 + Y 2
(1 + iY ) +
1− iY
e2 + ℓ
PC(Y )
1 + Y 2
e1 ∧ e2 . (D.33)
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