IFUM809FT
hepth/0411153
On Supersymmetric Solutions of Gauged Supergravity^{*}^{*}*Talk given at the “Workshop on Dynamics and Thermodynamics of Black Holes and Naked Singularities”, Politecnico di Milano, 1315 May 2004
Marco M. Caldarelli^{†}^{†}†
Dipartimento di Fisica dell’Università di Milano
and
INFN, Sezione di Milano,
Via Celoria 16, I20133 Milano.
We review the classification of supersymmetric solutions to minimal gauged supergravity in four dimensions. After a short introduction to the main features of the theory, we explain how to obtain all its solutions admitting a Killing spinor. Then, we analyze the rich mathematical structure behind them and present the supersymmetric field configurations. Among them, we find supersymmetric black holes, quarter and half BPS traveling waves, kink solutions, and supersymmetric Kundt and RobinsonTrautman solutions. Finally, we generalize the classification to include external sources, and show a particular solution describing a supersymmetric Gödeltype universe.
1 Introduction
Recently, significant steps have been made towards a classification of supersymmetric solutions of supergravity theories. Apart from the interest in understanting the mathematical structure lying behind these BPS configurations, such investigations are important due to the role supersymmetry has played in the developments in string theory. Having at hand a systematic approach to construct BPS solutions avoids the use of special ansätze and has lead to the discovery of many new supergravity backgrounds.
After the seminal work by Tod [1], who wrote down all metrics admitting supercovariantly constant spinors in , ungauged supergravity, progress in this direction has been made mainly during the last years using the mathematical concept of Gstructures [2]. This formalism has been applied successfully to several supergravity theories [3, 4, 5]. New interesting solutions have been obtained using these techniques, among which there are a maximally supersymmetric Gödel universe in five dimensions [4], supersymmetric AdS black holes in five dimensions [6], D1D5P black hole microstates [7] in the spirit of the proposal by Mathur et al. [8], and half BPS excitations of AdS configurations described by free fermions in the dual field theories [9]. Moreover, the results for the simpler, lowerdimensional supergravities can be uplifted to ten or eleven dimensions to describe some sectors of ten and elevendimensional supergravities.
In this article, we will review the classification of supersymmetric solutions to minimal gauged supergravity in four dimensions [5, 10, 11], which extends the early results [12, 13, 14, 15, 16, 17], where only plane waves in AdS and BPS solutions belonging to the general Petrov type D solution of Plebański and Demiański [18] were analyzed.
We will begin with a short introduction to minimal gauged supergravity in four dimensions and a presentation of the technique used to find all its supersymmetric solutions [5]. These fall in two cases, the “lightlike case” and the “timelike case,” to be defined below. In section 3 we shall present the complete classification of the lightlike solutions, showing that there is an enhancement of the supersymmetry for a subfamily of solutions. Next, in section 4, we shall present the general timelike solution, describe some properties of the set of equations governing it and solve them in some cases, obtaining interesting new solutions [11]. In section 5 we will extend the theory to include external sources, and build a supersymmetric Gödel universe[10]. The final section will be devoted to conclusions.
2 Minimal gauged supergravity
The gauged version of supergravity in four dimensions was found by Freedman and Das [19] and by Fradkin and Vasiliev [20]. In this theory, the rigid symmetry rotating the two independent Majorana supersymmetries present in the ungauged theory is made local by introduction of a minimal gauge coupling between the photons and the gravitini. Local supersymmetry then requires a negative cosmological constant and a gravitini mass term. The theory has four bosonic and four fermionic degrees of freedom; it describes a graviton , two real gravitini , and a Maxwell gauge field . As we said, the latter is minimally coupled to the gravitini, with coupling constant . The bosonic trucation of the lagrangian reads
(1) 
and describes the EinsteinMaxwell theory with a negative cosmological constant . We are interested in supersymmetric solutions of this model, i.e. bosonic solutions whose supersymmetry variation vanishes. A local supersymmetry transformation generated by an infinitesimal Dirac spinor produces a variation of the gravitino field equal to , where and the supercovariant derivative is defined by
(2) 
Hence, a bosonic background is invariant under some local supersymmetry transformations if there is some nontrivial solution to the Killing spinor equation
(3) 
This equation implies the following integrability condition,
(4)  
For a solution to be maximally supersymmetric, condition (4) must impose no algebraic constraint on the Killing spinor, and therefore the only maximally supersymmetric geometry is given by AdS with vanishing gauge field. An analogous result in five dimensions was obtained in [4].
To find the most general BPS solution we suppose that the geometry admits at least one Killing spinor , which can be used to construct some bosonic differential forms out of its bilinears. These forms are not all independent, and in fact there are some algebraic relations linking them, derived from the Fierz identities. Particularly important are the two oneforms and . A first consequence of the algebraic relations is that , and therefore the vector is either timelike or null. For simplicity, we will name timelike solutions the solutions which have a Killing spinor yielding a timelike vector and lightlike solutions the ones giving a null vector . If the solution preserves just one quarter of the supersymmetry, the timelike or lightlike character of the solution produces a rough classification. However, if there is some additional supersymmetry, the solution can fall in both classes, depending on the particular Killing spinor used to define the vector .
Next, the Killing spinor equation imposes a set of differential constraints for these forms. In particular, one finds and that allow the introduction of two preferred coordinates and , defined by and . Then, by cleverly using the algebraic and differential constraints and imposing the equations of motion for the gauge field, one obtains the most general form of the metric and gauge fields preserving some supersymmetry [5]. In the timelike case, the Einstein equations automatically follow from the integrability conditions for the Killing spinor, whereas in the lightlike case the component along of these equations has to be additionally imposed. Finally, one obtains a last equation requiring that the integrability conditions are satisfied, to ensure the actual existence of the Killing spinor.
In the following we will analyze the solutions that arise, and review the progress made towards a complete classification. We shall begin with the lightlike case, which is simpler and allows for an exhaustive classification. Then, we shall turn to the more complicated timelike case.
3 The lightlike case
The general lightlike supersymmetric solution is an electrovac traveling wave with metric and electromagnetic field given by [5]
(5)  
(6) 
Here, the arbitrary function defines the profile of the electromagnetic wave propagating on this metric, while is any solution of the inhomogeneous Siklos equation [21]^{1}^{1}1In this review, denotes the twodimensional flat laplacian in plane.
(7) 
The dependence of on describes the profile of the gravitational wave. The general solution to this equation reads [21]
(8) 
where is an arbitrary holomorphic function in .
This family of traveling waves enjoys a large group of coordinate transformations which preserve the form (5) of the line element. Under the diffeomorphism defined by
(9) 
where , and are arbitrary functions of , the metric keeps the same form (5) but with given by [21]
(10) 
Here, the prime denotes the derivative with respect to , while
(11) 
defines the Schwarzian derivative. These are not Killing symmetries, the metric changes, but solutions are brought into other solutions. The special diffeomorphisms with correspond to reparameterizations of the coordinate ; this transformation group is generated by a central extension of the Virasoro algebra [22].
To obtain a complete classification of these solutions one has to solve the Killing spinor equation and obtain the exact fraction of supersymmetry preserved by them. This can be done exactly (see [11]), and surprisingly it turns out that a subclass of these solutions has an enhanced supersymmetry. It is useful to consider separately the cases with and without gauge field excitations.
3.1 Purely gravitational waves
This case, with , has been extensively studied in [21]. Two cases are possible, and the vanishing of , with discriminates them.
The maximally supersymmetric case:
The condition is satisfied if and only if the spacetime is AdS [21]. In this case no condition is imposed on the Killing spinor by the integrability conditions, and one recovers the wellknown result that AdS is a maximally supersymmetric spacetime [23]. This is the unique maximally supersymmetric solution of the model under consideration [5].
One quarter BPS Lobatchevski waves:
When or , the spacetime describes an exact AdS gravitational wave [24]. The integrability conditions project out half of the components of the Killing spinor, but while solving the Killing spinor equation, an additional condition on the Killing spinor emerges, leaving just one quarter of the supersymmetries, as shown in [17]. This is a very simple example that shows that the vanishing of the supercurvature is not a sufficient condition to ensure the existence of Killing spinors.
3.2 The electromagnetic case
Let us now turn on the electromagnetic field and consider the case where . Generically, these solutions preserve one quarter of the supersymmetries. However, if the solution of the Siklos equation assumes a very particular form, then there is a supersymmetry enhancement. It was shown in [11] that for
(12) 
where and are arbitrary functions and and are given by
(13) 
the supergravity solution preserves exacly one half of the supersymmetries. Recently, the same phenomenon has been noted in the case of fivedimensional minimal gauged supergravity [26]. In (13), is an arbitrary real integration constant, which turns out to be the only physical parameter, since by an appropriate diffeomorphism of the form (9), explicitely given in [11], it is possible to put the metric in a canonical form where and
(14) 
Finally, it is important to distinguish between the and solutions. Siklos obtained a beautiful classification for spacetimes of the form (5), according to the number of independent Killing vectors [21]. It follows that for the spacetime admits a fivedimensional group of isometries, generated by five Killing vectors. We shall meet again this metric as the timelike solution (38) of section 4.2. On the other hand, when , the canonical form of falls in the class of [21], meaning that the only Killing vectors are and .
In any case, by computing the norm squared of the Killing vectors constructed from the Killing spinors, one checks that every one half supersymmetric lightlike solution is also a timelike solution.
SUSY 
purely gravitational solutions

electrovac spacetimes


1/4  Lobatchevski wave  
1/2  none  (14) 
3/4  none  none 
Max  AdS  none 
The complete classification of supersymmetric solutions in the lightlike case is summarized in table 1.
4 The timelike case
The general BPS solution for timelike reads [5]
(15)  
where ; , , and we defined , with and . The timelike Killing vector is given by and the functions , , , that depend on , are determined by the system
(16)  
(17)  
(18) 
where , and a prime denotes differentiation with respect to . (16) comes from the combined Maxwell equation and Bianchi identity, whereas (17) results from the integrability condition for the Killing spinor . Finally, the shift vector is obtained from
(19) 
with .
Before presenting the explicit solutions of the timelike case, let us study some general properties of the system (16) – (18).
Decomposing into its real and imaginary part, , one checks that the real part of eqn. (16) follows from (17) and (18), so that the remaining system is
(20)  
(21) 
together with . These equations can be derived from the action
(22) 
which enjoys various symmetries: first of all, we note that it is invariant under PSL transformations
(23) 
if transforms like a conformal field of weight two and as a Liouville field. As a consequence, has a connectionlike transformation behaviour. This symmetry acts in a nontrivial way on supergravity solutions, and can be used to generate new BPS solutions starting from known ones.
There is an additional infinitedimensional conformal symmetry, corresponding to holomorphic coordinate transformations in . However, it is easy to see that, from the fourdimensional point of view, these represent diffeomorphisms that preserve the conformal gauge for the twometric . Thus, unlike the PSL transformations above, this symmetry cannot be used to generate new solutions.
4.1 “Purely magnetic” solutions ( real)
Let us consider first solutions with real. Then, in the coordinates (15), the electric components of the field vanish. For this reason, we shall refer to these configurations as “purely magnetic” solutions. One has , and the only equation to solve is (21), that reduces to
(24) 
This is similar to the continuous Toda equation (or heavenly equation) which determines selfdual Einstein metrics admitting at least one rotational Killing vector [25].
This highly nonlinear equation is difficult to solve; however, under the simplifying assumption that is a function of alone, one finds
(25) 
with , and being real integration constants obeying . Equation (24) reduces then to a Liouville equation for ,
(26) 
describing the metric on euclidean twomanifolds of constant curvature .
When , this curvature is negative, and one obtains the antiNariai AdS spacetime, with a purely magnetic Maxwell field,
(27)  
This configuration preserves half of the supersymmetries, and admits an isometry superalgebra [15].
When , the supergravity solution is given by
(28)  
and its properties are determined by the sign of the curvature of the transverse twometric. We obtain thus the following classification:
Positive curvature:
Using polar coordinates on the plane, the metric and gauge field are given by (28), with the Liouville field
(29) 
Introducing new coordinates defined by , , we get
(30) 
As is identified modulo , we see that , so that the effect of the parameter is to introduce a conical singularity () or an excess angle () on the north and south poles of the twosphere. For there is no singularity, and the solution reduces to the onequarter supersymmetric magnetic monopole found by Romans [12]. The magnetic charge of the solution reads
(31) 
If , there is a magnetic fluxline that passes through , which causes the magnetic charge to be different from the value of Romans’ solution.
Vanishing curvature:
If , the function is harmonic, so that the twomanifold with metric is flat. The choice leads to the maximally supersymmetric AdS vacuum solution, which is the only configuration with maximal supersymmetry [5].
Negative curvature:
4.2 “Purely electric” solutions ( imaginary)
If one assumes instead the function to be purely imaginary, the field configurations have vanishing magnetic field in the coordinate system (15), and therefore we will refer to these solutions as the “purely electric” ones. In this case , and thus and are independent of . The equations (20) and (21) reduce to
(33)  
(34) 
This system follows from the two–dimensional dilaton gravity action
(35) 
if we use the conformal gauge . However, the equations of motion following from (35) contain also the constraints (whose trace yields (33)), and therefore they are more restrictive than the system (33), (34). Of course, every solution that extremizes the action (35) is a solution of our system, but not vice versa. It is interesting to note that (35) is similar to the action that arises from KaluzaKlein reduction of the threedimensional gravitational ChernSimons term [30], with the only difference that here is a fundamental field, whereas the field arising in [30] is the curl of a vector potential.
If we use the ansatz , equations (33) and (34) are satisfied for and . The former value of yields the BPS solution [5]
(36) 
(37) 
This metric has four Killing vectors acting transitively on the whole spacetime and represents a homogeneous, stationary and geodesically complete BPS spacetime endowed with a nonnull electromagnetic field. In the context of EinsteinMaxwell theory, this Petrov type I solution was obtained in [31].
For the other value of the exponent, , the fourdimensional geometry and the electromagnetic field strength are respectively
(38) 
(39) 
By means of a coordinate transformation, this solution can be recast in the electrovac AdS travelling wave of section 3 with , and therefore preserves half of the supersymmetries and admits five Killing vectors.
4.3 Solutions with .
If we allow only for a dependence in , the conformal factor of the transverse twometric is given by (25), with an integration constant. Defining
(40) 
the equations can be further simplified in two cases, according to which coordinates the function depends on.
4.3.1 The case
The simplest case of vanishing has been analyzed in section 4.1. By relaxing this condition and allowing for an dependence in , the BPS configurations have and given by (17), and are described by the system
(41)  
(42) 
where denotes an arbitrary constant and are real integration constants obeying .
It is interesting to observe that if we choose , , so that and , the system for and reduces to the same set of equations (34) and (33) that describes “purely electric” solutions. Therefore, to any “purely electric” solution we can associate a new BPS configuration. Let us apply this to the solutions presented in section 4.2. Starting from the Petrov type I solution (36) one obtains
(43)  
while the configuration related to the BPS solution (38) is
(44)  
A calculation of the Weyl scalars shows that the two spacetimes, as before, are of Petrov type I and N respectively. Note that these solutions can also be obtained from their “purely electric” counterparts by an appropriate PSL transformation.
4.3.2 Kink solutions and generalizations
More general solutions can be obtained in the case. As shown in [11], the system (41)(42) follows from the dimensionally reduced gravitational ChernSimons model considered in [30]. Therefore, to any solution of this model we can associate a BPS solution of the gauged supergravity. Using (for ) the “kink” solution [30], one obtains the supergravity solution
(45)  
Asymptotically for the gauge field goes to zero and the metric approaches AdS written in nonstandard coordinates [16], so that the “kink” solution (45) interpolates between two AdS vacua at .
Grumiller and Kummer were able to write down the most general solution of the dimensionally reduced gravitational ChernSimons theory using the fact that it can be written as a Poissonsigma model with fourdimensional target space and degenerate Poisson tensor of rank two [32]. This gives rise to new BPS supergravity solutions generalizing (45),
(46) 
with . In the special case we recover the kink solution considered above. This solutions has been recently further analyzed in [33], where it was interpreted as a soliton consisting of photons kept together by gravity.
Note that all these solutions are defined only for . One can now verify that the domain of the parameter can be extended also to the negative region. This yields the solution
(47) 
with .
4.3.3 The case
In this case must be a function of the coordinate alone, and the system of equations describing this subset of solutions is
(48)  
(49)  
(50) 
where is the curvature scalar of the twomanifold with metric . Without loss of generality we can set . The general solution for the complex function is therefore [5]
(51) 
where , and are complex integration constants (if these constants are real, we fall back in the “purely magnetic” case already considered in section 4.1). If we take , we obtain the solution AdS with magnetic flux on already considered in section 4.1.
The case was solved in [5], where it was shown that one recovers the supersymmetric ReissnerNordströmTaubNUTAdS solutions [16] with metric
(52)  
(53)  
(54) 
Here, denotes the NUT parameter, and and are the electric and magnetic charges respectively. These are fixed by the constants , and ,
The electromagnetic field strength is easily obtained from (16). One checks that the final solution belongs to the ReissnerNordströmTaubNutAdS class of solutions [16], with arbitrary nut charge and electric charge , whereas the magnetic charge and mass parameter are given by
(55) 
These are exactly the conditions on and found in [16], under which the RNTNAdS solutions preserve one quarter of the supersymmetry^{3}^{3}3The other sign for given in [16] can be obtained by replacing , .. Note that, taking more general solutions of the Liouville equation (49), one would obtain a larger class of BPS configurations, and in particular recover all purely magnetic solutions of section 4.1.
4.4 Harmonic solutions
Another rich class of solutions can be found if we choose to be harmonic, . Equation (16) gives
(56) 
where , and are arbitrary holomorphic functions of . Using the EddingtonFinkelsteinlike coordinate , the solution assumes the form
(57) 