CTPTAMU20/99
hepth/9908089
July 2, 2021
Matter Coupled AdS Supergravities
and Their Black Strings
N.S. Deger , A. Kaya , E. Sezgin and P. Sundell

Center for Theoretical Physics, Texas A&M University, College Station, TX 77843, USA

Physics Department, Karlstad University, S651 88 Karlstad, Sweden
Abstract
We couple copies of scalar multiplets to a gauged supergravity in dimensions which admits as a vacuum. The scalar fields are charged under the gauged symmetry group and parametrize certain Kahler manifolds with compact or noncompact isometries. The radii of these manifolds are quantized in the compact case, but arbitrary otherwise. In the compact case, we find halfsupersymmetry preserving and asymptotically Minkowskian black string solutions. For a particular value of the scalar manifold radius, the solution coincides with that of Horne and Horowitz found in the context of a string theory in dimensions. In the noncompact case, we find halfsupersymmetry preserving and asymptotically string solutions which have naked singularities. We also obtain two distinct supergravities coupled to copies of scalar multiplets either by the truncation of the model or by a direct construction.
Research supported in part by NSF Grant PHY9722090.
1 Introduction
Important advances have been made in our understanding of theory in space over the last few years [2, 4, 6]. In particular, evidence has been accumulated for a remarkable relation between certain gauged supergravities which admit space as vacua and appropriate conformal field theories defined on the boundaries of these spaces. An especially manageable example of this phenomenon arises in the context of correspondence. While interesting work has been done on the CFT aspects of this problem, a great deal remains to be done on the supergravity side. With this motivation in mind, and in view of their relative simplicity, in this paper we study the structure of matter coupled supergravity theories with and supersymmetry and their string solutions.
Since the group is reducible, namely, , the supersymmetry parameters could come in copies of and copies of , thus describing supersymmetry. The pure supergravity with supersymmetry was constructed long ago and it is a ChernSimons gauge theory based on the supergroup [8]. Later, the coupling of the case to copies of scalar multiplets which parametrize a Kahler manifold was constructed [10]. In this model, the scalars are neutral under the symmetry group . Supersymmetric solutions of this model have been studied and in particular, it has been shown [10] that the pure supergravity sector of the theory admits the BTZ black hole [12] as a supersymmetric solution. The model has features unlike the familiar gauged supergravity theories. In fact, a higher dimensional origin of the model, whether it is field theoretic one or string/Mtheoretic, is apparently not known.
There must exist, however, a class of (matter coupled) supergravity theories which describe various compactifications of string/Mtheories. In particular, the compactification of Type IIB string has been a subject of number of studies recently [18, 20, 22]. The full spectrum of this compactification is known, and the massless sector is expected to be described by a matter coupled supergravity with the gauge group . There exist other supersymmetric compactifications of supergravities in diverse dimensions down to , and in all these cases, we expect to find the gauged versions of the matter coupled Poincaré supergravities in dimensions constructed long ago by Marcus and Schwarz [24], with the matter sector consisting of scalar multiplets with an underlying (for certain values of implied by string theory) or structure, and their lower supersymmetric truncations.
Ultimately we would like to construct all the supergravities mentioned above in a unified framework, and to study their connection with the boundary conformal field theories. In this paper we take the first step in this direction. In particular, we construct an supergravity coupled to complex dimensional Kahler manifold, and its truncation. This paper is devoted to understanding of the supergravity aspects of these models and their string solitons. We expect that there exist compactifications of string/Mtheory giving rise to the supergravity theories studied here as their low energy limits.
As mentioned above there already exists a matter coupled supergravity constructed sometime ago by Izguierdo and Townsend [10]. However, this model differs from ours in a significant way, namely its scalar fields are neutral with respect to the symmetry group unlike in our model. Consequently, the model of [10] does not have a potential while our model leads to a rather elaborate potential. In fact, some of the properties exhibited by our model are quite similar to those which arise in gauged supergravity coupled to a restricted kind of Kahler sigma model in [14]. For example, the sigma model manifold arising can be either compact or noncompact. The gravitational coupling constant, the radius of and the radius of the sigma model manifold are not related to each other by supersymmetry, unlike the gauged supergravities with higher supersymmetries. Moreover, the radius of the sigma model manifold, when compact, is quantized in units of the gravitational coupling constant.
We also obtain a supersymmetric version of the results mentioned above by a consistent truncation of the model. We show that there exists a one parameter extension of this theory for the coupling of single scalar multiplet.
In this paper we also present string solutions of our models which preserve half supersymmetry, both for compact and noncompact sigma models. Interestingly enough, for the compact case there is a particular value of the radius such that the solution reduces to that of Horne and Horowitz [16] found in the context of low energy limit of a string theory in dimensions. The solutions exhibit an event horizon and are asymptotically Minkowskian for the compact sigma model. The solutions for the noncompact sigma model, on the other hand, have naked singularities and are asymptotically .
The plan of this paper is as follows. In Sec. 2, we describe the supergravity coupled to an complex dimensional Kahler sigma model. In Sec. 3, we specialize to the case of , namely and . The black string solutions and their properties will be discussed in Sec. 4. The supersymmetric matter coupled supergravity is presented in Sec. 5. Our results and a number of open problems raised by them are discussed in Sec. 6.
2 N=(2,0) AdS Supergravity Coupled to Scalar Multiplets
The supergravity multiplet consists of a graviton , two Majorana gravitini (with the spinor index suppressed) and an gauge field . The copies of the scalar multiplet, on the other hand, consists of real scalar fields and Majorana fermions .
For simplicity, we will take the sigma model manifold to be a coset space of the form where can be compact or noncompact and is the maximal compact subgroup of , where is the symmetry group. For concreteness, we shall consider
(2.1) 
Our results can be readily translated to the case of with
of and .
Let where form a representative of the coset . It follows that
(2.2)  
where correspond to the scalar manifolds . The , and vector indices are raised and lowered with the Kronecker deltas and the vector indices with the metric .
The gauged pullback of the MaurerCartan form on can be decomposed into the connections and , and the nonlinear covariant derivative as follows:
(2.3) 
where and the covariant derivative is defined as
(2.4) 
The antihermitian generator occurring in this definition is realized in terms of an matrix, which can be chosen as
In coupling to supergravity, we will also need the introduction of the “boosted matrix elements” defined as
(2.5) 
From these definitions and (2.3) it follows that
(2.6)  
where the covariant derivatives are defined as
(2.7) 
Recall that correspond to the scalar manifolds specified in (2.1).
Using the formulae given above and applying the standard Noether procedure we get the following matter coupled gauged supergravity Lagrangian up to quartic order fermion terms ^{1}^{1}1Our conventions are as follows: , and are symmetric, the charge conjugation matrix is unity, is symmetric and . A convenient representation is , . We define . Note that .
(2.8)  
which has the local supersymmetry
(2.9) 
We have set the gravitational coupling constant equal to one, but it can easily be introduced by dimensional analysis. The constant is the characteristic curvature of (e.g. is the inverse radius in the case of ) and the constant is the cosmological constant. The gauge coupling constant has been absorbed into the definition of . We emphasize that, unlike in a typical anti de Sitter supergravity coupled to matter, here the constants are not related to each other for noncompact scalar manifolds, while will be quantized in terms of in the compact case, as we shall see later.
The covariant derivatives are defined as
(2.10) 
The coefficients in front of the composite connection has been determined by supersymmetry.
Having defined the above covariant derivative, we can now see more clearly why the  and functions arise in the model. Firstly, the supersymmetric variation of the gravitino kinetic term involves the commutator
(2.11) 
This is where we see first the occurrence of the function . The dependent term arising here is cancelled by the variation of the ChernSimons term. The rest of the Noether procedure eventually involves the differentiation of the function which leads to the function , and its differentiation leads to the function .
It is straightforward to adapt all the formulae given above in terms of
complex scalars and Dirac spinors
and a Dirac gravitino , with the dreibein and vector
field , of course, remaining real. Typical sigma model manifolds
arising in this way are the compact and noncompact , if
we are only concerned about the local aspects of the symmetries
involved. Insisting that the model is globally well defined, some
restrictions will arise on the scalar manifold geometry, as was shown
long ago by Witten and Bagger in the context of supersymmetric
models coupled to supergravity in . These restrictions typically
occur when the scalar manifold is compact. Indeed, as in [14],
here too the compact scalar manifold turns out to be a Hodge manifold,
which is a certain type of Kahler manifold. An important consequence of
this is that the radius of the scalar manifold gets quantized in units
of the Planck length. This phenomenon is explained in detail in
[14] and therefore will not be repeated here. However, we shall
get back to the specifics of the quantization condition in the next
section where we consider the black string solutions of the model when
the scalar manifold is in effect taken to be . In the case of
the quantization condition does not arise.
Let us consider various limits of this model. Firstly, by rescaling and the matter scalar fields such that and then sending the inverse sigma model radius such that , , one obtains , supergravity with cosmological constant coupled to an sigma model. The pure , supergravity [8] can then be obtained by setting all the matter fields equal to zero. To obtain the Poincaré limit of the theory [24, 26, 28] , on the other hand, we start with the Lagrangian (2.8), rescale and then let . Once the Poincaré limit is taken, the supergravity fields can further be decoupled by setting the gravitini equal to zero and taking the metric to be flat Minkowskian. This yields an supersymmetric sigma model. A rigid supersymmetric limit does not seem to be possible in this model.
3 The Cases of and
The variables of the model presented above can easily be complexified so that the scalar manifold becomes generically or . In search of a string solution, it is convenient to set equal to zero all but one of the complex scalar fields in such a way that the model is consistently truncated to an or sigma model coupled to the supergravity with supersymmetry. The geometry can easily be accounted for as the infinite radius limit of .
The coset representative for and can be parametrized as
(3.1) 
Defining
(3.2) 
the key relations (2.6) take the form
(3.3) 
The functions and are computed from the definitions (2.5), which, for the cases at hand, are
(3.4) 
where the generators of and algebras are
(3.5) 
Representing by for and for , we obtain from (3.4)
(3.6) 
Similarly, the nonlinear covariant derivative and the connection are computed from the definitions (2.3), which for the cases we are considering, take the form
(3.7) 
from which, recalling the definitions (3.2), it follows that
(3.8)  
where
(3.9) 
In describing the sigma model manifolds, we have used a particular coordinate system. We have to ensure that this description makes sense globally. In fact, the coordinates are the stereographic projection of or onto the complex plane. In the case of this is a globally well defined coordinate system to cover the manifold. But in the case of , as is well known, one needs two patches in order to avoid the singularities at the north and south poles. Following the standard procedure, we cover the upper hemisphere with coordinate and the lower one by . We must then check that the action is well defined in the overlap region. To achieve this, we also need to transform the gauge field as . Under the combined transformation
(3.10) 
the quantities and transform as
(3.11)  
For these transformations to leave the action invariant, we must also transform the fermionic fields. Noting that these terms are given by
(3.12) 
where is the Lorentz covariant derivative, we find that the appropriate transformation rules for the fermions are
(3.13) 
where we have reintroduced the gravitational coupling constant . For these transformations to be single valued, we need to impose, á la Witten and Bagger [14], the quantization condition
(3.14) 
where is an integer.
4 The Black String Solution
We shall now seek string solutions of the model described in the previous section. To this end, let us note the bosonic part of the Lagrangian
(4.1) 
where the potential is given by
(4.2) 
and and are defined in (3.6). Note that . The resulting bosonic field equations are
(4.3)  
(4.4)  
(4.5) 
where
(4.6) 
We shall also need the supersymmetry transformation rules. Let us first define
(4.7) 
Dropping the hat for notational convenience, the transformation rules (2.9), applied to the case at hand, take the form
(4.8) 
where .
Before presenting the black string solutions,it is worthwhile to note that the theory admits various maximally symmetric vacua. For the case of ,the potential (4.2) has minimum at corresponding to a supersymmetric vacuum, a valley of minima at corresponding to a supersymmetric dimensional Minkowski vacuum and two valleys of maxima at , where , corresponding to nonsupersymmetric de Sitter vacua. Here is an arbitrary real scalar field. For the case of , we have the following extrema: (i) For , there is a maximum at which is a supersymmetric vacuum, (ii) for , there are two valleys of minima at where which are nonsupersymmetric vacua, (iii) for there is a minimum at which is a supersymmetric vacuum. The case (ii) similar in nature to a situation encountered in finding the extrema of the gauged supergravity theory [30].
Let us now consider the following ansatz for the metric
(4.9) 
where are functions of the transverse coordinate only. Next, we set
(4.10) 
Then, the supersymmetry condition implies that
(4.11) 
where the prime indicates differentiation with respect to , provided that we also impose the condition
(4.12) 
which means that we are seeking halfsupersymmetry preserving solution. The choice of minus sign is merely for convenience.
The supersymmetry conditions and are satisfied provided that
(4.13) 
The remaining condition determines the dependence of the spinor to be
(4.14) 
(4.15) 
where we have set a multiplicative integration constant equal to 2 for convenience. Thus, the metric takes the form
(4.16) 
It is straightforward to verify that all the field equations are satisfied by this metric and the ansatz (4.9).
The fact that is not determined by the equations of motion is a consequence of having freedom in reparametrizing the radial coordinate . Indeed, the function can be determined by performing a dependent coordinate transformation. A convenient such transformation is
(4.17) 
where is an integration constant. We next analyze the compact and noncompact cases separately.
4.1 The Case of ()
The inversion of (4.17) yields the dependence of
(4.18) 
where the tilde on has been dropped for notational convenience. In obtaining this result, we have chosen the positive root in (4.17). The negative root gives an expression for which diverges at . Note that implies in accordance with the fact that is the stereographic coordinate of .
(4.19) 
This metric has no horizons and there is a naked singularity. To see this, we first let and then define a new radial coordinate . The metric then takes the form
(4.20) 
The asymptotic geometry near is . The metric has a singularity at , while it is regular at other points. The fact that is a genuine singularity can be seen from the curvature scalar associated with this metric, given by
(4.21) 
which clearly diverges for . The implications of this naked singularity for the cosmic censorship conjecture remains to be investigated.
Finally, we find that the energy per unit length for the string metric (4.20) vanishes. Actually, the commutator of two supersymmetry transformations can be shown to vanish at radial infinity, but one cannot deduce from this alone that the energy vanishes. This is due to the fact that the result is a combination of the true Lorentz rotations and translations in . A more convenient method to pin down the energy for the case at hand is due to Hawking and Horowitz [32], and applying this method, we indeed find the result stated above, namely the vanishing of the energy for our solution.
4.2 The Case of ()
The inversion of (4.17) for the upper hemisphere yields
(4.22) 
For the lower hemisphere we obtain
(4.23) 
Note that in both cases, in accordance with the fact that
are the stereographic coordinates of such that and . In fact, and
constitute a well defined map from spacetime into .
(4.24) 
Note that for this case is quantized to be an integer. This metric is asymptotically Minkowskian. Moreover, there is a horizon at , and the near horizon geometry is . The Hawking temperature of this black string can be readily shown to be vanishing. Thus, we expect this solution to be quantum mechanically stable.
The curvature scalar associated with the metric (4.24) is
(4.25) 
which is regular at . This formula also shows that there is a singularity at . However, for some values of the parameter the singularity cannot be reached by the observers outside the horizon. To investigate this point, let us consider the geodesic equation. Let be tangent to an affinely parametrized geodesic, and let us define the conserved quantities associated with the two translations on the string worldsheet as , . Then the geodesic equation associated with the metric (4.24) takes the form
(4.26) 
where the dot denotes derivative with respect to an affine parameter, for null geodesics and for timelike geodesics. For , the geodesics can not reach the horizon. Indeed, there is a turning point corresponding to the vanishing of the right hand side of (4.26). For , a simple analysis of (4.26) near the horizon shows that when is an even integer the region is accessible, but not accessible when is an odd or half integer. For the former case, we need to extend the definition of to the region . However, Einstein equations imply that and thus for . On the other hand, we see from its definition that for any value of . Therefore, we can not extend the solution to the region when is an even integer.
To summarize, we have physically well defined black string solutions for
(4.27) 
In these cases the timelike or null geodesics can not penetrate the horizon located at , and the field need not be extended to the region .
For , the metric (4.24) coincides with the metric found by Horne and Horowitz [16] obtained from a different starting point, namely low energy limit of a string theory in dimensions described by the Lagrangian
(4.28) 
where and is a constant. In Einstein frame, this Lagrangian takes the form
(4.29) 
The metric (4.16) is a solution for this theory, with the dilaton given by . What we have shown here is that not only this metric is a solution of two rather different theories but it is also supersymmetric. We note that the string theory which should generate our matter coupled supergravity model remains to be determined.
Finally, we note that the mass per unit length for our general string solutions in the sigma model case can be conveniently deduced from the algebra of supercharges, since these solutions are asymptotically Minkowskian. A standard procedure which makes use of the Nester twoform (see, for example, [34, 36]) yields the result