Non-Abelian fields in AdS{}_{4} spacetime: axially symmetric, composite configurations

Non-Abelian fields in AdS spacetime:
axially symmetric, composite configurations

Olga Kichakova, Jutta Kunz, Eugen Radu and Yasha Shnir

Institut für Physik, Universität Oldenburg, Postfach 2503 D-26111 Oldenburg, Germany
Departamento de Fisica da Universidade de Aveiro and I3N, Campus de Santiago, 3810-183 Aveiro, Portugal
Department of Theoretical Physics and Astrophysics, BSU, Minsk, Belarus
BLTP, JINR, Dubna, Russia
July 12, 2019
Abstract

We construct new finite energy regular solutions in Einstein-Yang-Mills-SU(2) theory. They are static, axially symmetric and approach at infinity the anti-de Sitter spacetime background. These configurations are characterized by a pair of integers , where is related to the polar angle and to the azimuthal angle, being related to the known flat space monopole-antimonopole chains and vortex rings. Generically, they describe composite configurations with several individual components, possesing a nonzero magnetic charge, even in the absence of a Higgs field. Such Yang-Mills configurations exist already in the probe limit, the AdS geometry supplying the attractive force needed to balance the repulsive force of Yang-Mills gauge interactions. The gravitating solutions are constructed by numerically solving the elliptic Einstein-DeTurck–Yang-Mills equations. The variation of the gravitational coupling constant reveals the existence of two branches of gravitating solutions which bifurcate at some critical value of . The lower energy branch connects to the solutions in the global AdS spacetime, while the upper branch is linked to the generalized Bartnik-McKinnon solutions in asymptotically flat spacetime. Also, a spherically symmetric, closed form solution is found as a perturbation around the globally anti-de Sitter vacuum state.

1 Introduction

The study of solutions of the Yang-Mills (YM) theory in a curved spacetime geometry can be traced back at least to the early work [1]. Among other results, that study has given an exact solution of the YM equations in a fixed Schwarzschild black hole background. This shows that the non-trivial solutions to the full system of Einstein–Yang-Mills (EYM) equations are likely to exist, at least for large enough event horizon black hole radius.

Indeed, this has been confirmed ten years later, when several different authors have constructed asymptotically flat, black hole (BH) solutions within the framework of SU(2) EYM theory [2]. Although these BHs were static and spherically symmetric, with vanishing YM charges, they were different from the Schwarzschild one and, therefore, not characterized exclusively by their total mass. Unfortunately, soon after their discovery, it has been shown that these solutions are unstable [3], [4]. However, despite this fact, they still present a challenge to the standard ’no hair conjecture’ [5], [6].

These results have led to a revision of some of the basic concepts of BH physics based on the uniqueness and no-hair theorems. For example, the Israel�s theorem does not generalize to the non-Abelian (nA) case, since static EYM black holes with non-degenerate horizon turn out to be not necessarily spherically symmetric [7]. Moreover, in strong contrast to the Abelian case, the EYM hairy black holes in [2] do not trivialize in the limit of a vanishing horizon area111In fact, rather curious, the EYM particle-like solutions have been discovered [8] before their black hole generalizations., reducing to horizonless, globally regular, particle-like configurations, originally found by Bartnik and McKinnon in Ref. [8].

As a result, the subject of particle-like and hairy BH solutions in EYM theory has become an active field of research, with many new results being reported each year222 A detailed review of the situation ten years after the discovery of the EYM solutions in [2], [8] can be found in [9].. One interesting question addressed in this context is what happens if one drops the assumption of asymptotic flatness for the spacetime background. The case of the EYM system with a negative cosmological constant is of particular interest, the natural background of the theory corresponding to anti-de Sitter (AdS) spacetime. Solutions of various physical models in this geometry received recently much interest due to the conjectured anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. This is a concrete realization of the holographic principle, which asserts that a consistent theory of quantum gravity in -dimensions must have an alternate formulation in terms of a nongravitational theory in -dimensions. Despite the fact that string theory in AdS space is still too complicated to be dealt with in detail, in many interesting cases, it is sufficient to consider the low energy limit of the superstring theory, namely, supergravity. However, the gauged supergravity theories generically contain YM fields (although most of the studies in the literature have been restricted to the case of Abelian matter content in the bulk), and thus the interest in the study of the EYM system with .

The first result on nA fields in a globally AdS geometry can be found again in Ref. [1], where a non-trivial solution of the YM equations is exhibited in closed form333However, note that the Ref. [1] has considered only the case of a positive cosmological constant and a slightly different coordinate system as compared to (6).. This solution describes a globally regular, finite energy soliton, with a nonzero magnetic flux at infinity (despite the absence of a Higgs field), anticipating most of the basic properties of the (gravitating) nA fields in an AdS background.

The study of EYM solutions in a globally AdS background have started with Refs. [10], [11], where spherically symmetric BHs and solitons have been studied, again for the gauge group SU(2). As shown there, a variety of well known features of asymptotically flat self-gravitating nonabelian solutions are not shared by their AdS counterparts. Restricting for simplicity to purely magnetic configurations, one finds a continuum of particle-like and BH solutions describing nA monopoles with a non-integer magnetic charge (we recall that the asymptotically flat EYM configurations are magnetically neutral, forming a discrete sequence indexed by the node number of the magnetic gauge potential [2]). Moreover, perhaps most remarkable, some of the solutions are stable against spherically symmetric linear perturbations [12], [13]. Also, as already found in Ref. [1], the curved background geometry provided an extra attracting force, which makes possible the existence of finite mass, particle-like YM solutions already in the probe limit, in a fixed AdS spacetime. As discussed in [14], [15] the soliton and black hole solutions in [10], [11] possess interesting generalizations with higher gauge groups. A review of the EYM solutions in a globally AdS background can be found in Ref. [16].

We note that an even more intricate picture is found when studying EYM topological BHs [17]. For example, no globally regular particle-like limit of the solutions is found in this case. Moreover, as discovered in [18], [19], the planar hairy BHs describe gravity duals of wave superconductors. As a result, this type of EYM solutions enjoyed recently much interest. Also, the configurations possess higher dimensional generalizations [20], with the EYM planar black holes describing holographic wave superfluids [21]. However, these aspects are beyond the purposes of this paper, where we shall restrict ourselves to the case of the SU(2) gauge fields and the asymptotically AdS spacetime. One should remark that, even in the case of the global AdS spacetime, the study of YM solutions is still far from being complete. For example, very few things are known about non-spherically symmetric EYM-AdS solutions. Non-Abelian solitons which are axially symmetric only have been studied444Their black hole generalizations have been considered in [23]. in Ref. [22]. These configurations possess an azimuthal winding number and describe (non-topological) monopoles localized at the origin, sharing all basic properties of the spherically symmetric counterparts (which have ). Also, despite the generic presence of a net magnetic flux, they can be viewed as the natural AdS generalizations of the asymptotically flat EYM solitons in [24].

However, as discussed in [25], the EYM system with possesses a variety of other globally regular solutions, describing composite configurations, with several constituents555Note that no such solutions exist with Abelian matter fields only, the closest approximation there being the Majumdar-Papapetrou [26], [27] extremal black holes in Einstein-Maxwell theory.. In the notation of [25], these solutions are characterized by a pair of positive integers , where is related to the polar angle and to the azimuthal angle.

One should remark that the AdS solutions discussed so far in the literature cover the case only, that is, solutions with a single center. The main purpose of this work is to explicitly construct AdS configurations with , looking for new features induced by the different asymptotic structure of the spacetime.

Although some common features are present, the results we find for are rather different from those valid in the asymptotically flat case. Perhaps the most prominent new result is the existence of multi-center solutions with a net magnetic flux at infinity, despite the absence of a Higgs field (such solutions are absent for ). Similar to the spherically symmetric case [1], such nA configurations exist already in the probe limit ( no backrection). Moreover, when including the gravity effects, we establish the absence, for large enough values of , of configurations with a zero magnetic flux (which are the only ones existing in the asymptotically flat case).

This paper is structured as follows: in the next Section we introduce the gauge field ansatz and address the question of possible asymptotics for (purely magnetic) static, axially symmetric YM fields. In Section 3 we construct solutions with these asymptotics in the probe limit, for a fixed AdS background. Although being relatively simple, nevertheless this case appears to contain all the essential features of the gravitating solutions. The backreaction of the solutions on the spacetime geometry is considered in Section 4. There special attention is paid to a particular value of , for which the EYM system becomes a consistent truncation of the , gauged supergravity [28]; therefore such solutions can be uplifted to dimensions [29]. When we abandon this restriction, the variation of the gravitational coupling constant (which is the ratio between the Planck length and AdS length scales) reveals the existence of two branches of gravitating solutions which bifurcate at some critical value . These branches interpolate between the solutions in the global AdS spacetime and, for small values of the coupling constant on the upper branches, the generalized Bartnik-McKinnon solutions in the asymptotically flat spacetime [25], confined in the interior region, and outer configurations in the global AdS spacetime.

We give our conclusions and remarks in the final Section. The Appendix A contains a derivation of an exact solution of the EYM equations with negative cosmological constant as a perturbation around the globally AdS vacuum state.

2 SU(2) Yang-Mills fields on AdS

2.1 The model

We consider the action of the SU(2) YM theory

(1)

with the field strength tensor

(2)

and the gauge potential

(3)

being the gauge coupling constant. Also, are space-time indices running from 1 to 4 and the gauge index is running from 1 to 3.

Variation of (1) with respect to the gauge field leads to the YM equations

(4)

while the variation with respect to the metric yields the energy-momentum tensor of the YM fields

(5)

For the background metric, we shall consider the (covering-)AdS spacetime, written in global coordinates as

(6)

where are the radial and time coordinates, respectively (with and ), while and are angular coordinates with the usual range, parametrizing the two dimensional sphere . Also, is the AdS length scale, which is fixed by the cosmological constant,

(7)

2.2 The axially symmetric YM Ansatz

In this work we shall restrict to purely magnetic YM configurations and employ a gauge field ansatz in the parametrization666This is in fact a suitable reparametrization of the axially symmetric YM ansatz introduced for the first time by Manton [30] and Rebbi and Rossi [31], which is better suited for numerical purposes. Also, note that (8) is a consistent truncation of the most general YM ansatz, which contains 12 potentials. originally proposed in [24]

(8)

in terms of four gauge field functions which depend on and only. The SU(2) matrices factorize the dependence on the azimuthal coordinate , with

where are the Pauli matrices. The positive integer represents the azimuthal winding number of the solutions. For and , the usual spherically symmetric magnetic (singularity-free) YM ansatz is recovered.

This ansatz is axially symmetric in the sense that a rotation around the axis (with ) can be compensated by a suitable gauge transformation [32, 33]. However, note that the gauge transformation leaves the ansatz form-invariant [34]. Thus, to construct regular solutions we have to fix the gauge. The usual gauge condition [24] which is used also in this work is

(9)

A straightforward computation leads to the following expression of the non-vanishing components of the SU(2) field strength tensor (with ):

(10)

We are interested in singularity-free solutions of the YM equations (4) with a finite mass. For configurations in a fixed AdS background, the total mass is defined as the integral of the mass-energy density over a three-dimensional space,

(11)

From (5) and (8), one finds

(12)

where we denote

(13)

As known already from the study of spherically symmetric configurations [11], a generic feature of the generic YM solutions in an AdS background is that they possess a nonvanishing magnetic flux through the sphere at infinity. A measure of this flux is provided by the magnetic charge, , which, in the absence of the Higgs field, does not have a meaning of a topological charge of the configuration, thus it is allowed to be non-integer. A possible gauge invariant definition of which we shall employ in this work, is [35]

(14)

We have verified that in the spherically symmetric case, (14) agrees, up to a sign, with the magnetic charged expression in [11].

We close this part by noticing that the static axially symmetric YM configurations in a fixed AdS spacetime satisfy the virial identity777As usual, this virial identity is found by considering the scale transformation of the effective action of the model (which essentially coincides with (11)), for a given set of boundary conditions. Then must have a critical point at , which results in the virial relation (15).

(15)

This makes clear that the AdS geometry supplies the attractive force needed to balance the repulsive force of Yang-Mills gauge interactions.

2.3 The issue of boundary conditions at infinity

2.3.1 flat space case

Let us start with the more familiar case of gauge fields in a Minkowski spacetime background. For , a systematic study of axially symmetric YM-Higgs configurations has revealed the existence of two possible types of asymptotics of the YM fields, describing different ground states of the model. These asymptotics are indexed by a number , which is a positive integer.

Odd- conditions

For an odd number, , the YM fields possess a solution with

(16)

which describe an infinite energy embedded Abelian configuration with a singular origin. The configurations with these far field asymptotics carry a nonzero magnetic charge, (with computed according to (14)).

However, in a flat spacetime background, a magnetically charged configuration requires Higgs fields to exist. (or they are just embedded Abelian singular solutions). Indeed, as discussed in [36], the Yang-Mills-Higgs (YMH) system possesses regular, finite energy solutions whose gauge potentials approach (16) in the far field. The better known case are the , self-dual magnetic monopoles (which are in fact the only YMH closed form solutions). For , they describe composite (non-self-dual) monopole-antimonopole configurations with a net magnetic charge. A systematic study of these solutions can be found in [36].

Even- conditions

A different picture is found for . The corresponding ground state YM solution reads

(17)

Again, these asymptotics emerge from a systematic study of the axially symmetric YMH system [36]. The corresponding YMH solutions describe again (non-self dual) monopole-antimonopole chains. However, different from (16), the total magnetic charge vanishes in this case888Note that for any value of , the expression of the magnetic charge of the YMH solutions found by employing the Abelian ’t Hooft tensor agrees with that from (14). , .

Moreover, as found in [25], in strong contrast to the odd case, these configurations survive in the limit of a vanishing Higgs field, provided that the gravity effects are included. For example, the well-known Bartnik-McKinnon EYM particle-like solutions [8] are recovered for , . The values , lead to their axially symmetric generalizations in [24].

One should also mention that, as discussed in [36], the YM configuration (17) with corresponds to a gauge transformed trivial solution,

(18)

However, the configurations (16) describe a gauge transformed charge- Abelian multimonopole ():

(19)

where

(20)

Also,

(21)

for both and .

The relation with the gauge field paramerization in [36] and [25]

Although relying on the same ansatz proposed by Manton [30] and Rebbi and Rossi [31], the Ref. [36] expresses the YM potentials in a slightly different SU(2) basis,

(22)

with the number entering also the SU(2) matrices:

A direct comparison with (8) implies , while

Note also that the Ref. [25], dealing with pure EYM solutions, uses a version of (22) with and (and a set of boundary conditions at infinity resulting from (17), with ).

We would like to emphasize that the description of a given nA configuration in terms of , or in terms of are equivalent. The choice in this work for has the advantage to simplify somehow the emerging general picture, providing a unified framework. Also, it leads to slightly better numerical results for the gravitating solutions.

2.3.2 YM far field asymptotics in AdS spacetime

As we know already from the study in [11], [10] of the spherically symmetric case, the asymptotics of the YM fields are less constrained for . In the generic axially symmetric case, an obvious condition results from the requirement that , as given by (12), decays faster than as , as imposed by the assumption that the total mass is finite. This implies

(23)

with and , two regular functions which vanish as as imposed by the regularity of the configurations.

A general expansion of the YM potentials in AdS spacetime reads (with ):

(24)

where the functions depend on the angular coordinate only. Once an expression is chosen for , the functions (with ) are found by solving the YM equations in the far field, as a power series in . Note, however that the gauge condition (9) implies

(25)

while no obvious expressions are found for , (note that the regularity of the solutions implies ).

In what follows, among all possible sets, we shall restrict ourselves to those YM asymptotics which provide natural AdS generalizations of the flat spacetime boundary conditions (16), (17).

Odd- conditions

One can prove that (16) is still a solution of the YM equations for . However, in strong contrast to the (asymptotically) flat case, solutions with a non-zero magnetic charge exist even in the absence of a Higgs field and/or the gravity effects. This can easily be seen for the simplest case with , where the following spherically symmetric exact solution is known:

(26)

This is the exact solution found in [1], which describes a unit charge magnetic monopole with a finite mass . However, the results in [39] show the existence of (numerical) generalizations of this solution with the same behaviour at the origin, , and a relaxed set of boundary conditions at infinity, . As discussed in [22], these solutions admit axially symmetric generalizations with , possessing many similar properties.

A systematic study of possible boundary conditions compatible with regularity and finite energy requirements led us to the following AdS natural generalizations of (16):

(27)

with a constant which is not fixed apriori999Note that (27) is not a solution of the YM equations, unless . It describes instead the leading order behaviour of the YM potentials in the AdS spacetime, as given by ., the flat space boundary conditions being recovered for . For example, one takes

(28)

and

(29)

For the general asymptotics (27), one finds which implies

(30)

Therefore, the magnetic charge of the AdS configurations with the generic asynmptotics (27) is no longer an integer, a feature which occurs already in the spherically symmetric case. However, one can see that magnetically neutral solutions are found for .

Even- conditions

The situation is somehow different for even values of . It turns out that (17) provides the only possible set of boundary conditions compatible with the condition of a vanishing magnetic charge101010We did not find AdS generalizations of (17) with , which would be similar to the odd- conditions (27). In our approach, we consider a deformation of (17) with: (31) with a constant. The YM equations are supplemented with the zero-magnetic charge condition , , as . This results in two first order differential equations for and , which have closed form solutions. However, the functions and fail to satisfy the regularity conditions on the symmetry axis. Thus we conclude that (17) are the only boundary conditions compatible with the assumption of a zero-magnetic flux at infinity..

Of course, this does not exclude the existence of AdS deformations of (17). However, they would generically possess a nonzero magnetic charge. For example, we have studied in a systematic way solutions with

(32)

behaviour as , such that the conditions (17) are recovered for . For example, one has

(33)

and

(34)

This reveals an interesting feature: the lowest polar winding number solutions and , are in fact identical, since (28) results from (33) via the identification . This is related to the existence of a continuum of solutions (in terms of ), which is a pure AdS property. The solutions with higher satisfy different boundary conditions and are inequivalent111111This can be seen by comparing gauge invariant quantities, the energy density. .

The even- solutions with the asymptotics (32) carry a magnetic charge

(35)

Also, let us note that in contrast to the case, both (27) and (32) correspond to intrinsic nA configurations (for and , respectively).

3 The solutions in the probe limit

In this section we shall establish the existence of finite energy, regular YM configurations approaching at infinity the asymptotics (27) and (32), respectively. Different from the case of a (asymptotically) Minkowski spacetime background, such configurations exist already in the probe limit ( when neglecting the backreaction of the matter fields on the spacetime geometry). This approximation greatly simplifies the problem but retains most of the interesting physics. For example, the probe limit was implemented recently to analyse properties of axially symmetric solutions of the Yang-Mills-Higgs theory [45].

The problem has an intrinsic length scale (we recall ). Without any loss of generality, we fix , the total mass of solutions, as given by (11), being expressed in units of . The numerical approach employed in this case is similar to those described in Section 4 for gravitating configurations and we shall not discussed it here121212Note that the typical numerical accuracy is better for pure YM solutions, the typical numerical error estimates being on the order of .. We mention only that the solutions are found by directly solving the YM equations with a given set of boundary conditions.

3.1 The results

Fixing , the only input parameters of the problem are the integers and the continuous constant , which enters the boundary conditions at infinity (27), (32). We have studied a large number of configurations with and . Then we conjecture the existence of YM solutions in AdS background for any values131313This contrasts with the gravitating YM solutions in [25], which do not cover the whole space. of .

The YM solutions are found subject to the following boundary conditions:

(36)

at the origin, and (27), (32) at infinity.

Also, for solutions with parity reflection symmetry (the only type we consider in this paper), the boundary conditions at are

(37)

(therefore we need to consider the solutions only in the region ).

Regularity of the solutions on the symmetry axis imposes also

(38)

a condition which is verified from the numerical output.

For given , the solutions are found by varying the parameter which enters the boundary conditions at infinity. As expected, we have found a continuum of solutions in terms of .

It is not easy to extract some general characteristic properties of the solutions, valid for every choice of the parameters . However, we have found that the functions present always a considerable angle-dependence, except for the case (there one finds usually a small angular dependence for the potentials ). We have also noticed that the angular dependence generally increases with . The profiles of typical solutions are given in Figures 1-4, both as 3D-plots (with , ), and as a function of the radial coordinate for several different angles. There we show the gauge potentials together with the mass-energy density as given by .

Also, we have found that for all sets , in the absence of backreaction, the solutions exist for a single interval in , the mass of the solutions strongly increasing for large values of . One can see this in Figures 5, 6 for the spectrum of the solutions. There we plot the mass of the solitons in terms of the parameter which enters the far field asymptotics (27) and (32) (we recall that fixes the magnetic charge of solutions via for odd , and for even ). It is interesting to notice that the picture (as shown in Figures 5, 6), resembles the behaviour of non-gravitating Q-balls, whose frequency-mass diagram exhibits a similar pattern, see [49]. However, different from the frequency in that case, we could not derive analytical bounds for the parameter which enters the boundary conditions for the YM potentials.

As seen in Figures 5, 6, one important difference between odd- and even- solutions is that the former configurations are disconnected from the vacuum141414Since the and configurations coincide, this feature occurs also for the case.. That is, solutions with are found for only. This can be understood heuristically as follows. For any azimuthal winding number , the solutions with an even are found by varying the parameter in (32), starting with . However, in that case results in a similar set of boundary conditions at and at infinity. The only solution we could find in this case corresponds to the trivial one, , with . Then solutions with a small value of would be just deformations of this ground state and would possess a small mass. However, this does not hold for , since any value of is leading to a set of boundary conditions at infinity different from the the trivial solution (as set by the boundary conditions at ). As a result, the mass of the odd- solutions possesses always a nonzero minimal value.

An unexpected feature of the odd- solutions with large enough is their non-uniqueness. That it, three different solutions are found for given and some range of the parameter . This property is shown in Figure 6 for . To illustrate this behaviour, we also exhibit in Figure 7 the energy isosurfaces of the configurations at and the fixed value of the energy density .

One can understand this pattern as a manifestation of the composite structure of the configurations. Indeed, from Figure 7 we can clearly see that a typical solution consists of three constituents, each of them representing a soliton. Similarly, a configuration consists of two components. Since each of the components of the composite configuration possesses a magnetic dipole moment [25], whose magnitude increases with , the energy of the dipole-dipole interaction between the components becomes a significant part of the total energy. However this interaction energy can be both repulsive and attractive, depending on the orientation of the dipoles. Thus the lowest branch corresponds to the aligned triplet of dipoles, when the forces are most attractive (Figure 7, right plot), and two other branches correspond to the higher mass solutions with two other possible orientations of the triplet of dipoles.

Also, one can observe that, for and in (32), one finds zero magnetic charge configurations151515As noticed already, such solutions exist also for and ., which are the direct AdS counterparts of the solutions in [25].


Figure 1. The profiles of a typical solution with , .


Figure 2. The profiles of a typical solution with , .


Figure 3. The profiles of a typical solution with , .


Figure 4. The profiles of a typical solution with , .


Figure 5. The spectrum of solutions is shown for several odd- solutions.



Figure 6. The same as Figure 5 for even- solutions. One can notice the non-uniqueness of solutions for large enough values of the winding number .



Figure 7. The energy isosurfaces at the value of the energy density of the , solutions are shown for .

3.2 From AdS YM monopoles to flat space dynamical YM configurations

We close this section by remarking that the solutions discussed above may be relevant in yet another direction.

It is well known that the coordinate transformation (see, [46])

(39)

puts the AdS metric (6) in a conformally flat form

(40)

Then, due to the conformal invariance of the YM system, any solution of the YM equations in an AdS background results in a solution in a Minkowski spacetime161616 One can consider as well YM solutions in a fixed Einstein universe background, see [47].. For the case of the axially symmetric configurations discussed above, these configurations are described by a YM ansatz with a supplementary electric potential as compared to (8):

with the same functions and

(42)

One can see the gauge potentials acquire a dependence on the time coordinate , via the transformation (39).

Due to their numerical nature, the study of the physical properties of the corresponding Minkowski spacetime YM solutions is a difficult task, which is beyond the purposes of this work. Here we mention only that the AdS exact solution (26) corresponds to the flat spacetime meron configuration [48], written in a special gauge [1]. Thus we conjecture that at least the AdS solutions are likely to be relevant to the subject of multi-merons.

4 Including the backreaction: Einstein–Yang-Mills- solutions

Now we shall address the question on how the YM solutions discussed in the previous Section would deform the spacetime geometry, by including the backreaction and solving the full set of EYM equations.

4.1 The EYM action and field equations

We consider the Einstein-Yang-Mills action with a cosmological term:

(43)

When including the backreaction, apart from the YM equations (4), one solves also the Einstein equations,

(44)

This model has two length scales: the Planck one, , and the cosmological one, , which is fixed by the cosmological constant. These two length scales define the dimensionless ratio

(45)

4.2 Eym- system as a truncation of the gauged supergravity

For the generic EYM case, the value of is not fixed but an input parameter of the theory. However, among all values of the cosmological constant, there is a case of special interest. This corresponds to a consistent truncation of the SO(4) gauged supergravity in [28]. Note that this supergravity model can be viewed as a reduction of supergravity on [29]. Thus these particular EYM- solutions have a higher-dimensional interpretation.

The bosonic sector of SO(4) gauged supergravity contains two SU(2) fields and , a dilaton and an axion . In the conventions of [28], the Lagrangian density of the model reads

(46)

where the potential is

(47)

It is easy to verify that

(48)

is a consistent truncation of the general model (46). As a result, we end up with the EYM- Lagrangian

(49)

Working in conventions with a length scale set by (, taking in the numerics ), it follows that the generic EYM- Lagrangian reduces to (49) for the value of the coupling constant

(50)

(note also that ). As a result, all EYM solutions with the above particular ratio between Planck and cosmological length scales extremize also the action of the supergravity. Then, by using the formulas in [28] with two equal gauge fields, , one can uplift any such configuration to eleven dimensions. For example, the corresponding metric ansatz reads

(51)

where is the four dimensional line element, and , are SU(2) right invariant one forms on two 3-spheres . The corresponding expression of the matter fields can be found in [29], and we shall not display them here.

Also, note that, however, the solutions discussed in this work are generically not supersymmetric. The interesting task of constructing supersymmetric configurations would require a different approach171717To our knowledge, this task has not been considered yet in the literature. The supersymmetric solutions in [38] have been found for a different truncation of the SO(4) gauged supergravity model than (48)..

4.3 Solving the gravity field equations: the Einstein-De Turck approach

4.3.1 The metric ansatz

The previous work [22] has solved the EYM field equations by using a standard approach as originally proposed in [24] for asymptotically flat solutions. There one employs a metric ansatz with three unknown functions, and

(52)

(we recall ). The line-element (52) is inspired by the one in [24] and uses a quasi-conformal choice of the gauge for the -part of the metric ( , ). In this approach, one solves only a part of the full set of Einstein equations (44), namely

(53)

The remaining equations, and provide two constraints, which were used to test the numerical accuracy of the results.

Our choice in this work was to construct the solutions by employing the Einstein-De Turck approach. This approach has been proposed in [40], [41] (see also [42] for a review), and been employed recently in the study of various asymptotically AdS configurations181818Note, however, that only configurations in a Poincaré patch of AdS (and possibly with Abelian matter fields), have been considered so far in the literature.. This scheme has the advantage of not fixing a metric gauge, and leads to an overall better quality of the numerical results.

In this approach one solves the so called Einstein-DeTurck (EDT) equations

(54)

instead of (44), where is a vector defined as

(55)

and is the Levi-Civita connection associated to the spacetime metric that one wants to determine. Also, a reference metric is introduced, with the corresponding Levi-Civita connection. Solutions to (54) solve the Einstein equations iff everywhere on . To achieve this, we shall impose boundary conditions which are compatible with on the boundary of the domain of integration. Then, this should imply everywhere, a condition which is verified from the numerical output.

In this approach, we use a metric ansatz with two additional functions as compared to (52),