Self-gravitating solutions in Yang-Mills-Chern-Simons theory coupled to 3D massive gravity
Cristóbal Corral Daniel Flores-Alfonso Gastón Giribet Julio Oliva





cristobal.corral@uai.cl
Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Avenida Padre Hurtado 750, 2562340, Viña del Mar, Chile


[]danflores@unap.cl
Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Avenida Playa Brava 3256, 1111346, Iquique, Chile
Facultad de Ciencias, Universidad Arturo Prat, Avenida Arturo Prat Chacón 2120, 1110939, Iquique, Chile



gaston.giribet@nyu.edu
Department of Physics, New York University, 726 Broadway, New York, NY10003, USA.



juoliva@udec.cl
Departamento de Física, Universidad de Concepción, Casilla, 160-C, Concepción, Chile.




We study self-gravitating solutions of 3-dimensional massive gravity coupled to Yang-Mills-Chern-Simons gauge theory. Among these, there is a family of asymptotically Warped-Anti de Sitter black holes that come to generalize previous solutions found in the literature and studied in the context of WAdS_3/CFT_2. We also present self-gravitating solutions to the 3-dimensional Einstein-Yang-Mills theory, as well as other self-gravitating solutions in the presence of higher-curvature terms.


Self-gravitating solutions in Yang-Mills–Chern-Simons theory coupled to 3D massive gravity
    Julio Oliva
    
==========================================================================================





§ INTRODUCTION


The 3-dimensional gravity provides a tractable scenario to explore many aspects of gravity that are of insurmountable technical complexity in four and higher dimensions. Well known examples of this are the microscopic derivation of the black hole entropy beyond supersymmetry <cit.> and the explicit computation of the gravity partition function <cit.>. At classical level, the 3-dimensional models provide a useful playground to study the non-linear regime of gravity and gauge theories; in particular, in the context of black hole physics. This is mainly due to the existence of the Bañados-Teitelboim-Zanelli (BTZ) solution <cit.>, which describes black holes with physical properties that mimic those of their higher-dimensional analogs but, at the same time, resulting much more tractable. Being locally equivalent to 3-dimensional Anti-de Sitter (AdS_3) space <cit.>, the BTZ geometry also appears in any other theory that admits an AdS_3 vacuum. This includes 3-dimensional massive gravity, higher-spin theories, string theory and many others. A particularly interesting model to work with is the parity-even theory of massive gravity introduced in reference <cit.>, which is commonly known as New Massive Gravity (NMG). This is a higher-curvature theory that propagates two massive, spin-2 local degrees of freedom and exhibits interesting properties, such as the existence of a rich variety of black holes, apart from the BTZ solution. This includes generalizations of the BTZ black holes with a weakened AdS_3 asymptotics <cit.>, black holes with anisotropic scale invariance at infinity <cit.>, black holes in de Sitter (dS) space <cit.>, and black holes in Warped AdS_3 (WAdS_3) spaces <cit.>. All these solutions are of interest in the context of holography, as they were used to explore to what extent the AdS/CFT duality could be generalized beyond asymptotically AdS_3 spaces <cit.>. Here, we will focus on the WAdS_3 spaces.

The WAdS_3 spaces appear in connection to black hole physics: they emerge in the near-horizon region of Kerr black holes <cit.> and are crucial ingredients in the formulation of the Kerr/CFT correspondence <cit.>. Besides, the WAdS_3 spaces are at the root of the investigation of an entirely new class of conformal field theories (CFT) known as Warped-CFTs (WCFT) <cit.>; see also <cit.>. Consequently, any generalization of WAdS_3 solutions, e.g. by coupling gauge fields, is in principle of interest both for black holes physics and for holography. Here, we will show that, when coupled to non-Abelian gauge theory, NMG admits black holes that asymptote WAdS_3 spaces. These are similar to the WAdS_3 black holes studied in <cit.> and <cit.>, but, in contrast to those, the ones we construct here support a non-Abelian gauge field configuration. In fact, we will show that the gauge field may suffice to support the WAdS_3 black holes even in absence of higher-curvature terms. The model we will consider consists of Yang-Mills–Chern-Simons (YM-CS) gauge theory coupled to NMG. This results in a higher-derivative theory of gravity with a non-linear source, and that is the reason why, in order to solve the problem, we need a good strategy.

Our strategy goes as follows: Firstly, we recall that AdS_3 space can be written as a sort of Hopf fibration over its lower-dimensional analog, AdS_2, with the fiber being ℝ. This way of writing AdS_3 turns out to be particularly convenient to construct the 3-dimensional WAdS_3 spaces. The latter are easily obtained by introducing a real parameter (α) that deforms the fibration over AdS_2, warping the geometry. In this way, one obtains either a stretched or squashed versions of AdS_3, in a similar manner as how a squashed 3-sphere is obtained from the round S^3. Secondly, we can resort to the results of <cit.>, where the authors proved that the WAdS_3 black hole solutions previously found in <cit.> turn out to be discrete quotients of the spacelike stretched (empty) WAdS_3 space, in an analogous way as how the BTZ black hole happens to be a discrete quotient of Lorentzian AdS_3 <cit.>. This means that, if we were able to prove that WAdS_3 spaces appear as non-vacuum solutions to the non-Abelian theory coupled to 3-dimensional massive gravity, then we would be ipso facto proving that the theory also admits charged WAdS_3 black holes. 

This strategy sounds simple; however, when trying to carry this out, one encounters a difficulty: Although WAdS_3 spaces are solvable, they are not Einstein manifolds, i.e. they are not of constant curvature, and they are not even conformally flat. In addition, they are not compact and of negative curvature. All this implies that, while not intractable, the metric of WAdS_3 black holes is much more involved than that of stationary BTZ. Besides, the theory we are dealing with is of fourth order in the metric, and non-linear both in the gravitational and in the matter sector. Thus, in order to circumvent the intrinsic difficulty of the problem, we find it convenient to look for these solutions progressively: In section <ref>, we introduce the theory. In section <ref>, we look for YM-CS solution on the 3-sphere with non-trivial gauge field configuration for the SU(2) gauge field. The idea is to gain intuition from the compact case to select the right ansatz. In order to find such solutions, we propose a particularly friendly meron-type ansatz <cit.> for the gauge field, which is shown to suffice. Then, we appropriately introduce a real parameter and obtain a squashed 3-sphere solution in the theory. Finally, in section <ref>, after having gained some intuition about the squashing procedure, we perform the analytic continuation to the non-compact SU(1,1) group, which amounts to work out the WAdS_3 case, which has negative curvature. In this way, we finally find the WAdS_3 black holes that we are interested in. It turns out that the meron-type ansatz results in a useful trick to find exact solutions of YM-CS theory coupled to gravity. In section <ref>, which we reserve for some final remarks, we emphasize this point by presenting more new solutions of NMG coupled to YM-CS gauge theory.





§ MASSIVE GRAVITY COUPLED TO GAUGE THEORY


We will be concerned with a 3-dimensional theory that consists of NMG coupled to YM-CS matter fields. More precisely, the gravity action is given by the Einstein-Hilbert term with a cosmological constant, supplemented by the higher-curvature terms introduced in <cit.>. The matter sector is given by a non-Abelian SU(2) field governed by the YM action augmented with a CS term. This yields

    I =  κ∫_ℳ ^3x√(|g|)[R-2Λ - 1/μ^2(R_μν R^μν - 3/8R^2 )  +  1/2κ e^2(F_μνF^μν)] + 4π^2 k/e^2I_ CS ,

where κ=(16π G)^-1 is the gravitational coupling, Λ is the cosmological constant, and μ is a real, dimension-1 parameter that represents the graviton mass. In the matter sector, e denotes the coupling constant of the gauge theory. The non-Abelian field strength is defined as F_μν=∂_μ A_ν - ∂_ν A_μ + [A_μ,A_ν], which, expressed in terms of differential forms, yields the curvature 2-form F=12F_μνx^μ∧x^ν=A+12[A,A]. The gauge field decomposes as usual, A=A_ μx^μ =A^i_ μt_ix^μ, with μ=0,1,2 being the spacetime indices and i=1,2,3 the group indices; t_i are the generators of su(2) algebra. The action also includes a CS term 

    I_ CS = 1/8π^2∫_ℳ(A∧A + 2/3A∧ A∧ A ) ,
    
which is natural in dimension 3. We will omit here the discussion about the topological origin of (<ref>), the role of large gauge transformations, the quantization condition for k for compact groups, and all that. Also, while the solutions we will discuss here admit a straightforward generalization to the case in which a gravitational CS term <cit.> is also included, here we will not deal with it.

The field equations derived from the action above take the form

    
    ℰ^(g)_μν   ≡ R_μν - 1/2g_μνR + Λ g_μν  - 1/2μK_μν - 1/2κT_μν = 0 , 
    ℰ_(A)^μ   ≡∇_ν F^νμ + [A_ν,F^νμ] + k ε^μνλF_νλ = 0  ,


where

    K_μν   = 2 R_μν - 1/2∇_μ∇_ν R - 1/2g_μν R + 4R_μλνρR^λρ-3/2RR_μν - g_μνR_λρR^λρ + 3/8g_μνR^2, 
    
        T_μν   = -2/e^2(F_μλF_ν^λ - 1/4g_μνF_λρF^λρ) .

Despite the field equations (<ref>) are of fourth order in the metric, their trace is of second order. This is at the very root of the interesting properties of NMG, e.g. the absence of ghosts about its maximally symmetric vacua.

This theory admits many interesting solutions in vacuum, including hairy black holes <cit.>, warped black holes <cit.>, Lifshitz black holes <cit.>, gravitational waves <cit.>, and many other spacetimes; see <cit.> and references therein. Solutions in the presence of gauge fields, in contrast, are much harder to obtain, at least analytically, cf. <cit.>. In fact, due to the higher-derivative terms of NMG, a seemingly simple problem such as charging a BTZ black hole with a U(1) gauge field turns out to be rather difficult. Despite this difficulty, here we will be able to find solutions with non-Abelian gauge fields, enlarging in this way the set of known solutions of massive gravity in the presence of matter.





§ SELF-GRAVITATING SOLUTIONS WITH SU(2) GAUGE FIELD


As explained in the introduction, our first step will be finding solutions of positive curvature with non-trivial SU(2) field configuration. That is to say, we look for solutions to Eqs. (<ref>) by considering a su(2)-valued gauge connection A=A_μ^i t_ix^μ, where i=1,2,3 are indices of the group, t_i=-i/2τ_i are generators of the SU(2) satisfying the su(2) Lie algebra [t_i,t_j]=ϵ_ijkt^k, and τ_i are the Pauli matrices. Notice that we have chosen the generators to be anti-hermitian. The standard (anti)commutation relations of the latter yield (t_it_j) = -12δ_ij and (t_it_jt_k) = - 1/4ϵ_ijk. 

In the following, we construct different self-gravitating SU(2) gauge configurations that support special 3-dimensional geometries in NMG.




 §.§ The round 3-spheres


The simplest regular space with positive curvature in three dimensions is the 3-sphere, S^3. To obtain self-gravitating YM-CS configurations that support this geometry in NMG, it is convenient to introduce the left-invariant forms of SU(2), call them σ_i, which can be parameterized in terms of the Euler angles,

    
    σ_1    = cosψ ϑ + sinϑsinψ φ , 
    σ_2    = -sinψ ϑ + sinϑcosψ φ , 
    σ_3    = ψ + cosϑ φ ,


where 0≤ϑ≤π, 0≤φ<2π, and 0≤ψ≤ 4π. These 1-forms satisfy the Maurer-Cartan equations of SU(2), namely σ_i + 12ϵ_ijkσ^j∧σ^k=0. The metric of S^3 can be written in terms of the left-invariant forms of SU(2) as  follows

    s^2 = ρ^2 δ_ij σ^i ⊗σ^j = ρ^2[(ψ + cosϑφ)^2 + θ^2 + sin^2ϑφ^2] ,

where ρ∈ℝ_+ is a constant that denotes the radius of S^3. The metric is invariant under the action of the SO(4) isometry group. For the non-Abelian field A, we assume an ansatz aligned along the left-invariant forms, i.e.

    A = A_μ^i t_i x^μ = λ σ^i t_i ,

where SU(2) indices are raised and lowered with the Cartan-Killing metric. Despite the remarkably simple ansatz for A, the non-linearity of the gauge theory yields a non-vanishing field strength, provided as 0 ≠λ≠ 1; namely

    F = λ/2(λ-1)ϵ^ijk t_i σ_j∧σ_k .

In Euclidean 4-dimensional space, solutions of this sort were first studied in Ref. <cit.>, for which the YM equations demand λ=1/2. Such configurations were dubbed merons, and present localized fractional topological charge; see also the early works  <cit.>. Black hole solutions in dimension 4 with similar gauge field configurations were recently studied in <cit.>.

Inserting the ansätze for the metric and the gauge fields into the field equations (<ref>), we find that the latter are solved if

    
    λ   = 1/2 + kρ , 
    ρ^2_±   =  1/4(4κΛ e^2-k^4){2κ e^2 - k^2 ± e√(κ[4(Λ+κ e^2 - k^2) - 1/μ(4κΛ e^2-k^4 )])} ,


for 4κΛ e^2-k^4≠0. Notice that, in dimension 3, the presence of the CS term produces a shift in the value λ=1/2. This was first observed in the context of the 3-dimensional Einstein-YM-CS theory in Ref. <cit.>. In fact, our solution (<ref>) reduces to those found in <cit.> in the limit  μ→∞. Notice that, for ρ_±^2 to be positive, the parameters of the theory have to be constrained. Moreover, in the topological limit e→ 0, k→ 0 with |k/e |<0, where the matter content reduces to CS, one recovers the pure gauge solutions λ =0 and λ=1. If the CS coupling k vanishes, there exist a perturbative branch for which Λ→0, i.e.

    λ = 1/2 ,        ρ^2_- = 1/16μ - 1/16κ e^2 + 𝒪(Λ) .


The particular case k^4=4κΛ e^2 needs to be studied separately. In that case, the solution becomes

    λ = 1/2 + kρ ,      ρ^2 = κ e^2 - μ/8μ(2κ e^2 - k^2 )=k^4-4μΛ/16μ k^2(k^2-2Λ ) .

The gauge field configuration that supports this solution has a nontrivial topological charge, what can be seen by integrating the CS term (<ref>) evaluated on the configuration (<ref>). This yields

    I_ CS = 1/2 + 3kρ/2 - 2k^3ρ^3 ,

which receives contributions from the CS term.

Now that we have shown that NMG coupled to YM-CS theory admits a 3-sphere with a non-trivial gauge field on it, we will introduce a squashing parameter and try to see how the solution can be gently deformed. 



 §.§ The squashed 3-sphere


Deforming one of the directions of the 3-sphere, say σ_3, by adding a warping factor α∈ℝ_+ leads to a space of non-constant curvature. The metric of such space is that of the squashed 3-sphere, which can be locally written as

    s^2 = ρ^2(σ_1^2 + σ^2_2 + α^2σ_3^2 ) = ρ^2[θ^2 + 
    sin^2θ ϕ^2+α^2(ψ+cosθ ϕ)^2] .

Here, ρ and α denote the radius and the squashing parameter, respectively. This metric is no longer invariant under the isometry group SO(4)=SU(2)× SU(2) /ℤ_2; its symmetry group is broken down to SU(2)× U(1) by the introduction of the squashing; α can be seen as a symmetry-breaking parameter. The squashed 3-sphere is an η-Einstein space, which means that it is equipped with a contact 1-form, η=η_μx^μ, and it satisfies the condition <cit.>

    R_μν =  a η_μη_ν + b  g_μν ,

for some a and b. These geometries have been studied in the context of NMG in Ref. <cit.>. In the case of the squashed 3-sphere, the contact form is η=σ_3 and the values of a and b are

    a=α^2(α^2-1),      b=2-α^2/2ρ^2.

Since these constants only depend on α^2 and the radius, one can perform the analytic continuation α→ iα and obtain a squashed analog of the dS_3 space.

To obtain a squashed 3-sphere solution with non-trivial SU(2) gauge field configuration in NMG, we might feel tempted to take a simple ansatz such as

    A    = 2λ [( sinϕ θ +cosϕcosθsinθ ϕ)t_1
        
            - (cosϕ θ -sinϕcosθsinθ ϕ)t_2
        - sin^2θ ϕ t_3] .

This choice, however, leads to a gauge field with a trivial topological charge, as the gauge connection depends only on two coordinates of the squashed 3-sphere. It is possible to obtain a solution with non-trivial topological charge by considering an anisotropic version of (<ref>). This has been explored in Ref. <cit.> and it is easy to prove that the same class of solutions works in NMG equations. However, for simplicity, here we will focus on the case of vanishing topological charge.

Inserting  Eq. (<ref>) and the ansatz (<ref>) into the field equations (<ref>), we find that the latter reduces to a system of algebraic equations that admit the following solution

    λ = 1/2  ,         ρ^2 = α^2/4k^2 ,

with the coupling constants being related as follows 

    Λ   = 
    -k^2(21 α^6 e^2κ -93 α^4 e^2κ +88 α^2 e^2κ -42 α^2 k^2-16 e^2κ +40 k^2)/2 e^2κ α^2(21 α^4-25 α^2+4) , 
    μ   = 
    -k^2κ e^2(21 α^4-25 α^2+4)/2 α^4 e^2κ -2 α^2 e^2κ +4 k^2 .

A real, Lorentzian version of the solution can also be obtained by performing the analytic continuation α→ iα.




§ WARPED ADS BLACK HOLE


Having studied the case of the 3-sphere and the squashed 3-sphere, we are ready to present the solution for the WAdS_3 black hole. As explained in the introduction, the argument goes as follows: AdS_3 space can be written as Hopf fibration, just like in the case of the 3-sphere, by simply replacing trigonometric functions in  (<ref>) by their hyperbolic counterparts. Then, the WAdS_3 spaces are nothing but the α-deformations of AdS_3, exactly as done in (<ref>) for the squashed 3-sphere. Having solved the case of positive curvature, the hyperbolic analog can easily be obtained by continuing the gauge theory to the SU(1,1) group. Then, taking into account the observation of <cit.>, which states that the WAdS_3 black holes of <cit.> are discrete quotients of WAdS_3, we are ready to derive the solutions we are looking for. Following the same steps as before, but adapting them to the hyperbolic case, we are led to find the adequate relations between the coupling constants of the theory and the squashed parameter α. We also take k=0. Then, we can consider the stationary ansatz

    s^2 = -f(r) t^2 + r^2/h(r) + ρ^2(r)(θ + N^θ(r) t)^2  ,

together with the following ansatz for the gauge field 

    A = λ^ijω_iη_j  ,

where λ^ij=(λ_(1),λ_(2),λ_(3)) is in the adjoint representation of SU(1,1); λ_(i) are constants, and ω_i are the left-invariant forms of SU(1,1). As it happens for NMG in vacuum <cit.>, the field equations (<ref>) are solved by the following functions

    f(r)    = ℓ^2(r-r_+)(r-r_-)/r[2α√(r_+r_-)+(α^2-1)r + r_+ + r_- ] , 
    
        h(r)    = (r-r_+)(r-r_-)/ρ_0^2 , 
    ρ^2(r)    = rℓ^4[2α√(r_+r_-)+(α^2-1)r + r_+ + r_- ]/4ρ_0^2 , 
    
        N^θ(r)    = 2ρ_0[√(r_+r_-) + α r ]/rℓ[2α√(r_+r_-)+(α^2-1)r + r_+ + r_- ] .

where Λ=-ℓ^-2. r_± are two integration constants that, provided 0≤ r_- ≤ r_+, represent the locations of the outer and the inner horizons of the black hole, cf. <cit.>. We also find that the following relations must be obeyed

    ρ_0^2    = [λ_(2)-λ_(1)λ_(2)]^2-α^2[λ_(1)-λ_(2)^2]^2/2κ e^2α^2(α^2-1) +  21α^2-4/8μ , 
    1/ℓ^2   = -1/ρ_0^4[21α^4-72α^2+16/64μ
       + α(9α^4-17α^2-24)√(9α^2+16)-27α^4(α^2-1 ) + 128α^2+32 /128e^2κ(α^2-1)] , 
    λ_(1)   = 1 - 2λ_(2)^2 , 
    λ_(2)^2    = λ_(3)^2 = α/8(√(9α^2+16)-3α) .

Notice that all these parameters get fixed in terms of α which, in turn, is a function of the AdS_3 radius through (<ref>).

The field configuration (<ref>) can be seen to be finite at the horizon and to exhibit an asymptotic behavior similar to the one obtained in <cit.> for the U(1) gauge field in asymptotically WAdS_3 spaces. 

The conserved charges associated with the WAdS_3 black holes have been computed in <cit.>; see also references therein and thereof. A similar computation works here.

We can also parametrize the left-invariant forms of SU(1,1) as follows

    
    ω_1    = u + sinhστ , 
    ω_2    = cosh uσ - sinh u coshστ , 
    ω_3    = sinh uσ - cosh u coshστ ,


with (u,σ,τ) being scalar functions of the coordinates (t,r,θ); they are explicitly given by 


    u    = ℓ^2/4αρ_0^2√(4-α^2/3)[2t - (α√(3/4-α^2)(r_++r_-) + ℓ√(r_+r_-)/ρ_0)θ] 
            - arctanh[r_++r_–2r/r_+-r_-(ℓ^2(r_+-r_-)/4ρ_0^2 θ) ] , 
    σ   = arcsinh[2√((r-r_+)(r-r_-))/r_+-r_-cosh(ℓ^2(r_+-r_-)/4ρ_0^2 θ) ] , 
    τ   = arctan[2√((r-r_+)(r-r_-))/2r-(r_++r_-)sinh(ℓ^2(r_+-r_-)/4ρ_0^2 θ) ] ,
    

which are defined for the range 1≤α<2 <cit.>. Coordinates (u,σ,τ) are useful to describe locally WAdS_3 spaces as fibrations and explore their properties.

Notice that the limit α→1 of the solution above is smooth and yields λ_(i)=1/2 for i=1,2,3. In this limit, we have

    ρ_0^2 = 85κ e^2+μ/40κ e^2 μ          1/ℓ^2 = 35κ e^2μ(25κ e^2 + μ)/(85κ e^2 + μ)^2 .

This is the limit in which the squashing effect disappears and the solution is locally equivalent to AdS_3. Another limit that is worth looking at is e→∞, where the solution above is seen to agree with the WAdS_3 black hole of NMG in vacuum, originally found in Ref. <cit.>.

For arbitrary values of the parameters, the solution (<ref>)-(<ref>) yields a non-vanishing stress-energy tensor, T_μν, which can be written as a linear combination

    T_μν = c_1  g_μν + c_2  K_μν

where K_μν is the tensor defined in Eq. (<ref>), and where the coefficients are

    
    c_1    = -1/8ρ_0^4α^2(α^2-1)(21α^2-4)[21α^6(λ_(1)-λ_(2)^2)^2 + 16λ_(2)^2(λ_(1)-1)^2(1-5α^2)  
            + α^4(63λ_(1)^2λ_(2)^2-20λ_(2)^4-86λ_(1)λ_(2)^2 -20λ_(1)^2 + 63λ_(2)^2  ) ] , 
    
        c_2    = 4[λ_(2)^2(λ_(1)-1)^2 - α^2(λ_(1)-λ_(2)^2)^2 ] /21α^6 - 25α^4+4α^2 ,


which can be seen to be finite in the limit α→ 1. Equation (<ref>) implies that the YM field shifts the cosmological constant Λ and the mass parameter μ. From the expressions for c_1 and c_2 one can verify that the WAdS_3 solution may exist even if the higher-curvature terms of NMG are not present, i.e. in the limit μ→∞. That is to say, the YM terms suffice to support the black hole background in Einstein gravity. In some cases, the meron-type ansatz renders the YM configuration locally equivalent to an Abelian gauge field <cit.>; in the case of the WAdS_3 black holes this seems to be different: while the WAdS_3 black holes are known to exist in the Einstein gravity coupled to the U(1) gauge field theory <cit.>, in that case, the CS term is needed to support the background. It is also worth mentioning that the solution we derived here persists when the gravitational CS term of Topologically Massive Gravity is also included. This merely induces a shift in the relation among the coupling constants of the theory and the parameters of the solution.




§ OTHER SOLUTIONS AND FINAL REMARKS


Here, we have constructed exact black hole solutions of 3-dimensional massive gravity sourced by a non-Abelian gauge field. The solutions asymptote WAdS_3 at large distance and coincide with the NMG solution in vacuum found in <cit.> in the appropriate limit. This result might be relevant to the study of WAdS_3/(W)CFT_2 correspondence <cit.> and it would be interesting to study the implications in that context. Among the solutions we found, there are self-gravitating black holes in Einstein–Yang-Mills theory. 

The theory we considered throughout the paper was NMG coupled to YM-CS gauge theory. This theory is of fourth order and is non-linear both in the gravitational and the matter sector, which makes the problem of solving the field equations analytically quite intricate. This demanded from us to find an adequate ansatz that resulted in a simplification of the equations. We considered a meron-type ansatz for the gauge field, which renders the problem tractable analytically. This ansatz turns out to be particularly useful to find exact solutions to this theory. In order to illustrate that and to show how the results of this paper can be extended, let us present some additional cases below. 

One relatively simple example is the Nil geometry, which is one of the eight Thurston geometries that appear in connection to the geometrization conjecture. It can be associated to a Bianchi II space endowed with a transitive nilpotent group of diffeomorphisms. Locally, it remains invariant under the action of the Heisenberg group; see Refs. <cit.>. We write its Euclidean metric as

    s^2 = ρ^2[ x^2+ y^2+( z+x y)^2] ,

where we denote ρ as the radius.

There are two, non-flat, left-invariant metrics on the Nil group with Lorentzian signature <cit.>. One of them corresponds to fiber over Euclidean space,

    s^2 = ρ^2[ x^2+ y^2-( t+x y)^2],

and the other is a fiber over Lorentzian space,

    s^2 = ρ^2[- t^2+ x^2+( y+t x)^2].

They are sometimes referred to as warped flat spaces <cit.>. As pointed out in Ref. <cit.>, the first of these metrics admits the interpretation of a cross-section of the cylindrical Som-Raychauduri spacetimes <cit.>. On the other hand, metric (<ref>) can be interpreted as a Kundt spacetime, a natural generalization of the pp-waves; see <cit.>.


Using an ansatz for the gauge field similar to the one used in the previous sections, we can show that NMG coupled to YN-CS theory also admits Nil manifolds as solutions. To show this, let us focus on the solution (<ref>) and take the SU(2) gauge field

    A    = 2λ[(-sin y x/√(1-x^2)+xcos y√(1-x^2) y)t_1 
            + (cos y x/√(1-x^2)+xsin y√(1-x^2) y)t_2  - (1-x^2)y  t_3 ] .

Inserting this ansatz together with (<ref>) into the field equations (<ref>), we find that the latter reduces to an algebraic system with the following solution

    λ   = 1/2 ,    ρ^2    = 1/4k^2 ,    Λ   = k^2(κ e^2+2k^2)/2κ e^2 ,    μ   = 21κ e^2 k^2/2(κ e^2 - 2k^2 ) ,

which holds provided κ e^2-2k^2≠0. In fact, if the latter condition is not satisfied, there are no solutions for μ≠0. Notice also that this solution does not exist in the absence of the CS term. A similar solution for Eq. (<ref>) exists, which is also valid for κ e^2-2k^2=0.

Another interesting type of solutions of NMG that support non-trivial gauge field configuration is the AdS-waves, which are analogs of pp-waves but in AdS_3, cf. <cit.>. The metric os these solutions takes the form

    s^2 = ρ^2/y^2[-2uv-F(u,y)u^2+y^2] ,

where u∈ℝ and v∈ℝ are null coordinates, ρ∈ℝ_+ is a constant, and F(u,y) is a function that represents the profile of the wave and is obtained by solving the field equations. Locally AdS_3 corresponds to the particular case F=const. In these coordinates, we consider a convenient parameterization of the SU(1,1) gauge field, namely

    A =    2λ[ (2(v-1)/y^2 u -  1/y y
        )η_1 + ( v(v-2)/y^2 u + v/2-  v-1/y y
        )η_2
          - ( v^2-2v+2/y^2 u + v/2-  v-1/y y
        )η_3] .

Inserting these ansätze into the YM-CS equations (<ref>), we find that the matter equations are solved if

    λ = 1/2 - kρ/2           F(u,y) = c(u) y^1-kρ ,

where c(u) is an arbitrary function of the null coordinate u. Inserting these solutions into the remaining field equations, we find that they are solved by


    ρ^2    = κ e^2 - 2μ/2[κ e^2(k^2-μ) - k^2μ] , 
    Λ   = -4μ^3(5κ e^2+4k^2) + 4μ^2(k^4 -5κ e^2 k^2 - κ^2 e^4  ) - 7μκ e^2 k^4 + 4κ^2 e^4 k^4/4μ(κ e^2-2μ)^2 .
    


This manifestly shows that the strategy followed throughout this paper turns out to be useful for finding exact, analytic solutions of this type of non-linear, higher-derivative theories.  


The authors thank Fabrizio Canfora for discussions. The work of C.C. is supported by  Agencia Nacional de Investigación y Desarrollo (ANID) through FONDECYT Regular grants No 1240043, 1240048, 1230112, and 1210500. D.F-A. is supported by ANID under FONDECYT grant No. 3220083. J.O. acknowledges the financial support of FONDECYT Regular grant No 1221504.







