Exact \textit{pp}-waves, (A)dS waves and Kundt spaces in the Abelian-Higgs model
Fabrizio Canfora, Adolfo Cisterna, Diego Hidalgo, Julio Oliva
aa r X i v : . [ h e p - t h ] F e b Exact pp -waves, (A)dS waves and Kundt spaces in theAbelian-Higgs model Fabrizio Canfora , Adolfo Cisterna , , , Diego Hidalgo , , , Julio Oliva Centro de Estudios Científicos (CECs), Casilla 1469, Valdivia, Chile. Vicerrectoría Académica, Toesca 1783, Universidad Central de Chile, Santiago, Chile Dipartimento di Fisica, Universita di Trento, Via Sommarive 14, 38123 Povo (TN), Italy. TIFPA - INFN, Via Sommarive 14, 38123 Povo (TN), Italy. Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción, Chile, Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile. [email protected], [email protected], [email protected], [email protected]
February 11, 2021
Abstract
We find new exact solutions of the Abelian-Higgs model coupled to General Relativity, charac-terized by a non-vanishing superconducting current. The solutions correspond to pp -waves, AdSwaves, and Kundt spaces, for which both the Maxwell field and the gradient of the phase of thescalar are aligned with the null direction defining these spaces. In the Kundt family, the geometryof the two-dimensional surfaces orthogonal to the superconducting current is determined by the so-lutions of the two-dimensional Liouville equation, and in consequence, these surfaces are of constantcurvature, as it occurs in a vacuum. The solution to the Liouville equation also acts as a potentialfor the Maxwell field, which we integrate into a closed-form. Using these results, we show that thecombined effects of the gravitational and scalar interactions can confine the electromagnetic fieldwithin a bounded region in the surfaces transverse to the current. A very important step towards a deep understanding of a classical field theory is a proper understandingof its classical solutions. For a generic field theory, this may seem an insurmountable task since the spaceof solutions are infinite-dimensional, nevertheless, for General Relativity (GR), important classificationschemes are available which allow defining classes of solutions, contributing to the understanding oftheir potential realization in nature [1].One of the most relevant field theory (both at the classical and quantum level) is the Abelian-Higgs model (the Maxwell-Ginzburg-Landau theory) which can describe successfully many importantsemi-classical features of superconductors (see [2] and [3] for detailed reviews: in the following, we willconsider the relativistic version of the theory). A further important phenomenological implication of1his theory is the presence of vortices discovered by Abrikosov, Nielsen and Olesen in [4, 5]. Theseare some of the many reasons why the minimal coupling of the Abelian-Higgs model with GR hasbeen deeply investigated (see [3] and references therein). Moreover, a no-hair theorem was provedin [6], which can be circumvented for horizons pierced by a vortex both in the static case [7], as wellas for stationary black holes [8], and for planar AdS black holes [9]. In the holographic setup, thissystem allows constructing holographic superconductors, where near the horizon of a black hole thescalar acquires a vev [10–12, 14], which can be understood as arising due to an instability triggered bya violation of the effective Breitenlohner-Freedman bound [15] in the AdS near horizon geometry ofextremal Reissner-Nordström black hole. Finally, this system also finds applications in the holographicdescription of superfluidity (see e.g. [13, 14, 16]). Given the relevance of this field theory, it is ofuttermost importance to continue shedding light on the structure of its space of solutions. Thispaper is devoted to such a task. In the present manuscript, we will study, with analytic methods,the gravitational consequences of the presence of a superconducting current in the Einstein-Maxwell-Ginzburg-Landau theory.Of course, one may wonder why to insist on finding analytic solutions if these equations can besolved numerically. Indeed, numerical techniques were already available in the literature of the eightiesand nineties to analyze these configurations in the gravitating Abelian-Higgs model (see [3] and refer-ences therein). Despite this, there are indisputable arguments that strongly suggest that, whenever itis possible, we should strive for analytic solutions. For example, much of what we currently know aboutblack hole physics in GR, and instantons and monopoles in gauge theories arose from a careful studyof the available analytic solutions like the Kerr solution in the former and non-Abelian monopoles andinstantons in the latter. Consequently, an analytic tool to analyze the gravitational effects of super-conducting currents in the model relevant to our present study can greatly enlarge our understandingof this system. Secondly and more concretely, our analysis discloses a nice mechanism that, at leastin principle, can confine the electromagnetic field in the two-dimensional surfaces orthogonal to thesuperconducting currents.One may think that pursuing an analytic approach in this non-linear system is hopeless. Neverthe-less, the methods developed in [17–22] to propose a proper ansatz, allowed to construct analytic gaugedsolitons in the gauged Skyrme model thanks to a suitable choice of variables which enables to partiallydecouple the field equations. These were extended in [23] to include the minimal coupling with GRand here we show that they are suitable to analyze Einstein-Maxwell-Ginzburg-Landau theory, as well.In Section II, we present the model and describe properties of the superconducting current sup-porting the solutions of the following sections. In Section III, we construct the pp -wave as well as theAdS-wave solutions and for the latter, in a particular case, we can integrate the whole system in anexplicit, closed manner. Section IV is devoted to the construction of the Kundt solutions, characterizedby the existence of a null, geodesic, congruence that is not covariantly constant but has vanishing op-tical scalars. Liouville equation naturally emerges in the constant u, v sector and we obtain non-trivialsolutions for both, the positive and negative cosmological constant value. Finally, we provide someconclusions in Section V. 2 The model
The gravitating Abelian-Higgs model is described by the action S [ g, Ψ , A ] = Z d x √− g (cid:18) R − − F µν F µν − D µ Ψ ( D µ Ψ) ⋆ − V (Ψ) (cid:19) , (1)where g is the determinant of the metric, R is the Ricci tensor scalar, Λ the cosmological constant,and we have set πG = 1 . The scalar field Ψ is complex, and Ψ ⋆ denotes its complex conjugate. Theelectromagnetic field strength is given by F µν = ∂ µ A ν − ∂ ν A µ , with A µ the electromagnetic potential.In (1), we have introduced the gauge covariant derivative of the field with charge q and its conjugatewith charge − q as D µ Ψ = ∂ µ Ψ + iqA µ Ψ , ( D µ Ψ) ⋆ = ∂ µ Ψ ⋆ − iqA µ Ψ ⋆ , (2)and hereafter ∇ µ denotes the covariant derivative constructed with the Christoffel symbol. In theAbelian-Higgs model, the self-interacting potential V (Ψ) of the complex scalar field is given by V ( | Ψ | ) = λ (ΨΨ ⋆ − ν ) , (3)where ν is a real constant and λ > . The field equations that follows from varying the action (1) are R µν − g µν R + Λ g µν = T µν , (4a) ∇ µ F µν = J ν , (4b) ∇ µ ∇ µ Ψ + iq ∇ µ A µ Ψ + 2 iqA µ ∇ µ Ψ − q A µ A µ Ψ − ∂∂ Ψ ⋆ V ( | Ψ | ) = 0 . (4c)The stress-energy tensor T µν is the sum of two contributions T µν = T ( A ) µν + T (Ψ) µν , (5)associated to the Maxwell and the scalar field, respectively, given by T ( A ) µν = 12 (cid:18) F µα F ν α − g µν F αβ F αβ (cid:19) , (6) T (Ψ) µν = 12 ( D µ Ψ ( D ν Ψ) ⋆ + D ν Ψ ( D µ Ψ) ⋆ − g µν ( D α Ψ ( D α Ψ) ⋆ + V (Ψ))) . (7)In (4b), the particle number current is given by J µ = iq (( D µ Ψ) ⋆ Ψ − D µ Ψ Ψ ⋆ ) . (8)In the following, we focus on two new families of independent solutions to this model. Firstly, weconstruct new charged pp -waves and (A)dS waves and show that they are controlled by an integrablesystem. Then, inspired by an extension of these solutions we will construct new charged spacetimesthat contain a two-dimensional sector whose conformal factor leads to the Liouville equation in two3imensions. We will see that this function plays the role of a potential and source of the Maxwellequation and the remaining Einstein equations, respectively.Before proceeding with the construction of the exact solutions, a few remarks are in order regardingthe persistent character of the U (1) currents. In [24] the deep and consequential idea of superconductingstrings was proposed. This idea (which was further generalized, for instance, in [25–48] and referencestherein) shed light on the highly non-trivial gravitational effects of superconducting currents. Thesereferences partly motivated the present analysis to build the simplest possible analytic example ofgravitational fields sourced by currents with the characteristics listed here below. As far as the presentanalysis is concerned, the relevant features of the U (1) persistent current [24] for our construction are:• The U (1) current (whose gravitational effects are under examination) should survive even in thelimit of zero gauge potential.• The corresponding residual current J (0) µ (in the limit A µ = 0 ) should have the form J (0) µ = Γ ∂ µ Ω , (9)where Γ is a function which cannot vanish everywhere while the function Ω is defined only modulo π : Ω ∼ Ω + 2 π .As far as the function Γ is concerned, the simplest case corresponds to Γ = cte : in the following, wewill consider configurations in which this option is realized. While for the function Ω , we will considerconfigurations in which the fact that Ω is defined only modulo π is manifest. In particular, from Eqs.(11) and (27) below it is clear that the function Ω is defined only modulo π and that the current isproportional to ∂ µ Ω . pp and AdS waves In order to simplify the presentation of the new solutions obtained in this section, we separate theanalysis of the
Λ = 0 case, from that with non-vanishing Λ . pp -waves The metric for a pp -wave in Brinkmann coordinates reads ds = − F ( u, x, y ) du − dudv + dx + dy . (10)This geometry is characterized by possessing a covariantly constant vector ∂ v , which being non-twisting,is orthogonal to the two-dimensional, planar hypersurface spanned by the coordinates ( x, y ) . In vac-uum, Einstein equations imply that the wave profile F ( u, x, y ) can be separated as an arbitrary functionof the coordinate u , times a harmonic function on ( x, y ) . On the other hand, these spacetimes areconsistent with the backreaction produced for example by a conformal source [49]. Metrics of the form410) also play an important role in holography since they emerge, for example, as supersymmetricconfigurations by taking a suitable Penrose limit of the AdS × S solution of Type-IIB SUGRA [51].Here, in the context of the gravitating Abelian-Higgs model (1), we focus on the spontaneouslybroken phase, but maintaining the phase of the scalar turned on, and we impose that both, the gradientof the scalar ∂ µ ψ as well as the gauge field A µ , to be aligned with the covariantly constant vector ∂ v that defines the pp -wave (10). This kind of strategy to decouple the field equations describing gaugedsolitons in the low energy limit of QCD, minimally coupled to Maxwell equations, has been introducedin [17–23]. These conditions lead to ψ = ρ e i Ω( u ) , and A = a ( u, x, y ) du , (11) A = ∇ µ A µ = A · ∇ ψ = 0 , (12)which have also been useful in the construction of static and rotating solutions in vector Galileontheories [50].Under these circumstances, considering a constant value of ρ = ν = cte , we obtain an effectivesystem of equations given by G µν = 12 F µλ F λν − g µν F + ρ [ ∇ µ Ω ∇ ν Ω + q ( A µ ∇ ν Ω + A ν ∇ µ Ω) + q A µ A ν ] , (13) ∇ µ F µν = 2 qρ ∇ ν Ω + 2 q ρ A ν , (14)and the Klein-Gordon equation is automatically satisfied when ρ is constant. As expected, on thespontaneously broken phase, the vector field A µ acquires a mass which can be read from (14) leadingto m A = 2 q ρ . (15)Defining ω ( u ) = ∂ u Ω( u ) , one can show that the whole system for the gravitating Abelian-Higgs model,in this sector reduces to the following two equations (cid:18) ∂ ∂x + ∂ ∂y (cid:19) a ( u, x, y ) − qρ ( qa ( u, x, y ) + ω ( u )) = 0 , (16) (cid:18) ∂ ∂x + ∂ ∂y (cid:19) F ( u, x, y ) − (cid:18) ∂∂x a ( u, x, y ) (cid:19) − (cid:18) ∂∂y a ( u, x, y ) (cid:19) − ρ ( qa ( u, x, y ) + ω ( u )) = 0 . (17)Remarkably, we have arrived to an integrable system. The equation (16) is a screened Poisson equationfor the gauge field component a ( u, x, y ) , which can be integrated in terms of a convolution of the Greenfunction for this operator and the source ω ( u ) , which is the phase of the complex scalar field. Clearly,the effective mass of the vector field m A given in (15) is responsible for the screening. Once this equationis integrated, equation (17) transforms into a Poisson equation for the pp -wave profile F ( u, x, y ) , whichagain, can be integrated using the corresponding Green function.Notice that one may want to remove the phase ω ( u ) by a gauge transformation ψ → ψe − iqξ ( x µ ) and A µ → A µ + ∂ µ ξ . This can be achieved locally, but since the phase depends on the null direction u , in5rder to remove it, one must in general implement a large gauge transformation. Such transformationscan modify the global interpretation of the solution, thus we prefer not to remove ω ( u ) . In this case the spacetime takes the form ds = ℓ x (cid:0) − F ( u, x, y ) du − dudv + dx + dy (cid:1) , (18)where we have defined Λ = − /ℓ , with ℓ the AdS radius. These Siklos spacetimes correspond to aconformal transformation of the pp -wave (10). Now, the field equations reduce to: x (cid:18) ∂ ∂x + ∂ ∂y (cid:19) a ( u, x, y ) − x ℓ qρ ( qa ( u, x, y ) + ω ( u )) = 0 , (19) (cid:18) ∂ ∂x + ∂ ∂y − x ∂∂x (cid:19) F ( u, x, y ) − x ℓ (cid:18) ∂∂x a ( u, x, y ) (cid:19) + (cid:18) ∂∂y a ( u, x, y ) (cid:19) ! − ρ ( qa ( u, x, y ) + ω ( u )) = 0 . (20) In the presence of the cosmological term, the Maxwell equation (19) is not an autonomous equationanymore, nevertheless it can be integrated and leads to a ( u, x, y ) = ω ( u ) (cid:20) x / (cid:0) A ( y ) x ν + B ( y ) x − ν + ( C J ν ( cx ) + C Y ν ( cx )) (cid:0) C e cy + C e − cy (cid:1)(cid:1) − q (cid:21) , (21)where J ν and Y ν are the Bessel functions of the first and second kind, respectively, A ( y ) and B ( y ) arearbitrary linear functions of y and C i =1 ,..., and c are integration constants. We have also defined ν = 12 p ℓ q ρ . (22)Even though we have been able to integrate the Maxwell’s equation in a closed form, for non-vanishingconstants C i , the metric profile F ( u, x, y ) cannot be integrated in a closed manner. To move forward,we therefore set C = C = 0 as well as A ( y ) = A and B ( y ) = B . Under these conditions theelectromagnetic field (21) reduces to a ( u, x, y ) = ω ( u ) (cid:20) x / (cid:0) A x ν + B x − ν (cid:1) − q (cid:21) , (23)and the AdS-wave profile reads, F ( u, x, y ) = ω ( u ) h(cid:16) D e hx (1 − hx ) + D e − hx (1 + hx ) (cid:17) ( D sin ( hy ) + D cos ( hy ))+ 18 (cid:18) E + E x + 1 ℓ (cid:18) A (1 + 2 ν )(3 + 2 ν ) x ν + B (2 ν − ν − x − ν (cid:19)(cid:19)(cid:21) . (24)6ere again, D ,..., , h and E , are integration constants. We can see that the effect of the charge onthe function F ( u, x, y ) induces a quite non-trivial profile.Before finishing this section, it is interesting to notice that the expression for ν in (22) can bewritten in terms of the effective mass of the vector field on the broken phase, m A given in (15), as ν = s − m A m BF , (25)where m BF = − (2 ℓ ) − is the Breitenlohner-Freedman bound for a spin 1 field on AdS . Therefore, asexpected, the x dependence of equation (23) is reminiscent of that for a massive vector on AdS (seee.g. equation (25)-(26) of [52]). Let us consider now consider an extension of the pp -wave ansatz, given by ds = (cid:0) f ( x, y ) + f v + Λ v (cid:1) du − dv du + e βh ( x,y ) (cid:0) dx + dy (cid:1) , (26)where u = t + w , v = t − w is a null coordinate, x, y, w are Cartesian-like coordinates, and f and β arearbitrary constants. This spacetime belongs to the Kundt family since it can be checked that the nullcongruence generated by ∂ v is not covariantly constant, but nevertheless it has vanishing expansion,shear and twist.The techniques developed in [17–22] are particularly suitable to analyze gravitating solitons whosemetrics have the form in Eq. (26) (see the analysis in [23]). Again, the complex scalar field adopts aharmonic dependence in u , and with a constant amplitude, given by Ψ( x µ ) = ν e i Ω( u ) , Ω( u ) = u . (27)We also assume that the Maxwell field has the following form A = A u ( x, y ) du , A u ( x, y ) = a ( x, y ) − q . (28)With this ansatz, the Klein-Gordon (4c) is automatically satisfied, while Maxwell equations reduce to (cid:18) ∂ ∂x + ∂ ∂y − q ν e βh ( x,y ) (cid:19) a ( x, y ) = 0 . (29)The only non-trivial Einstein field equations (4a) for this configuration are (cid:18) ∂ ∂x + ∂ ∂y (cid:19) h ( x, y ) = − β e βh ( x,y ) , (30a) (cid:18) ∂ ∂x + ∂ ∂y (cid:19) f ( x, y ) = − ρ ( x, y ) , (30b)7ith ρ ( x, y ) = 2 ν e βh ( x,y ) a ( x, y ) + 1 q (cid:18) ∂a ( x, y ) ∂x (cid:19) + (cid:18) ∂a ( x, y ) ∂y (cid:19) ! . (31)Equation (30) correspond to a Liouville equation for h ( x, y ) , namely the conformal factor of the two-dimensional space spanned by the coordinates ( x, y ) in the metric (26), and a Poisson equation for thefunction f ( x, y ) . Thus, as it happens in [23] in the case of Einstein-Maxwell coupled to a Non-LinearSigma Model, the present ansatz allows a useful partial decoupling of the field equations. In particular,Eq.(30a) allows a direct integration for h ( x, y ) . This equation actually implies that the induced metricon the u, v = constant surfaces is of constant curvature Λ , as it occurs in vacuum [1]. Then, once h ( x, y ) is known, one can solve the Maxwell equation in Eq.(29) for a ( x, y ) since it reduces to a Schrödinger-like equation in which e βh ( x,y ) plays the role of the potential. Eventually, once h ( x, y ) and a ( x, y ) areboth known, one can solve the remaining equation, Eq.(30b) for f ( x, y ) , since the source term ρ ( x, y ) is explicitly known once h ( x, y ) and a ( x, y ) determined. This hierarchical decoupling is the key of thestrategy developed in [17–22]. Therefore, following this logic, we start considering the general solutionof (30a), given by [53, 54] e βh ( x,y ) = 4Λ g ′ ( z )¯ g ′ (¯ z )( g ( z )¯ g (¯ z ) + 1) , if Λ > , (32a) e βh ( x,y ) = − g ′ ( z )¯ g ′ (¯ z )( g ( z )¯ g (¯ z ) − , if Λ < , (32b)where g ( z ) is any meromorphic function of z = x + iy , with at most simple poles, and dg/dz = 0 forall z in a simply connected domain. On the other hand, since the coordinates x, y are Cartesian, thePoisson equation possesses the particular solution [55] f ( x, y ) = 12 π Z ∞−∞ Z ∞−∞ ρ (¯ x, ¯ y ) ln p ( x − ¯ x ) + ( y − ¯ y ) ! d ¯ xd ¯ y . (33)As mentioned, the Liouville equation on h ( x, y ) , implies that the manifold spanned by the coordi-nates ( x, y ) is of constant curvature Λ . Therefore, locally, there is always a change of coordinates thatallows rewriting the metric (26) as ds = (cid:0) f ( µ, φ ) + f v + Λ v (cid:1) du − dv du + dµ − Λ µ + µ dφ . (34)In these coordinates, the equation for the electromagnetic field a ( r, θ ) reads d a ( µ, φ ) dµ + (1 − µ ) µ (1 − Λ µ ) da ( µ, φ ) dµ − q ν a ( µ, φ )(1 − Λ µ ) + 1 µ (1 − Λ µ ) d a ( µ, φ ) dφ = 0 . (35)The general solution to this equation is a ( µ, φ ) = X m a m sin ( mφ + δ m ) G m ( µ ) , (36)8here the function G m ( µ ) can be integrated in terms of Legendre functions. Here a m and δ m areintegration constants.For the Λ < case, setting Λ = − , the radial coordinate µ goes from [0 , ∞ [ , and the solution for G m ( µ ) which is non-divergent as µ → ∞ reads G Λ < m ( µ ) = µ − (1+2 ν ) 2 F (cid:18) − m ν , m ν , ν, − µ − (cid:19) , (37)where F stands for the Gauss hypergeometric function and ν was defined in (22). In this case thetwo-dimensional surfaces at u, v = constant are hyperbolic spaces with origin at µ = 0 . Even thoughthe behavior at µ → ∞ is regular, these solutions have a singular behavior near the center µ = 0 ofthe hyperbolic space, since near such point, one can see that G Λ < m ( µ ) = Aµ m (1 + O ( µ )) + Bµ − m (1 + O ( µ )) , (38)and one can see that both constant A, B are always non-vanishing. In spite of this behavior, one cancheck that the curvature invariants
R, R αβγδ R αβγδ , R γδαβ R τσγδ R αβτσ are actually constant, thereforethere is no singular backreaction on the geometry. Interestingly, the equation for G m in this case canbe written as a Schrödinger-like equation of the form − d G m ( s ) ds + 2 q ν sinh ( s ) G m ( s ) = − m G m ( s ) , (39)where we have introduced the inversion µ = (sinh( s )) − which maps the range µ ∈ (0 , ∞ ) to s ∈ ( ∞ , .This is a Schröedinger-like equation in a generalized Pöschl-Teller potential, which belongs to a classof exactly solvable, shape invariant potentials [58, 59]. The potential being positive, clearly impliesthat there cannot be solutions that are regular at both boundaries of the domain s ∈ ( ∞ , , whichis consistent with the asymptotic expansion of (37) around µ = 0 presented in (38). Nevertheless, asalso mentioned above, the backreaction on the geometry of this Maxwell field is regular.When Λ > , the range of the µ -coordinate in (34) is µ ∈ ] − , . Setting
Λ = 1 in this case, anddefining µ = sin( θ ) , leads to the following solution G Λ > m = sin | m | ( θ ) F (cid:18) | m | − q − q ν + 14 , | m | q − q ν + 14 , | m | , sin ( θ ) (cid:19) , (40)which is regular at the poles located at θ = 0 and θ = π .Finally, it is also instructive to see explicitly how these cases emerge from a suitable choice ofthe arbitrary function g ( z ) of the general solution of the Liouville equation in (32) and (32a). Forconcreteness, let us focus on the case with negative cosmological constant, normalized as Λ = − ,namely the case corresponding to Eq. (32a). Choosing g ( z ) = z and ¯ g (¯ z ) = ¯ z in (32a) leads to thefollowing metric for the Kundt spaces 9 s = ( f ( x, y ) + f v + Λ v ) du − dv du + 4 ( dx + dy )(1 − ( x + y )) , (41)which after the change of coordinates x = µ − (cid:16)p µ − (cid:17) cos( φ ) , y = µ − (cid:16)p µ − (cid:17) sin( φ ) , (42)leads to the metric ds = ( f ( x, y ) + f v + Λ v ) du − dv du + dµ µ + µ dφ , (43)that we have used in (34) and (35). We have constructed three new families of analytic solutions of the gravitating Abelian-Higgs model,characterized by a non-vanishing superconducting current. The first two families of solutions corre-spond to exact gravitational waves: pp and (A)dS waves. In these families the null vector characterizingboth the pp -wave and the (A)dS-wave is aligned with the superconducting current. Then, we havestudied a class of solutions that belong to the family of Kundt spaces, and as in vacuum, the two-dimensional geometry of the surfaces orthogonal to the superconducting currents is determined bythe two-dimensional Liouville equation. Such surfaces can have either positive or negative Gaussiancurvature depending on the sign of the cosmological constant. This sector possesses a remarkableproperty: the arbitrary analytic function characterizing the solution of the two-dimensional Liouvilleequation (which determines the geometry of two-dimensional surfaces transverse to the superconduct-ing current) can be chosen in such a way that the corresponding Maxwell equations reduce consistentlyto a Schrödinger-like equation in a generalized Pöschl-Teller potential. Requiring suitable boundaryconditions for the Maxwell field within this sector, for a negative cosmological constant, the com-bined effects of the gravitational and scalar interactions can confine the electromagnetic field withina bounded region of the surfaces transverse to the current itself. This result opens the interestingpossibility to analyze the properties of test electromagnetic fields propagating within these familiesof analytic solutions of the Abelian-Higgs model using the well known properties of the Pöschl-Tellerpotential [59]. We hope to come back on this feature in the future.
Acknowledgements
We thank Eloy Ay´on-Beato and Francisco Correa for enlightening comments on related topics. F. C.,A. C. and J. O. have been funded by Fondecyt Grants 1200022, 1210500 and 1181047, respectively. DHis partially founded by ANID grant It would be nice constructing an exhaustive classification of Kundt solutions in this model, along the lines of [60].
10y the Chilean Government through the Centers of Excellence Base Financing Program of ANID. Thiswork is also partially funded by Proyecto de Cooperación Internacional 2019/13231-7 FAPESP/ANID.
References [1] H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, “Exact solutionsof Einstein’s field equations,” doi:10.1017/CBO9780511535185; J. B. Griffiths and J. Podolsky,“Exact Space-Times in Einstein’s General Relativity,” doi:10.1017/CBO9780511635397[2] N. Manton, P. Sutcliffe,
Topological Solitons
Cambridge University Press, Cambridge, 2007.[3] A. Vilenkin, E.P.S Shellard,
Cosmic Strings and Other Cosmological Defects , Cambridge Univer-sity Press (1994).[4] A. A. Abrikosov,
Journal of Physics and Chemistry of Solids , 199 (1957).[5] N. Nielsen and P. Olesen, Nucl. Phys.
B61 , 45 (1973).[6] E. Ayon-Beato, Phys. Rev. D , 104004 (2000) doi:10.1103/PhysRevD.62.104004[arXiv:gr-qc/9611069 [gr-qc]].[7] A. Achucarro, R. Gregory and K. Kuijken, Phys. Rev. D , 5729-5742 (1995)doi:10.1103/PhysRevD.52.5729 [arXiv:gr-qc/9505039 [gr-qc]].[8] A. M. Ghezelbash and R. B. Mann, Phys. Rev. D , 124022 (2002)doi:10.1103/PhysRevD.65.124022 [arXiv:hep-th/0110001 [hep-th]].[9] M. H. Dehghani and T. Jalali, Phys. Rev. D , 124014 (2002) doi:10.1103/PhysRevD.66.124014[arXiv:hep-th/0209124 [hep-th]].[10] S. S. Gubser, Phys. Rev. D , 065034 (2008) doi:10.1103/PhysRevD.78.065034 [arXiv:0801.2977[hep-th]].[11] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Phys. Rev. Lett. , 031601 (2008)doi:10.1103/PhysRevLett.101.031601 [arXiv:0803.3295 [hep-th]].[12] G. T. Horowitz and M. M. Roberts, Phys. Rev. D , 126008 (2008)doi:10.1103/PhysRevD.78.126008 [arXiv:0810.1077 [hep-th]].[13] Y. Brihaye and B. Hartmann, JHEP , 002 (2010) doi:10.1007/JHEP09(2010)002[arXiv:1006.1562 [hep-th]].[14] D. Arean, P. Basu and C. Krishnan, JHEP , 006 (2010) doi:10.1007/JHEP10(2010)006[arXiv:1006.5165 [hep-th]]. 1115] P. Breitenlohner and D. Z. Freedman, “Positive Energy in anti-De Sitter Backgrounds and GaugedExtended Supergravity,” Phys. Lett. B , 197-201 (1982) doi:10.1016/0370-2693(82)90643-81042 citations counted in INSPIRE as of 01[16] Y. Brihaye and B. Hartmann, Phys. Rev. D , 126008 (2011) doi:10.1103/PhysRevD.83.126008[arXiv:1101.5708 [hep-th]].[17] F. Canfora, Eur. Phys. J . C78 , 929 (2018).[18] L. Aviles, F. Canfora, N. Dimakis, and D. Hidalgo,
Phys. Rev.
D96 , 125005 (2017).[19] F. Canfora, M. Lagos, S. H. Oh, J. Oliva and A. Vera,
Phys. Rev.
D98 , 085003 (2018).[20] F. Canfora, N. Dimakis, and A. Paliathanasis,
Eur.Phys.J.
C79 , 139 (2019).[21] F. Canfora, S. H. Oh, and A. Vera,
Eur. Phys. J.
C79 , 485 (2019).[22] F. Canfora, M. Lagos and A. Vera, Eur. Phys. J. C , no. 8, 697 (2020).[23] F. Canfora, A. Giacomini, M. Lagos, S. H. Oh, A. Vera, Eur.Phys.J.C 81 (2021) 1, 55.[24] E. Witten, Nucl. Phys.
B249
557 (1985).[25] A. Hanany, and D. Tong,
JHEP , 037 (2003);
JHEP , 066 (2004).[26] R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi, and A. Yung,
Nucl. Phys.
B673 , 187 (2003).[27] M. Shifman, and A. Yung,
Phys. Rev.
D70 , 045004 (2004).[28] A. Gorsky, M. Shifman, and A. Yung,
Phys. Rev.
D71 , 045010 (2005).[29] M. Eto, M. Nitta, and N. Yamamoto,
Phys. Rev. Lett. , 161601 (2010).[30] A. Gorsky, M. Shifman, and A. Yung,
Phys. Rev.
D83 , 085027 (2011).[31] R. L. Davis, and E. P. S. Shellard,
Nucl. Phys.
B323 , 209 (1989).[32] R. H. Brandenberger, B. Carter, A. C. Davis, and M. Trodden,
Phys. Rev.
D54 , 6059 (1996).[33] A. C. Davis, and W. B. Perkins,
Phys. Lett.
B393 , 46 (1997).[34] L. Masperi, and M. Orsaria,
Int. J. Mod. Phys.
A14
Astropart. Phys. , 173 (1998).[36] E. Radu and M. S. Volkov, Phys. Rept. , 101 (2008).[37] Y. Lemperiere, and E. P. S. Shellard,
Phys. Rev. Lett. , 141601 (2003).[38] Y. Lemperiere, and E. P. S. Shellard, Nucl. Phys.
B649 , 511 (2003).1239] T. Vachaspati,
Phys.Rev.Lett. , 141301 (2008).[40] J. Ye, K. Wang, and Y.-F. Cai,
Eur.Phys.J.
C77 , 720 (2017).[41] A. Gruzinov, and A. Vilenkin,
JCAP , 029 (2017).[42] I. Yu. Rybak, A. Avgoustidis, and C.J.A.P. Martins,
Phys.Rev.
D96 , 103535 (2017); Erratum:
Phys.Rev.
D100 , no.4, 049901 (2019).[43] B. Hartmann, F. Michel, and P. Peter,
Phys.Rev.
D96
JCAP , 040 (2014).[45] K. Miyamoto, and K. Nakayama,
JCAP , 012 (2013).[46] B. Hartmann, and F. Michel,
Phys.Rev.
D86 , 105026 (2012).[47] Y.-F. Cai, E. Sabancilar, D. A. Steer, and T. Vachaspati,
Phys.Rev.
D86 , 043521 (2012).[48] E. Trojan, and G. V. Vlasov,
Phys.Rev.
D85 , 107303 (2012).[49] E. Ayon-Beato and M. Hassaine, Phys. Rev. D , 064025 (2007) doi:10.1103/PhysRevD.75.064025[arXiv:hep-th/0612068 [hep-th]].[50] A. Cisterna, M. Hassaine, J. Oliva and M. Rinaldi, Phys. Rev. D , no.10, 104039 (2016)doi:10.1103/PhysRevD.94.104039 [arXiv:1609.03430 [gr-qc]].[51] D. E. Berenstein, J. M. Maldacena and H. S. Nastase, JHEP , 013 (2002) doi:10.1088/1126-6708/2002/04/013 [arXiv:hep-th/0202021 [hep-th]].[52] W. S. l’Yi, [arXiv:hep-th/9808051 [hep-th]].[53] D. G. Crowdy, General solutions to the 2D Liouville equation , Internat. J. Engrg. Sci. 35 (1997)141-149.[54] J. Liouville, Sur l’equation aux differences partielles d log λdudv ± λ a = 0 , J. Math. Pure et Appl., 1reSerie (1853) 71-72.[55] A. D. Polyanin and V. E. Nasaikinskii, Handbook of linear partial differential equations for engi-neers and scientists (second edition) . CRC Press, Taylor & Francis Group. 2015[56] A. O. Barut, A. Inomata and R. Wilson, J. Phys. A , 4083 (1987) doi:10.1088/0305-4470/20/13/017[57] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, JHEP , 015 (2008) doi:10.1088/1126-6708/2008/12/015 [arXiv:0810.1563 [hep-th]].[58] S. Flügge, Practical Quantum Mechanics , Springer-Verlag, Berlin, 1971, Vol. 1, p. 94.1359] F. Cooper, A. Khare and U. Sukhatme, “Supersymmetry and quantum mechanics,” Phys. Rept. , 267-385 (1995) doi:10.1016/0370-1573(94)00080-M [arXiv:hep-th/9405029 [hep-th]].[60] M. Ortaggio, Class. Quant. Grav.35