Spinning $\sigma$-model solitons in $2+1$ Anti-de Sitter space

We obtain numerical solutions for rotating topological solitons of the nonlinear $\sigma$-model in three-dimensional Anti-de Sitter space. Two types of solutions, $i)$ and $ii)$, are found. The $\sigma$-model fields are everywhere well defined for both types of solutions, but they differ in their space-time domains. Any time slice of the space-time for the type $i)$ solution has a causal singularity, despite the fact that all scalars constructed the curvature tensor are bounded functions. No evidence of a horizon is seen for any of the solutions, and therefore the type $i)$ solutions have naked singularities. On the other hand, the space-time domain, along with the fields, for the type $ii)$ solutions are singularity free. Multiple families of solutions exhibiting bifurcation phenomena are found for this case.


Introduction
Asymptotically AdS solutions are of current interest due to their application to holography and their possible indication of phase transitions in the boundary field theory. [1] Examples of such solutions are AdS black holes, [2] AdS solitons, [3] and their hairy extensions. [4], [5], [6], [7] Here we show the existence of asymptotically AdS 3 σ-model solitons. Their stability requires the fields to be rotating. Nonrotating asymptotically AdS 3 σ-model solitons were previously shown not to exist. [8] This also is evident from a simple scaling argument. While the σ-model Lagrangian is scale invariant in two spatial dimensions for static field configurations, this is no longer the case in a background anti-de Sitter space. Rather, there is a contribution which scales like r 2 , leading to an attractive force in addition to the gravitational attraction. The absence of any stabilizing forces, is thus consistent with the nonexistence of static solutions. The above arguments do not apply for rotating field configurations. We show that, as a result, there exist rotating topological solitons which approach AdS 3 in the large distance limit. Asymptotically flat self-gravitating solitons in the 2 + 1 dimensional nonlinear σ-models have been known to exist for a long time. [9]. Analogous self-gravitating solutions, or skyrmions, in 3+1 dimensions are well known. [10] Spinning solutions have also been considered. [11] Solutions with large winding number (corresponding to baryon number) have been proposed to model dense stars. [12], [13], [14] Singularities and horizons can arise for the latter solutions in space-times with various cosmological constants. Such solutions are hairy black holes, and they have been extensively studied. [10], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28] It is of interest to know if the 2 + 1 dimensional nonlinear σ−model also admits solutions with horizons at finite distances.
Here we examine the standard nonlinear σ−model coupled to gravity with a negative cosmological constant. Our ansatz for σ−model fields in 2 + 1 dimensions is suitable for the construction of solitons with arbitrary winding number. Using numerical methods we obtain two types of rotating soliton solutions with integer winding number. They are due to the existence of two types of space-time metrics near the origin. From either space-time metric one gets that all scalars constructed from the curvature tensor are bounded at the origin. Nevertheless, the origin is a casual singularity for one case, which we denote by i) and not the other, which we denote by ii). The singularity for i) closely resembles that of a BTZ black hole. Here it is a naked singularity because the solutions have no horizons. ‡ There are no space-time singularities (or horizons) for solutions ii), and therefore they are topological solitons. The σ−model fields are everywhere well defined for solitons i) even though the domain has a singularity, i.e., the fields have a well-defined limit at the casual singularity. Thus the solitons i) are also restricted to distinct topological sectors. From their asymptotic form at spatial infinity, the solutions can be labeled by the same parameters, namely mass and angular momentum, as those of a BTZ black hole, in addition to parameters associated with the matter content. An alternative mass and angular momentum can be assigned to the solitons using collective coordinate techniques. Collective coordinate quantization leads to the usual spectrum for a rigid rotor in two spatial dimensions.
We denote the nonlinear σ−model fields by Φ a , a = 1, 2, 3, constrained on S 2 , Φ a Φ a = 1. The action for Φ a coupled to 2 + 1 gravity is where G is the three-dimensional version Newton's constant (here in dimensionless units), Λ is the cosmological constant and λ is a Lagrange multiplier. S GHY is the Gibbons-Hawking-York term [29] on the boundary at spatial infinity r → ∞ h is the determinant of the induced metric on the boundary, and K is the trace of the extrinsic curvature, , wheren µ is the unit vector normal to the boundary. S AdS is the infinite AdS vacuum action, which we subtract off in order for the gravity contribution to the action to be finite. Φ a → constant in order for the matter contribution to the action to be finite. Therefore just as in flat space the domain for the nonlinear-sigma model on any time-slice is S 2 , and topologically distinct field configurations result. We demand that Φ a has a unique limit everywhere on S 2 , including at the point associated with the origin, which may or may not be a causal singularity. We label the topological sectors by the winding number where the integral is on any time-slice and n is normalized to be an integer. ǫ abc and ǫ ij denote totally antisymmetric tensors, and i, j, .. = 1, 2 are spatial indices.
In section two we write down the ansatz for the metric tensor and Φ a and give asymptotic solutions near spatial infinity and the origin. Some numerical solutions are presented in section three. Collective coordinate quantization is shown in section four. The question of the existence of black hole solutions with nonlinearσ-model hair is examined in section five, while some brief concluding remarks are given in section six.

Asymptotic solutions
We parametrize the two-dimensional space by polar coordinates (r, φ), and the time by t. Our ansatz for the metric tensor is expressed in terms of three radial functions A, B and Ω, while the σ-model fields Φ a are written in terms of one radial function χ and a fixed angular velocity ω, The functions A, B and χ are dimensionless, while Ω and ω have units of inverse-time. Without any loss of generality we can set χ(∞) = 0. Then for fields in the n th topological sector, χ(0) = nπ. Upon substituting (4) and (5) into the action (including the Gibbons-Hawking-York term) we get where κ = 8πG, and we set Λ = − 1 ℓ 2 . It is convenient to introduce the dimensionless radial variable x = r/ℓ. Then whereΩ = ℓ Ω andω = ℓ ω, and the prime denotes a derivative with respect to x. Upon extremizing the action with respect to variations in A, B,Ω and χ, we get respectively.
Next we write down the solutions to (8) in the asymptotic regions x → ∞ and x → 0.

x → ∞
For the asymptotic region x → ∞ we demand that χ → 0 and that we recover anti-de Sitter space in the limit. The large distance behavior for A, B,Ω and χ can be determined from (8): where M, J,Ω ∞ and ν are constants, the first two being the mass and angular momentum parameters, respectively. The solution is consistent with the standard large distance behavior of the metric tensor for three-dimensional anti-de Sitter space with a localized matter source. [4] The Ricci scalar tends towards the AdS 3 value of −6 in the limit. The constantΩ ∞ can always be eliminated by transforming to the co-rotating frame at spatial infinity, whereω in the ansatz (5) gets replaced byΩ ∞ +ω. Conversely, we can transform to a frame where the σ−model fields are static by replacingΩ ∞ byΩ ∞ +ω.

x → 0
Two possible power series expansions for A, B,Ω and χ exist near the origin. Two of the functions, A andΩ, are singular at the origin for one solution, while all functions have a finite limit for the other. For the former, A,Ω ∼ 1 x 2 , as x → 0. More specifically, near the origin the solution has the form , as well as the remaining components of the metric tensor, are bounded at the origin, All scalars constructed from the curvature tensor are bounded in the x → 0 limit, e.g. the Ricci scalar tends toward Nevertheless, a causal singularity exists at the origin for this solution. The metric tensor near the origin closely resembles that of the BTZ black hole. [2]. For the numerical solutions discussed in section 3 there are no horizons at finite x, and so the singularity is naked for all such solutions.
The power series solution (11) is not valid for J 0 = 0. For this case one has the alternative power series solution where all functions have a finite limit. This solution is parametrized by M 0 ,Ω 0 and χ 1 . The invariant length near the origin takes the form For a Lorentzian space-time near the origin we need that M 0 < 0. WhenΩ 0 = 0, any t−slice approaches flat Euclidean space as x → 0. WhenΩ 0 = 0, the space-time near the origin is rotating. In either case the space-time is singularity free.

Numerical solutions
We have not found any analytic solutions to (8) away from the asymptotic regions and therefore rely on numerical methods. We numerically integrate (8) subject to the asymptotic expressions (9) near the AdS 3 boundary to obtain A, B,Ω and χ at finite x. For topological solitons χ(0) must be an integer multiple of π. The topological solitons solutions can be parametrized by the constants J, M , ν andΩ ∞ appearing in (9). One strategy for obtaining solutions is to first fix three of the parameters (along with κ andω), and then apply shooting methods to tune the remaining one such that χ → nπ as x → 0, where the winding number n is equal to a nonzero integer. Near the origin the solutions must satisfy either (10) or (12), corresponding to type i) or ii) solutions, respectively. The parameters appearing in (10) or (12) can then be determined numerically from M, J, ν andΩ ∞ . Conversely, given J 0 , M 0 , B 0 ,Ω 0 and χ 2 of the expression (10) or M 0 ,Ω 0 and χ 1 of (12) (and the winding number n) we can numerically determine the parameters M, J, ν andΩ ∞ describing the large distance behavior.

Type i) solutions
Numerical solutions satisfying (10)  On the other hand, we find no solutions when bothω and J vanish, which is consistent with the no-go result in [8]. In addition, we find novel solutions where both the mass and angular momentum parameters vanish, M = J = 0, and one with M = −1, J = 0. If one takes these as parameters for the BTZ black hole, the former would correspond to a zero mass black hole and the latter would correspond to anti-de Sitter space. An example of a soliton with M = J = 0 occurs for κ =ω = 1, ν ≈ .77, and a soliton with M = −1, J = 0 occurs for κ =ω = 1, ν ≈ 1.061. As required, A approaches x 2 as x → ∞ for all of the above examples, while it does not pass through zero for any x. The latter behavior indicates that there are no horizons. A and Ω go as 1/x 2 near the origin.

Type ii) solutions
Numerical solutions ii) can also be found for which all of the functions are bounded, including at the origin where they approach (12). These solutions cover a smaller region in parameter space than i) since they correspond to the limiting case of J 0 → 0 in (11). By integrating from x → ∞ using (9) and from x → 0 using (12) we can match the four functions A, B, Ω and χ, along with their derivatives, at finite x to arbitrary accuracy. An example of such a solution is shown in figure 6(a)     π to 0 as x goes from 0 to ∞. On the other hand, solutions can also be found where χ(x) has multiple nodes, as is illustrated in figure 6(b) for solutions with zero, one and two nodes. The depicted n = 1 solitions have common values for κ, ω and M 0 , and differing values of the parameters χ 1 andΩ 0 .
The space of nonsingular solutions can be parametrized by ω, κ, M 0 ,Ω 0 and χ 1 . We span the parameter space for the zero, one and two node solutions in figures 7, 8(a) and 8(b). Keeping κ and ω fixed, we plot −χ 1 versusΩ 0 in figure 7. χ 1 (Ω 0 ) is seen to be multi-valued, with a cusp singularity occuring for the zero node solutions at some minimum value ofΩ 0 . (Analogous behavior has been noted for self-gravitating Skyrmions. [16]) −χ 1 (κ) [with ω and M 0 held fixed] is plotted in figure 8(a) and −χ 1 (−M 0 ) [with ω and κ held fixed] is plotted in figure 8(b). Finite domains are seen for both of these functions, implying upper bounds on the allowed values for κ and −M 0 . We also get no soliton solutions in the limiting cases of κ → 0 and −M 0 → 0. [The latter limit corresponds to the function A vanishing at the origin, indicating a horizon in the zero radius limit. So if solutions existed with −M 0 → 0 they could coincide with the zero radius limit of the horizon of a hairy black hole. Thus the absence of such solutions is consistent with not finding any black hole solutions with σ−model hair, which is what we report in section five.]

Collective coordinate quantization
The collective coordinate quantization of the soliton allows for an alternative definition of the mass and angular momentum of the soliton. We denote them by M and J , respectively. Both can be computed from the action (1) evaluated for the soliton. A Chern-Simons term [30], [31] can be included in the total action, and this will produce a contribution which is linear in the angular velocity, in addition to those   coming from (1). However, such contributions do not affect the energy spectrum, and so we will not consider the Chern-Simons term.
In the collective coordinate approach one replacesω with a dynamical angular velocityψ, with the caveat that its variation is sufficiently small so that it doesn't significantly change the values of the mass M or the moment of inertia I of the soliton. M is defined as theψ-independent contribution to the soliton action, while I/2 is the coefficient of the quadratic contribution inψ. As indicated above, there is also a linear contribution. Thus the soliton action can be written where I, α and M are given by the radial integrals The infinite AdS vacuum action S AdS was subtracted from M. The angular momentum of the soliton is J = Iψ. From the asymptotic behavior (9) as x → ∞ and the behavior (10) or (12)  The Hamiltonian for the system is The angular momentum J is related to the canonical momentum p ψ by J = p ψ −α. Its Poisson bracket with the U (1) phase e iψ is then In passing to the quantum theory the spectrum of the operatorĴ corresponding to J is not unique, the eigenvalues being integers plus an arbitrary constant. [30], [31] It obeys the commutator where e iψ is the operator corresponding to e iψ . The algebra has the Casimir operator exp 2πī hĴ , whose eigenvalues are phases e iφ0 which label different irreducible representations in the quantum theory. The spectrum forĴ is thenh times an integer m plus an arbitrary phase constant,hm + φ0h 2π , and so from (16) the energy eigenvalues are Of course the energy spectrum depends on an additional integer n, the winding number, since I and M do.

The question of hairy BTZ black hole solutions
The functions A(x) and B(x) were positive for all of the numerical solutions obtained previously by integrating either from x → 0 or x → ∞. Thus none of these solutions developed horizons. We can instead assume a priori the existence of at least one horizon. In the case of multiple horizons, let x H > 0 denote the the location outer most one. Then A(x H ) = 0. A consistent solution of (8) near the horizon, x − x H << 1, can be obtained by demanding thatΩ(x H ) = −ω. A power series expansion for the functions A, B,Ω and χ can then be determined from three independent parameters, say B H = B(x H ), χ H = χ(x H ) andΩ 1 =Ω ′ (x H ), as well as x H . Up to first order in x − x H , follows that sin(2χ H ) < 0. Given these inequalities on the horizon parameters, along with the conditions (21), we can then integrate the equations of motion (8) from x H to x → ∞. Upon so doing we were unable to recover the asymptotic solution (9) at x → ∞, and hence we did not find any black hole solutions with nonlinear σ-model hair.

Concluding remarks
We obtained numerical solutions for two types, i) and ii), of rotating self-gravitating topological solitons of the nonlinear σ-model where the space-time approaches AdS 3 in the large distance limit. Upon embedding the solutions in 3 + 1 dimensions, they can be interpreted as cosmic strings. For the type i) solution, any time slice of the space-time domain has a causal singularity, which is analogous to the BTZ black hole singularity. On the other hand, the space-time domain is singularity free for type ii) solutions. χ(x) for such solutions exhibit an arbitrary number of nodes. No evidence of a horizon was seen for any of the solutions. Therefore these solutions are not hairy black holes, and furthermore the type i) solutions have naked singularities.
Among the lines of inquiry that remain to be investigated is the search for black hole solutions with nonlinear σ-model hair, analogous to the known 3 + 1 dimensional black hole solutions with Skyrme hair. This may require the inclusion of higher order derivative terms, analogous to the Skyrme term, in the nonlinear σ-model action. While the solitons obtained here are topologically stable, the question of whether or not they are stable under local fluctuations needs to be determined. Finally, it is worthwhile to understand the role that these new three-dimensional AdS solutions may or may not play for the two-dimensional space-time boundary field theory.