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Dynamical sectors for a spinning particle in AdS_3

We consider the dynamics of the motion of a particle of mass M and spin J in AdS_3. The study reveals the presence of different dynamical sectors depending on the relative values of M, J and the AdS_3 radius R. For the subcritical M^2 R^2-J^2>0 and supercritical M^2 R^2-J^2<0 cases, it is seen that the equations of motion give the geodesics of AdS_3. For the critical case M^2R^2=J^2 there exist extra gauge transformations which further reduce the physical degrees of freedom, and the motion corresponds to the geodesics of AdS_2. This result should be useful in the holographic interpretation of the entanglement entropy for 2d conformal field theories with gravitational anomalies.

Point particles in spacetime offer a simplified setting to study the interplay between the geometry of spacetime and quantum mechanics. As noted long ago by Deser, Jackiw and 't Hooft, point particles in 2 + 1 gravity are naked conical singularities in which the mass is related to the angular deficit. The energy-momentum tensor has Dirac deltas with support at the position of the particles, which induces delta singularities in the curvature at those points. Infinite curvature concentrated at an isolated point corresponds to a conical singularity produced by the removal of a wedge. These conical defects do not affect the local geometry on open sets that do not include the singularities, but they change the global topology [1].
Point particles in AdS 3 are also related to black holes. Since point particles are conical singularities, they are obtained by identification in AdS 3 by a global spacelike Killing vector with a fixed point in a similar way as the 2+1 black hole is obtained by identifications in the universal covering of AdS 3 [2]. The spacetime geometry of a conical singularity in AdS 3 is identical to the BTZ geometry, where the mass of the black hole is minus the mass of the point particle, and therefore a point particle of mass M in 2+1 can be viewed as a black hole of mass −M [3].
The 2+1 black hole can have angular momentum J and by the same token, a point particle can be endowed with spin. An important quantity that characterizes both black holes and point particle states where R is the AdS radius. Nonextremal black holes and spinning point particles correspond to κ < 0 (subcritical case). The extreme (critical) geometries, κ = 0, have additional special features, like admitting globally defined Killing spinors (supersymmetric BPS states). Figure 1 displays the different sectors of 2+1 BH-particle states in the M -J plane. The AdS 3 geometry, without identifications, is the point J = 0, M R = −1; black holes cover region I (κ ≤ 0, M > 0); point particles are described by III (κ ≤ 0, M < 0). The supercritical regions II (κ > 0) are unphysical states. States below the hyperbola κ = −1, M < 0 correspond to angular excesses rather than defects, and also display delta-like curvature singularities at r = 0. On the other hand, it is also possible to think of a particle as a probe that does not significantly affect the spacetime geometry around it, a localized perturbation with negligible back reaction. The appropriate setting to describe this would be an action for a relativistic point particle whose trajectories, in the absence of other interactions, are geodesics of the spacetime background. In that case M and J are parameters in the action and κ is an intrinsic feature, while for the black hole they are constants of motion that depend on the initial conditions. The geodesic equation is a local statement and is therefore the same for a particle on a patch of AdS 3 or around a 2+1 black hole. The only difference would be in the orbits, since they depend on the global properties of the manifold. Hence, the conserved quantities for the different orbits could be the same, e.g. energy and angular momentum, but their specific values would determine the class of geodesics that the particle traces.
Motion of particles in AdS 3 can also be related to the holographic entanglement entropy. As shown in [4], the entanglement entropy in AdS D is related to minimal (D − 2)-surfaces in AdS D . For the case of AdS 3 , the minimal surface is the length of the geodesics, and these are the trajectories of particles in AdS 3 . The holographic description of the entanglement entropy for conformal field theories in two dimensions with gravitational anomalies [5] has been studied in [6].
Motivated by these observations, we consider the dynamics of a particle of mass M and spin J in AdS 3 . The Lagrangian equations of motion reveals the presence of different dynamical sectors depending on M, J and R. For the subcritical (κ < 0) and supercritical (κ > 0) cases it is seen that the equations of motion give the corresponding geodesics of AdS 3 . For the critical case (κ = 0) there exists an extra gauge transformation which further reduces the physical degrees of freedom. 1 The orbits correspond to geodesics of AdS 2 .
The presence of dynamical sectors appears also in a (2+1)D harmonic oscillator system with exotic Newton-Hooke symmetry. The system displays three different phases depending on the values of the parameters [11]. The reduced phase space description reveals a symplectic structure similar to that of Landau problem in the non-commutative plane [12,13]. There is a close relation between the (2+1)D exotic Newton-Hooke symmetry and the non-commutative Landau problem [14]. 2 The non-relativistic limit of a spinning particle in AdS 3 shows the Newton-Hooke symmetry and gives the (2+1)D exotic harmonic oscillator system.
Since the AdS 3 algebra can be written as a sum of two chiral so(2, 1) factors, we construct a Lagragian for spinning particles in terms of chiral coordinates. The relation between the chiral and non-chiral variables is given in terms of a differential equation that can be solved perturbatively. The equations of motion for the critical case in the chiral formulation describe orbits that are geodesics of AdS 2 .
The subcritical case (κ < 0) includes a special representative, J = 0, while M = 0 is a representative for the supercritical case (κ > 0). In each case we explicitly give the coordinate transformation that takes the Lagrangian into one described by the corresponding representative. We expect the points J = 0 or M = 0 to be described in terms of non-commutative coordinates, in a similar way as for the flat case (see e.g. [16]).
Some of the particle states have a lowest energy (BPS) bound. In particular, the critical sector with κ = 0, with M > 0 saturates a BPS bound and therefore is a candidate to be a supersymmetric configuration, with 1/4 SUSY, if the system is embedded in a supersymmetric model. These particles belong to the BH sector of the J-M plane. The subcritical configurations with M < 0 do not have a lowest energy bound but an upper one, and therefore do not seem to correspond to stable BPS states. As shown in [3], the critical states with M < 0 also admit globally defined Killing spinors. Nevertheless, a complete correspondence with the BH spectrum should not be expected since we assume the particle as a probe that does not modify the spacetime background.
Summing up, we prove the existence of sectors in the dynamics of a spinning particle in AdS 3 . The critical sector has half the physical degrees of freedom due to existence of an extra, kappa-like bosonic gauge symmetry. The presence of sectors is in correspondence with the spectrum of BH. Our results shed light on the holographic interpretation of the entanglement entropy for CFT 2 with gravitational anomalies [6], for which the left c L and right c R central charges are different, and one will have sectors for these class of theories. The critical sector corresponds to chiral CFTs with only left or right moving sector, while the sign of c L c R determines the subcritical (+) and supercritical (−) sectors. Any particle with spin J in a locally AdS 3 metric will exhibit the same kind of dynamical sectors, but they will be absent, for instance, for local dS 3 geometries. The analysis should also be useful in the study of the motion of anyons around a BTZ black hole.
The paper is organized as follows. Section II presents the Lagrangian, discusses the dynamical sectors and the equations of motion. Section III introduces the chiral variables and analyzes the dynamical sectors in this description. The relation between chiral and nonchiral variables is established in Section IV by means of a set of nonlinear differential equations that can be solved perturbatively. Section V presents an explicit biparametric transformation, in terms of the chiral variables, that connects Lagrangians with different M , J parameters in the super-and subcritical sectors. Finally, Section VI uses the chiral form of the Lagrangian to discuss the existence of BPS energy bounds for several values of the parameters.

II. ADS3 ACTION AND EQUATIONS OF MOTION
The action of a massive spinning particle in AdS 3 is constructed from the coordinates of the world line x a (τ ) and a generic Lorentz transformation Φ a b (τ ). To lowest order in derivatives, the action is [16,17] where e a m (x) and ω ab m (x) are the dreibein and spin connection of AdS 3 . The Lagrangian is given by the Maurer-Cartan (MC) form (non-linear realization [18]) for the coset G/H = SO(2, 2)/SO(2). The AdS 3 generators verify the so(2,2) algebra and the stability group H is generated by M 12 . We consider a local parametrization of the coset element The Lorentz transformation U can be expressed as The dreibein and spin connection in this parametrization are obtained from the MC form Ω 0 = −ig −1 0 dg 0 = P a e a + 1 2 M ab ω ab , e 0 = coshx 1 coshx 2 dx 0 , e 1 = coshx 2 dx 1 , e 2 = dx 2 , ω 02 = coshx 1 sinhx 2 dx 0 , ω 12 = sinhx 2 dx 1 , ω 01 = sinhx 1 dx 0 , wherex a = x a /R. The MC form associated to g is Ω = −ig −1 dg = P a L a + 1 2 M ab L ab , where The Lagrangian (1) is constructed from the pullback of the (pseudo) invariant forms L 0 and L 12 as The EL equations of motion can be written as If J 2 = M 2 R 2 these equations become L a = 0 and L 0a = 0. They relate the spin variables v a to the coordinates of the world line by and yield also the geodesic equation, for the metric g mn = e m b η ba e n a . Note that (9) has no contribution from the spin variables v a because the physical states in configuration space are given by x a only and the v a are not independent local degrees of freedom, as can be seen from (8) as well as from the Hamiltonian analysis. In the reduced phase space, the velocityẋ m is proportional to the momentum, and therefore the coordinates do not exhibit zitterbewegung.
In the critical case J 2 = M 2 R 2 , only two of the equations in (7) are independent and there are new gauge symmetries, besides diffeomorphisms, that reduce the number of degrees of freedom from 4 to 2. The sectors are also present in local AdS 3 or warped AdS 3 , but not for dS 3 .
In the critical case (µ − = 0), the equations of motion are L a + = 0 and give the geodesics in AdS 2 , while the X a − 's are gauge degrees of freedom since they are absent from the Lagrangian .
In the non-relativistic limit X a ± → X a ± /ω, µ ± → ω 2 µ ± , the Lagrangian takes the Newton-Hooke form [11] up to a divergent total derivative, and the equations of motion (18) become those of two harmonic oscillators.

IV. NON-CHIRAL TO CHIRAL VARIABLES
The relation between chiral and non-chiral coordinates is established by comparing the corresponding expressions for the coset element in the two parametrizations. Introducing an auxiliary parameter t to rescale v i in the coset expressions (3) and (14), one finds = e ±ij ± 0 X 0 (t) e ±ij ± 1 X ±1 (t) e ±ij ± 2 X ±2 (t) e ij ± 0 α(t) .(19)