Dynamical sectors for a spinning particle in ${\mathrm{AdS}}_3$

We consider the dynamics of a particle of mass $M$ and spin $J$ in ${\mathrm{AdS}}_3$. The study reveals the presence of different dynamical sectors depending on the relative values of $M$, $J$ and the ${\mathrm{AdS}}_3$ 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 ${\mathrm{AdS}}_3$. For the critical case $M^2{R}^2=J^2$ there exist extra gauge transformations which further reduce the physical degrees of freedom, and the motion corresponds to the geodesics of ${\mathrm{AdS}}_2$. This result should be useful in the holographic interpretation of the entanglement entropy for two-dimensional conformal field theories with gravitational anomalies.

Figure 1

$J$-$M$ plot for locally ${\mathrm{AdS}}_3$ geometries: Physical black holes ($MR\ge |J|$), point particles ($MR\le -|J|$) and unphysical states ($|M|R\le |J|$). The critical cases $|M|R=\pm |J|$ correspond to extremal configurations.