Circular strings, magnons, plane waves and local quenches in BTZ

TL;DR

This work constructs a precise map between time-like geodesics at the origin of AdS$_3$ and in-falling geodesics in BTZ, enabling BTZ realizations of circular strings, giant magnons, and plane waves with dispersion relations identical to their AdS$_3$ counterparts. By tracking $SL(2,\mathbb{R})$ charges and Virasoro constraints under the AdS$_3$–BTZ map, the authors show the BTZ solutions obey the same dispersion relations and plane-wave spectra, reflecting the local AdS$_3$ structure of BTZ. The bulk solutions are given holographic duals as local quenches in the thermal CFT, carrying energy density, $R$-charges, and a marginal operator vev; the inclusion of transverse velocity yields asymmetric left/right pulses, enriching the local-quench phenomenology. These results strengthen the AdS$_3$/CFT$_2$ intuition for black-hole spacetimes, suggesting broad applicability to higher-dimensional hyperbolic black holes and informing entanglement dynamics in quenched thermal systems.

Abstract

We show that string theory on the geometry $BTZ\times S^3\times M$ supported with either Neveu-Schwarz flux or Ramond flux admits states which obey identical dispersion relations to those of classical solutions like circular strings, giant magnons, or plane wave excitations in the geometry $ AdS_3 \times S^3 \times M$. Here, $M$ can be $T^4$, $K3$, or $S^3\times S^1$. This is made possible by the map, which takes the particle at the origin of $AdS_3$ with angular momentum along one of the angles of $S_3$ to a particle falling into the BTZ horizon. We use this map to construct circular strings, magnons, as well as plane waves in the BTZ geometry. We show that the $SL(2, R)$ charges of these states on $AdS_3$ and that of the corresponding states in the BTZ geometry are related by a boost. The dual description of these states in the BTZ geometry are local quench in the thermal CFT. These quenches carry energy density, $R$-charges, non-trivial expectation value of the marginal operator dual to the dilaton and move on the light cone in CFT. In general, the left and the right moving quenches are not symmetric.

Figures (4)

• Figure 1: Expectation value of the energy density of the local quench. Panel (a) shows $\langle T_{\tau\tau} \rangle - \frac{c\pi^2}{3\beta^2}$ at $\tau = 0$, while panel (b) shows it at $\tau = 3$. Both plots are generated for the parameter values $\Delta = 2$, $\eta_1 - \eta_2 = 0.6$, and $\eta_1 + \eta_2 = 1$ with $\frac{2\pi }{\beta} =1$. The profile initially consists of a single pulse, which subsequently splits into two pulses propagating at the speed of light. The height and width of the pulses are controlled by the parameters $\eta_1$, $\eta_2$, and the conformal dimension $\Delta$ of the primary operator. The pulses become symmetric when $\eta_2 =0$.
• Figure 2: Profile of the expectation value of the charge at different times. Panel (a) shows $\langle j^\tau \rangle$ at $\tau = 0$, while panel (b) shows $\langle j^\tau \rangle$ at $\tau = 3$. Both plots are generated for $\Delta = 2$, $\eta_1 - \eta_2 = 1$, and $\eta_1 + \eta_2 = 0.6$ with $\frac{2\pi }{\beta} =1$. Here, we have interchanged the value of the parameters so that the pulse at $\tau =0$ is mostly positive. The qualitative behaviour of the profile remains the same as in the case of the energy density. The height and width of the pulses are determined by the parameters $\eta_1$, $\eta_2$, and the conformal dimension $\Delta$ of the primary operator.
• Figure 3: Profile of the expectation value of the current at different times. Panel (a) shows $\langle j^x \rangle$ at $\tau = 0$, while panel (b) shows $\langle j^x \rangle$ at $\tau = 5$ with $\frac{2\pi }{\beta} =1$. Both plots are generated for $\Delta = 2$, $\eta_1 - \eta_2 = 0.6$, and $\eta_1 + \eta_2 = 1$. The qualitative behaviour of the profile remains the same as in the previous cases.
• Figure 4: Expectation value of the negative of scalar operator $\Psi$. Panel (a) shows $\langle \Psi \rangle$ at $\tau = 0$, while panel (b) shows $\langle \Psi \rangle$ at $\tau = 2$. Both plots are generated for $\Delta = 2$ , $\eta_1 + \eta_2 = 2$, and $\eta_1 - \eta_2 = 0.6$ . Note the qualitative contrast between these profiles and those shown in the previous figures. The profile initially consists of a single pulse at all times, though its amplitude decreases and the peak shifts either to the left or to the right depending on the parameter values. For the parameters chosen here, the peak shifts to the right.