On decoding the string from interfaces in 2d conformal field theories

Abstract

General solutions of a gravitational junction between two copies of a three-dimensional Einstein manifold $\mathcal{M}$ correspond to the solutions of the non-linear Nambu-Goto equation for a string in $\mathcal{M}$. We show that, for the junctions in three-dimensional anti-de Sitter spacetimes constituted by tensile strings, which are dual to interfaces between thermal states in conformal field theories, the solutions of the Nambu-Goto equation describing the junction correspond to state-dependent wave-packets, which are perfectly reflected at the interface to future null infinity without shape distortion when incident from past null infinity. These wavepackets are realized by state-dependent half-sided conformal transformations and affect the expectation value of the displacement operator. We further show that the entanglement entropy of an interval straddling the interface deciphers the stringy modes of the dual junction even in the tensionless limit. We also demonstrate that the strong sub-additivity of entanglement entropy is satisfied and is saturated for symmetric intervals generally.

Figures (3)

• Figure 1: (a) A plot of $\mathbb{h}^{-1}(\rho)$ with the parameters in \ref{['Eq:paramchoices']}. The branch point at $\rho=1$ is the dashed red line and the branch point at $\rho=\sqrt{\frac{\kappa-1}{\kappa+1}}=\frac{1}{\sqrt3}$ is the blue dashed line. The function, its first, second and third derivatives are continuous. (b) Penrose diagram representing the perfect reflection in the dual interface CFT in two-dimensional Minkowski space. The solid black lines are the null infinities $\mathcal{I}^\pm$. The solid blue line is the interface. The departure, from the thermal state, of the state on $\mathcal{I}^-$ on the right CFT in the form of a wavepacket is indicated by the magenta function. The state departs from the thermal state for $\tilde{x}^+>-\frac{\ln3}{4\mu}$. The region of $\mathcal{I}^-$ where the state is thermal is shown in red. Similarly, on $\mathcal{I}^+$ we have the non-trivial state (wavepacket deformation) for $\tilde{x}^->-\frac{\ln3}{4\mu}$, and the thermal state (red) otherwise. As shown by the arrows, the non-trivial excitations on $\mathcal{I}^-$ hit the interface, are perfectly reflected, and then travel to $\mathcal{I}^+$. Causality of this process is manifest.
• Figure 2: The four regions $A \cup B$, $A$, $B$ and $A\cap B$ involved in the SSA are shown with (a) and (b) denoting two different configurations. The dotted black lines are the light cone directions. The purple line is the trajectory of the interface.
• Figure 3: The two spacetimes L and R are glued along the worldsheet in blue. $t$ is the boundary time and $z$ is the radial coordinate. A geodesic (solid red) connecting the two end-points (red circles) of the boundary interval is shown. The geodesic intersects the worldsheet at the magenta points. The left and right magenta points have the same worldsheet coordinates $\tau_*, \sigma_*$ but have different images in the left and right spacetimes. The black circle denotes $x=0$, where the worldsheet intersects the boundary.