A Killing Vector Treatment of Multiboundary Wormholes

TL;DR

This work develops a global Killing-vector framework to construct multiboundary wormholes in $AdS_3$, deriving explicit Killing vectors for a static three-boundary geometry and showing the $t=0$ slice realizes the full Teichmüller moduli $T(3,0)$. The authors extend the construction to arbitrary $(n,g)$ via a sequence of dilatations, pinching, and foldings, and they incorporate rotation through left-right lifts of the Killing data, identifying a single effective angular velocity $\Omega$. Horizon lengths $L_1,L_2,L_3$ are computed in terms of identification parameters, with $L_3$ shown to be independent of $L_1$ and $L_2$ and to depend on a twist parameter in a controlled way, confirming the three moduli are independent. The framework provides a global, constructive approach to studying multipartite entanglement and holographic complexity in AdS/CFT, with potential extensions to warped $AdS_3$ geometries and broader multiboundary topologies.

Abstract

The two-sided BTZ black hole has been instrumental in elucidating several aspects of AdS/CFT. Similarly, multiboundary wormholes provide a useful and rich arena in which probing questions of quantum gravity can be posed and explored. In this work, we find the explicit forms of the Killing vectors needed to construct three-boundary wormholes, with and without rotation, as quotients of AdS$_3$. We ensure that our method captures the full moduli space of such wormholes and elaborate on the generalization of our procedure to more exotic multiboundary spaces, including higher genus.

Figures (10)

• Figure 1: The Riemann surface obtained by quotienting the upper half-plane by dilatation (\ref{['ibtz']}). The fundamental domain is bounded by the red semicircles, while the fixed point of (\ref{['ibtz']}) is the center of the semicircles. This is the $t = 0$ slice of the two-sided BTZ.
• Figure 2: The three-boundary and one-boundary, one-genus Riemann surfaces as quotients of the two-boundary Riemann surface. The three-boundary surface is obtained by "pinching" one of the boundaries into two, while the one-boundary, one-genus surface is obtained by "folding" one of the boundaries onto the other.
• Figure 3: The action of the general orientation-reversing isometry on two arbitrary semicircular geodesics in $\mathbb{H}$. Also defined are the centers and radii of the semicircles.
• Figure 4: The fundamental domain of the three-boundary Riemann surface. The color-coded dashed lines $L_{1,2,3}$ are the minimal periodic geodesics, whose lengths are the three physical parameters of the system. The variables $\lambda$, $R_0$, $R$, $c_1$, and $c_2$ represent parameters for the picture.
• Figure 5: The Riemann surface obtained by quotienting the upper half-plane by dilatation (\ref{['ibtz']}), with three of the resulting periodic geodesics drawn. By symmetry, we can see that the vertical geodesic is extremal.