Lecture 19: Islands on Branes
Islands can be understood most simply in a doubly holographic setup. Here, we consider a CFT in d-dimensions that lives on a half space with a boundary that hosts a (d-1)-dimensional CFT. The entire system has a bulk dual as a theory of gravity in (d+1)-dimensional AdS that is terminated by a brane.The entropy of regions of the nongravitational boundary can be computed by the standard RT prescription except that RT surfaces are allowed to end on the brane. When the dominant RT surface ends on the brane, the entanglement wedge of the boundary region contains a part of the brane, which is just the island.
Such systems can also be used to demonstrate a puzzle and its resolution via islands. Consider two copies of the setup above, placed in a thermofield double state. Consider the time-evolution of the entropy of the union of a region on one boundary and a region on the other boundary. We started to analyze one minimal surface that contributes a growing term to this entropy. If this minimal surface were to dominate forever, it would lead to a puzzle since the entropy would be unbounded.This puzzle will be resolved by the phase-transition between dominant RT surfaces.
