  We study the dual description of the self-dual orbifold, a locally AdS_3
spacetime which is a circle fibration over AdS_2 and arises as the near-horizon
limit of the extreme BTZ black hole. The geometry has two boundaries; we argue
that this should correspond to a saddle-point for two copies of a chiral CFT
living on these two boundaries in an entangled state. This picture arises
naturally in the near-horizon limit, but there is a potential inconsistency
with the bulk physics because of causal connections between the boundaries. We
discuss a possible resolution of this puzzle. We also construct geometries
which asymptotically approach the self-dual orbifold on a single boundary.
These geometries (which contain mild singularities) enable us to explore other
states of the dual chiral CFT. One of the geometries corresponds to the ground
state of this CFT and can be obtained as a particular near-horizon limit of the
BTZ M=0 black hole. The self-dual orbifold is a finite temperature version of
this geometry.
