What is a chiral 2d CFT? And what does it have to do with extremal black holes?

The near horizon limit of the extremal BTZ black hole is a``self-dual orbifold of AdS_3. This geometry has a null circle on its boundary, and thus the dual field theory is a Discrete Light Cone Quantized (DLCQ) two dimensional CFT. The same geometry can be compactified to two dimensions giving AdS_2 with a constant electric field. The kinematics of the DLCQ show that in a consistent quantum theory of gravity in these backgrounds there can be no dynamics in AdS_2, which is consistent with older ideas about instabilities in this space. We show how the necessary boundary conditions eliminating AdS_2 fluctuations can be implemented, leaving one copy of a Virasoro algebra as the asymptotic symmetry group. Our considerations clarify some aspects of the chiral CFTs appearing in proposed dual descriptions of the near-horizon degrees of freedom of extremal black holes.