  Pure theories of AdS$_3$ quantum gravity are conjectured to be dual to CFTs
with sparse spectra of light primary operators. The sparsest possible spectrum
consistent with modular invariance includes only black hole states above the
vacuum. Witten conjectured the existence of a family of extremal CFTs, which
realize this spectrum for all admissible values of the central charge. We
consider the quantum corrections to the classical spectrum, and propose a
specific modification of Witten's conjecture which takes into account the
existence of "small" black hole states. These have zero classical horizon area,
with a calculable entropy attributed solely to loop effects. Our conjecture
passes various consistency checks, especially when generalized to include
theories with supersymmetry. In theories with $\mathcal{N}=2$ supersymmetry,
this "near-extremal CFT" proposal precisely evades the no-go results of
Gaberdiel et al.

High Energy Physics - Theory;General Relativity and Quantum Cosmology