  We continue our study of factorizing theories of dilaton gravity,
characterized by a universal bilocal interaction. All such factorizing theories
can be shown to have discrete spectra, distinguished only by their local
dilaton potentials. We show how such theories can be used to construct all
alpha-states in the Hilbert space of baby universes of ordinary JT gravity.
Large classes of these theories with different local potentials are found to be
non-perturbatively equivalent and have identical discrete spectra. This is a
concrete example of how different bulk descriptions can give rise to the same
boundary theory. Such equivalences manifest themselves as null states, which
have to be quotiented out in order to construct a proper baby universe Hilbert
space. Our results also allow us to revisit the mechanism discussed by Coleman,
Giddings, and Strominger, and concretely link ensemble averaging to the
appearance or disappearance of spacetime wormholes. We then investigate JT
gravity deformed only by the universal bilocal interaction. In this theory, the
only terms that do not cancel in a topological expansion are disks, which
capture perturbative fluctuations around a two-dimensional black hole saddle.
We find that this theory of black holes has an evenly spaced spectrum, instead
of a quantum chaotic one. We present a dual quantum mechanical system with
exactly the same discrete spectrum, and propose that this is an example of a
new holographic duality between a two-dimensional theory of quantum gravity and
a conventional quantum mechanics.
