  We present a new holographic duality between q-Schwarzian quantum mechanics
and Liouville gravity. The q-Schwarzian is a one parameter deformation of the
Schwarzian, which is dual to JT gravity and describes the low energy sector of
SYK. We show that the q-Schwarzian in turn is dual to sinh dilaton gravity.
This one parameter deformation of JT gravity can be rewritten as Liouville
gravity. We match the thermodynamics and classical two point function between
q-Schwarzian and Liouville gravity. We further prove the duality on the quantum
level by rewriting sinh dilaton gravity as a topological gauge theory, and
showing that the latter equals the q-Schwarzian. As the q-Schwarzian can be
quantized exactly, this duality can be viewed as an exact solution of sinh
dilaton gravity on the disk topology. For real q, this q-Schwarzian corresponds
to double-scaled SYK and is dual to a sine dilaton gravity.
