  Being interested in the compatibility of Asymptotic Safety with Hilbert space
positivity (unitarity), we consider a local truncation of the functional RG
flow which describes quantum gravity in $d>2$ dimensions and construct its
limit of exactly two dimensions. We find that in this limit the flow displays a
nontrivial fixed point whose effective average action is a non-local functional
of the metric. Its pure gravity sector is shown to correspond to a unitary
conformal field theory with positive central charge $c=25$. Representing the
fixed point CFT by a Liouville theory in the conformal gauge, we investigate
its general properties and their implications for the Asymptotic Safety
program. In particular, we discuss its field parametrization dependence and
argue that there might exist more than one universality class of metric gravity
theories in two dimensions. Furthermore, studying the gravitational dressing in
2D asymptotically safe gravity coupled to conformal matter we uncover a
mechanism which leads to a complete quenching of the a priori expected
Knizhnik-Polyakov-Zamolodchikov (KPZ) scaling. A possible connection of this
prediction to Monte Carlo results obtained in the discrete approach to 2D
quantum gravity based upon causal dynamical triangulations is mentioned.
Similarities of the fixed point theory to, and differences from, non-critical
string theory are also described. On the technical side, we provide a detailed
analysis of an intriguing connection between the Einstein-Hilbert action in
$d>2$ dimensions and Polyakov's induced gravity action in two dimensions.
