A microscopic Normal Matrix Model for $(A)dS_2$

We describe the duality between the gravitating $c=1$ (compact) Sine-Gordon model and a normal matrix model. From a two-dimensional quantum gravity perspective and due to the periodic nature of the potential, this model admits both anti-de Sitter and de-Sitter saddles, similarly to simpler models of Sine-Dilaton gravity, as well as more complicated interpolating "wineglass wormhole" geometries. From a string theory perspective the Euclidean de-Sitter (genus zero) saddles are related to the presence of a classical entropic contribution associated to the target space geometry. The gravitating Sine-Gordon model corresponds to a well defined CFT by construction and the eigenvalues of the dual normal matrix model are supported in a compact region of the complex plane. The duality with the normal matrix model is operationally defined even for a finite, but sufficiently large matrix size $N$, depending on the precise observable to be determined. We define and study a "microscopic" version of the large-N limit that allows us to recover non-perturbative results for all physical observables.