An effective matrix model for dynamical end of the world branes in Jackiw-Teitelboim gravity

We study Jackiw-Teitelboim gravity with dynamical end of the world branes in asymptotically nearly AdS$_2$ spacetimes. We quantize this theory in Lorentz signature, and compute the Euclidean path integral summing over topologies including dynamical branes. The latter will be seen to exactly match with a modification of the SSS matrix model. The resolution of UV divergences in the gravitational instantons involving the branes will lead us to understand the matrix model interpretation of the Wilsonian effective theory perspective on the gravitational theory. We complete this modified SSS matrix model nonperturbatively by extending the integration contour of eigenvalues into the complex plane. Furthermore, we give a new interpretation of other phases in such matrix models. We derive an effective $W(\Phi)$ dilaton gravity, which exhibits similar physics semiclassically. In the limit of a large number of flavors of branes, the effective extremal entropy $S_{0,\text{eff}}$ has the form of counting the states of these branes.