Matrix Quantum mechanics and quantum averaging of JT gravity

I will discuss the matrix quantum mechanics with potential corresponding to an arbitrary spectral curve. This can be seen as a generalization of the duality between JT gravity and a particular matrix integral. Using the recently developed techniques of quantum generalised hydrodynamics, the effective theory of the eigenvalue density fluctuations is a simple 2D free-boson BCFT on a curved background. The ensemble average over random matrices then corresponds to a quantum expectation value. Using this formalism we reproduce non-perturbative results for matrix integrals (the ramp and the plateau in the spectral form factor). We also compute the entanglement between eigenvalues, matching a previous result by Hartnoll and Mazenc for the c = 1 matrix model and extending it to the general case. The hydrodynamical theory provides a clear picture of the emergence of spacetime in two dimensional string theory. The talk is based on an upcoming paper with G. di Ubaldo.