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Description
The sine-Liouville gravity, or the sine-Gordon model coupled to 2D gravity, is discretised as a dilute vertex model on random triangulations, which in turn has a dual description as a large N matrix model referred here as vertex matrix model (VMM). It is shown that in the scaling limit, the spectral curve of the vertex matrix model is analytically connected to the spectral curve of Matrix Quantum Mechanics (MQM), which is known to give a non-perturbative formulation of sine-Liouville gravity. The spectral curve has three critical points, one described by pure gravity and the other two described by $c=1$ gaussian matter fields compactified on circles with two different radii. The flow connecting the two $c=1$ critical points is the gravitational analogue of the massless flow in the imaginary coupled sine-Gordon theory. Unlike MQM, the spectral curve of the VMM has a neat interpretation in terms of boundary observables. The disk partition function and the bulk one-point function for fixed boundary length are not FZZT type and are expressed in terms of certain generalisation of the K-Bessel functions.
