Periodic Dilaton Gravity

I will discuss dilaton gravity with a sine potential, where periodic shifts of the dilaton (that leave the potential invariant) are treated as a redundancy. Most things I will say hold true for generic periodic potentials, but the sine potential is special because that theory has a microscopic holographic realization as the double-scaled SYK model. The periodicity of the dilation discretizes the length of Cauchy slices. For closed cauchy slices, this discretization of the physical Hilbert space corresponds with a discretization of the length of the neck of trumpets. The KG inner product gives a positive definite inner product on this Hilbert space, which we use to show that the double trumpet matches the spectral correlation of a one-cut matrix integral. This shows that our theory is a path integral formulation of q-deformed JT gravity. The no-boundary state is not a physical state. Instead, the disk is the sum of physical trumpets with a Gaussian wavefunction, and is therefore normalizable. If one would not treat the dilaton shift symmetry as a redundancy, the theory is ill defined. For instance, even at the semiclassical level the partition function would be divergent.