Equivalence of two quantization procedures for JT gravity with positive cosmological constant

Establish whether the two proposed quantization procedures for the classical phase space of Jackiw–Teitelboim gravity with positive cosmological constant are equivalent: (1) quantization by defining the Hilbert space as square-integrable functions on the fully gauge-fixed quotient space (G \ 1)/{~Ad, ~Inv} with wavefunctions vanishing at the identity class, and (2) quantization by starting from L2(\widetilde{SO^+}(1,2)) and then imposing invariance under the codifferential actions Ad* and Inv* together with the constraint P q = P to obtain the physical Hilbert space.

Background

After constructing the classical phase space of JT gravity with positive cosmological constant, the authors propose two natural routes to quantization. The first uses a fully gauge-fixed description in terms of a cotangent bundle over the quotient space of identifications, suggesting a direct L2 quantization on that singular space with a vanishing condition at the identity class. The second starts from the smooth universal cover group and imposes constraints (Ad* and Inv* invariance and Pq = P) to project to the physical Hilbert space.

They express the expectation that both routes should agree but indicate that they have not yet performed a systematic comparison, leaving open the task of proving or disproving their equivalence and clarifying any technical subtleties arising from the non-Hausdorff and singular structure of the quotient.