The Radon transform on the Poincare disc shows that the SYK four-point boundary condition theta = 3 pi / 4 is the unique one compatible with the antipodal identification of kinematic de Sitter space.
The Probe of Curvature in the Lorentzian AdS$_2$/CFT$_1$ Correspondence
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We establish the Lorentzian AdS$_2$/CFT$_1$ correspondence from a reconstruction of all bulk points through the kinematic-space approach. The OPE block is exactly a bulk local operator. We formulate the correspondence between the bulk propagator in the non-interacting scalar field theory and the conformal block in CFT$_1$. When we consider the stress tensor, the variation probes the variation of AdS$_2$ metric. The reparameterization provides the asymptotic boundary of the bulk spacetime as in the derivation of the Schwarzian theory from two-dimensional dilaton gravity theory. Finally, we find the AdS$_2$ Riemann curvature tensor based on the above consistent check.
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Radon Transforms and the SYK model
The Radon transform on the Poincare disc shows that the SYK four-point boundary condition theta = 3 pi / 4 is the unique one compatible with the antipodal identification of kinematic de Sitter space.