Random Euclidean CFT$_2$ from Triangulated dS$_3$ Gravity

Abstract

We propose a random Euclidean CFT description of three-dimensional de Sitter gravity, with fixed central charge and primary spectrum but fluctuating operator product coefficients. Our construction starts from compact regions assembled from generalized hyperbolic tetrahedra. We analytically continue the gravitational amplitudes computed from these triangulations to de Sitter parameters and use them to infer OPE correlations. We recover matching normalized amplitudes from a complex Liouville cosmological wavefunction. For the tetrahedral $K_4$ filling, the non-Gaussian OPE cumulant carries an oscillatory hyperbolic-volume phase. We reconstruct the associated Lorentzian de Sitter spacetime and show that angle derivatives of this phase yield the rapidities of boost holonomies around its initial singularity. Its spacelike future boundary carries a positive-definite conformal metric, while the wavefunction obeys Euclidean conformal Ward identities. The physical CFT spectrum is organized by integer spin, and the scalar restriction of the two-chiral fusion coefficient yields the Virasoro $6j$ square. We construct a positive finite OPE ensemble that reproduces the theta normalization and Virasoro tetrahedral cumulant while satisfying the retained sphere-crossing equations.