Liouville description of conical defects in dS₄, Gibbons-Hawking entropy as modular entropy, and dS₃ holography

We model the back-reaction of a static observer in four-dimensional de Sitter spacetime by means of a singular ℤ_q quotient. The set of fixed points of the ℤ_q action consists of a pair of codimension two minimal surfaces given by 2-spheres in the Euclidean geometry. The introduction of an orbifold parameter q > 1 permits the construction of an effective action for the bulk gravity theory with support on each of these minimal surfaces. The effective action corresponds to that of Liouville field theory on a 2-sphere with a finite vacuum expectation value of the Liouville field. The intrinsic Liouville theory description yields a thermal Cardy entropy that we reintrepret as a modular free energy at temperature T = q⁻¹, whereupon the Gibbons-Hawking entropy arises as the corresponding modular entropy. We further observe that in the limit q → ∞ the four-dimensional geometry reduces to that of global dS₃ spacetime, where the two original minimal surfaces can be mapped to the future and past infinities of dS₃ by means of a double Wick rotation. In this limit, the Liouville theories on the minimal surfaces become boundary theories at zero temperature whose total central charge equals that computed using the dS₃/CFT₂ correspondence.