Liouville description of conical defects in dS$_4$, Gibbons-Hawking entropy as modular entropy, and dS$_3$ holography

TL;DR

This work presents a framework to incorporate the back-reaction of a static observer in four-dimensional de Sitter space by modeling the geometry with a Z_q orbifold, producing two antipodal conical defects. Each defect carries a two-dimensional Liouville theory on a surrounding S^2 with a fixed Liouville vev, and the thermal Cardy entropy of these theories coincides with a modular free energy whose modular entropy reproduces the Gibbons–Hawking entropy; the central charge scales as c_q = (1 - 1/q) (3 L^2 / G_4). In the limit q -> ∞, the four-dimensional geometry reduces to global dS_3 and the two defects map to the past and future boundaries, yielding a dS_3/CFT_2 central charge c = 3 L / (2 G_3) and aligning with established holographic results. Overall, the paper provides a microscopic, defect-based interpretation of de Sitter entropy and reveals a holographic bridge between 4D defect Liouville theories and 3D boundary CFTs.

Abstract

We model the back-reaction of a static observer in four-dimensional de Sitter spacetime by means of a singular $\mathbb Z_q$ quotient. The set of fixed points of the $\mathbb Z_q$ 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 $q\to\infty$ the four-dimensional geometry reduces to that of global dS$_3$ spacetime, where the two original minimal surfaces can be mapped to the future and past infinities of dS$_3$ 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 dS$_3$/CFT$_2$ correspondence.