Timelike-bounded dS4 holography from a solvable sector of the T2 deformation

Abstract

Recent research has leveraged the tractability of \( T\overline{T} \) style deformations to formulate timelike-bounded patches of three-dimensional bulk spacetimes including dS₃. This proceeds by breaking the problem into two parts: a solvable theory that captures the most entropic energy bands, and a tuning algorithm to treat additional effects and fine structure. We point out that the method extends readily to higher dimensions, and in particular does not require factorization of the full T ² operator (the higher dimensional analogue of \( T\overline{T} \) defined in [1]). Focusing on dS₄, we first define a solvable theory at finite N via a restricted T ² deformation of the CFT₃ on S² × ℝ, in which T is replaced by the form it would take in symmetric homogeneous states, containing only diagonal energy density E/V and pressure (-dE/dV) components. This explicitly defines a finite-N solvable sector of dS₄/deformed-CFT₃, capturing the radial geometry and count of the entropically dominant energy band, reproducing the Gibbons-Hawking entropy as a state count. To accurately capture local bulk excitations of dS₄ including gravitons, we build a deformation algorithm in direct analogy to the case of dS₃ with bulk matter recently proposed in [2]. This starts with an infinitesimal stint of the solvable deformation as a regulator. The full microscopic theory is built by adding renormalized versions of T ² and other operators at each step, defined by matching to bulk local calculations when they apply, including an uplift from AdS₄/CFT₃ to dS₄ (as is available in hyperbolic compactifications of M theory). The details of the bulk-local algorithm depend on the choice of boundary conditions; we summarize the status of these in GR and beyond, illustrating our method for the case of the cylindrical Dirichlet condition which can be UV completed by our finite quantum theory.