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

A bstract Recent research has leveraged the tractability of $$ T\overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> style deformations to formulate timelike-bounded patches of three-dimensional bulk spacetimes including dS 3 . 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 $$ T\overline{T} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> defined in [1]). Focusing on dS 4 , we first define a solvable theory at finite N via a restricted T 2 deformation of the CFT 3 on S 2 × 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 dS 4 /deformed-CFT 3 , 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 dS 4 including gravitons, we build a deformation algorithm in direct analogy to the case of dS 3 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 AdS 4 / CFT 3 to dS 4 (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.