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

Abstract

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.

Topics & Concepts