Recent research has leveraged the tractability of $T\bar T$ 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 does not require factorization of the full $T^2$ operator (the higher dimensional analogue of $T\bar T$ 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 ${S}^2\times \mathbb{R}$, 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 defines a finite-N solvable sector of $dS_4/\text{deformed-CFT}_3$, 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.
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Universitat de Barcelona（巴塞罗那大学）；Inamori Research Institute for Science（稻盛科学研究所）
Temporal Entanglement from Twist Correlators in 2d Conformal Field Theory and Holography
INFN, Sezione di Firenze（意大利国家核物理研究所佛罗伦萨分部）；Institute of Theoretical Physics and Mark Kac Center for Complex Systems Research Jagiellonian University（雅盖隆大学理论物理研究所与马克·凯克复杂系统研究中心）；Dipartimento di Fisica e Astronomia, Università di Firenze（佛罗伦萨大学物理与天文学系）；Department of Physics and Astronomy, Ghent University（根特大学物理与天文学系）
