Toward a Unified de Sitter Holography: A Composite $T\bar{T}$ and $T\bar{T}+Λ_2$ Flow

Abstract

In de Sitter (dS) holography, both the dS/CFT correspondence and the dS static patch holography have been extensively studied. In these two holographic frameworks, the dual field theories are defined on spacelike and timelike boundaries, respectively, where the inward motion of the holographic boundary into the bulk corresponds to the $T\bar{T}$ and $T\bar{T}+Λ_2$ deformations in the respective dual field theories. In this work, we develop a unified framework for these two dS holographic models by introducing a composite flow that incorporates both $T\bar{T}$ and $T\bar{T}+Λ_2$ deformations. We propose that this composite flow corresponds to the inward motion of a spacelike boundary from the asymptotic infinity of dS spacetime, traversing the cosmological horizon and approaching the worldline of a static observer. This proposal is supported by the computation of the quasi-local energy and the holographic entanglement entropy within the dS static spacetime and its extened geometry.

Figures (7)

• Figure 1: \ref{['dsstaticcoordinate']} The blue region denotes the static patch ($r<\ell_{\rm dS}$), the red region corresponds to the extended spacetime ($r>\ell_{\rm dS}$), and the interface represents the cosmological horizon at $r=\ell_{\rm dS}$. \ref{['dsstaticflow']} The red dashed line indicates the spacelike boundary, while the blue dashed line indicates the timelike boundary. Arrows represent the motion of these boundaries, which is opposite to the direction of their outward-pointing unit normal vectors.
• Figure 2: The energy flow of the $T\bar{T}$ deformation with $L=2\pi$ and $J=0$. One finds that when the deformation parameter $\lambda_{\text{dS}}$ exceeds $\frac{1}{E_0}$, the energy becomes complex.
• Figure 3: Energy flow of the $T\bar{T}+\Lambda_2$ deformation with $L=2\pi$ and $J=0$.
• Figure 4: Killing vector field. The Killing vector $\xi^{\mu}$ is timelike in the static region and spacelike in the extended region.
• Figure 5: Dimensionless energy in dS spacetime. We take $G=\ell_{\rm dS}=1$. The red dashed line denotes the cosmological horizon $r=\ell_{\rm dS}$.