The gravitational dynamics of kinematic space

TL;DR

The paper identifies the dynamics of kinematic space for a 2D CFT with Jackiw-Teitelboim (JT) gravity, clarifying how the Liouville entanglement field plays the role of the conformal mode and the modular Hamiltonian acts as the dilaton. By recasting the JT first law as a maximal entanglement principle, the authors derive JT equations of motion from entanglement considerations and show that coupling a boundary CFT to JT gravity yields the boundary kinematic space as an AdS$_2$ JT system. They develop a coherent framework that unifies bulk and boundary kinematic spaces, linking entanglement data to geometric data via holographic and dimensional-reduction insights, including a differential-entropy interpretation of the vacuum dilaton. The work suggests deep connections between entanglement, gravity in two dimensions, and the structure of kinematic space, with potential implications for T$ar{T}$ deformations and holographic dualities beyond the present setting.

Abstract

We show that the dynamics of the kinematic space of a 2-dimensional CFT is gravitational and described by Jackiw-Teitelboim theory. We discuss the first law of this 2-dimensional dilaton gravity theory to support the relation between modular Hamiltonian and dilaton that underlies the kinematic space construction. It is further argued that Jackiw-Teitelboim gravity can be derived from a 2-dimensional version of Jacobson's maximal vacuum entanglement hypothesis. Applied to the kinematic space context, this leads us to the statement that the kinematic space of a 2-dimensional boundary CFT can be obtained from coupling the boundary CFT to JT gravity through a maximal vacuum entanglement principle.

Figures (5)

• Figure 1: Interval $x \in [u,v]$ (in blue) on a constant time slice of a CFT$_2$ in vacuum state $|0\rangle_X$. The kinematic space construction involves 1) promoting the $t=0$ time-slice of the CFT to past infinity of kinematic space, 2) identifying the interval endpoints $u$ and $v$ with kinematic space lightcone coordinates, and 3) using the one-interval entanglement formula to define a hyperbolic metric on kinematic space through \ref{['ds2K']}. The yellow triangle is a sketch of the emergent dS$_2$ kinematic space, superimposed here on the picture of the CFT background.
• Figure 2: Interval [$P$, boundary $x=0$] (in blue) on a constant time slice of a bCFT$_2$ in vacuum state $|0\rangle_X$. It is specified by the location of the point $P$ in CFT lightcone coordinates $(x^+_P,x^-_P)$. The kinematic space construction involves 1) promoting the $x=0$ boundary of the CFT to spacelike infinity of kinematic space, 2) identifying $x^+$ and $x^-$ with kinematic space lightcone coordinates, and 3) using the one-interval entanglement formula to define a hyperbolic metric on kinematic space through equation \ref{['ds2Kb']}. The yellow triangle is a sketch of the emergent AdS$_2$ kinematic space $K_\partial$, superimposed here on the picture of the bCFT background. Note that the kinematic space lightcone coordinates are just given by the bCFT lightcone coordinates, which is different from the situation in figure \ref{['figK']} for a CFT without boundary.
• Figure 3: To write down the expressions for $S$ and $H_{mod}$ through the interval [$P$, boundary $x=0$], it is instructive to consider the doubled interval stretching from the point $P$ to its mirror image denoted $-P$. The formula \ref{['entbCFT']} counts the entanglement contribution from right-moving degrees of freedom through the doubled interval, or the full entanglement through [$P$, boundary $x=0$]. We employ here the Penrose diagram representation of flat space with future and past null infinity at 45 degree angles.
• Figure 4: Left: The AdS$_2$-black hole solution of JT (yellow) and the Poincaré solution (largest triangular region), described by Poincaré covering coordinates ${\rm{t}}$ and ${\rm{z}}$. The Killing vector $\xi_{JT}$ associated with the Killing horizon of the black hole solution vanishes at the point $\{{\rm{t}} = 0,{\rm{z}} = \frac{1}{\sqrt{\mu}} \}$ labeled by the dot. The boundary is at ${\rm{z}}=0$. The coordinate axes refer to global AdS$_2$ coordinates, see e.g. Spradlin:1999bnMaldacena:2016upp. Right: The AdS$_2$-black hole solution has a metric Killing vector $\xi_g$, with flow lines in green, that vanishes in $P$. The point $P$ at location $(X^+,X^-)$ or $({\rm{t}}={\rm{t}}_0,{\rm{z}}=R)$ in Poincaré covering coordinates marks the boundary $\partial \Sigma$ of the interval $\Sigma = \text{[$P$, boundary ${\rm{z}}=0$]}$ (in blue). Imposing reflective boundary conditions at ${\rm{z}}=0$, the interval can be effectively doubled to a region with diamond-shaped domain of dependence $\diamond$.
• Figure 5: Left: Dilaton as differential entropy \ref{['Sdiffdef']} from considering the parent asAdS$_3$ theory with possible horizon at $r=r_+$. This relates the dilaton with the entanglement of the interval in blue, different from the interpretation \ref{['Phi0int1']} that relates the dilaton with the entanglement of the interval in blue in figure \ref{['figJTsol']} (right). Right: Differential entropy measures 'entanglement' of the strip of width $2 z$.