A further support for the statement that the action of the HRT-area generates a kink transformation in the pure AdS$_3$ gravity

TL;DR

The work provides a canonical-gravity derivation of how the HRT area flow acts on boundary data in AdS$_3$, showing that it implements a boundary-condition-preserving kink transformation. By expressing the boundary stress tensor in terms of initial data via holographic renormalization, the authors reproduce the stress-tensor shock found in prior work and establish a concrete link between the HRT observable and kink-like spacetime deformations. The analysis also clarifies the dual roles of the twist flow and a Connes cocycle-related kink, illustrating a broader algebraic structure in AdS$_3$ gravity. The results strengthen the interpretive bridge between geometric entropy functionals, modular Hamiltonians, and canonical gravity, with potential extensions to higher-derivative theories and generalized entropy functionals.

Abstract

It is stated that the action of the Hubeny-Rangamani-Takayanagi (HRT) area, which is viewed as an observable, generates a kink transformation in gravity. In this paper, we provide a further support for the kink transformation statement by performing a relevant computation. Specifically, by acting the kink transformation to the boundary stress tensor, we compute the bracket between the HRT area with the boundary stress tensor. Here we represent the boundary stress tensor in terms of initial data on the Cauchy surface under holographic renormalization. Our computation reproduces the same result as the one in arXiv:2203.04270 derived from a different approach.

Figures (3)

• Figure 1: A match of foliation: (a) illustrates the foliation of bulk spacetime, where the gray surface denotes the Cauchy surface $\Sigma_T$ with a constant $T$, the yellow region denotes the IR cutoff surface $\Gamma_\epsilon$ and the red dotted line denotes the intersection between both surfaces;(b) illustrates the foliation of the asymptotic boundary $\Gamma^{(0)}$, where the red dotted line denotes the Cauchy surface $\Sigma^{(0)}$ with a constant $T^{(0)}$. We can match them by setting $T=T^{(0)}$.
• Figure 2: An interpretation of the action of the HRT area flow on the initial data in the case of planar asymptotic boundary: (a) illustrates the kink transformation of Cauchy surface in bulk spacetime, where the HRT area flow induces a relative boost between the left and right sides of the Cauchy surface $\Sigma_T$;(b) illustrates the preservation of the Cauchy surface $\Sigma^{(0)}$ on the asymptotic boundary $\Gamma^{(0)}$ under the action of the HRT area flow (Or precisely the action of dual Modular Hamiltonian). Before the action of the HRT area flow, we have matched them by setting $T=T^{(0)}$.
• Figure 3: A kink transformation dual of connes cocycle flow. The first step shows a kink transformation and its boundary dual, connes cocycle flow. The second step shows a coordinate transformation. Under the new coordinates $\{\widetilde{x}^\mu\}$, the Cauchy surface is located at $\widetilde{t}=0$.