Gravity Dual of Connes Cocycle Flow

TL;DR

The paper proposes the kink transform as a bulk holographic dual to Connes cocycle flow across a planar boundary cut, realized by a one-sided boost of initial data about the RT surface and glued to a precursor slice. $| ilde{ ext{ψ}}_s angle$ in the bulk is shown to reproduce the one-sided CC flow on the boundary, including the associated entanglement-wedge structure and boundary stress-tensor shocks, via precise shock matching. A key novel prediction is a bulk-derived shock in ⟨T_{xx}⟩ that complements the known T_{tx} and T_{vv}/T_{uu} shocks, suggesting universality and robustness of CC flow in holographic CFTs. The framework remains valid under higher curvature corrections and offers avenues to generalize beyond flat cuts, with implications for JLMS, QNEC, and holographic descriptions of coarse-graining.

Abstract

We define the "kink transform" as a one-sided boost of bulk initial data about the Ryu-Takayanagi surface of a boundary cut. For a flat cut, we conjecture that the resulting Wheeler-DeWitt patch is the bulk dual to the boundary state obtained by Connes cocycle (CC) flow across the cut. The bulk patch is glued to a precursor slice related to the original boundary slice by a one-sided boost. This evades ultraviolet divergences and distinguishes our construction from one-sided modular flow. We verify that the kink transform is consistent with known properties of operator expectation values and subregion entropies under CC flow. CC flow generates a stress tensor shock at the cut, controlled by a shape derivative of the entropy; the kink transform reproduces this shock holographically by creating a bulk Weyl tensor shock. We also go beyond known properties of CC flow by deriving novel shock components from the kink transform.

Figures (7)

• Figure 1: Kink transform. Left: a Cauchy surface $\Sigma$ of the original bulk $\mathcal{M}$. An extremal surface $\mathcal{R}$ is shown in red. The orthonormal vector fields $t^a$ and $x^{a}$ span the normal bundle to $\mathcal{R}$; $x^a$ is tangent to $\Sigma$. Right: The kink transformed Cauchy surface $\Sigma_{s}$. As an initial data set, $\Sigma_{s}$ differs from $\Sigma$ only in the extrinsic curvature at $\mathcal{R}$ through Eq. \ref{['onlychange']}. Equivalently, the kink transform is a relative boost in the normal bundle to $\mathcal{R}$, Eq. \ref{['boostangle']}.
• Figure 2: The kink-transformed spacetime $\mathcal{M}_s$ is generated by the Cauchy evolution of the kinked slice $\Sigma_{s}$. This reproduces the left and right entanglement wedges $D(a)$ and $D(a')$ of the original spacetime $\mathcal{M}$. The future and past of the extremal surface $\mathcal{R}$ are in general not related to the original spacetime.
• Figure 3: Straight slices $\Sigma$ (red) in a maximally extended Schwarzschild (left) and Rindler (right) spacetime get mapped to kinked slices $\Sigma_s$ (blue) under the kink transform about $\mathcal{R}$.
• Figure 4: On a fixed background with boost symmetry, the kink transform changes the initial data of the matter fields. In this example, $\mathcal{M}$ is Minkowski space with two balls relatively at rest (red).The kink transform is still Minkowski space, but the balls collide in the future of $\mathcal{R}$ (blue).
• Figure 5: A boundary subregion $A_0$ (pink) has a quantum extremal surface denoted $\mathcal{R}$ (brown) and an entanglement wedge denoted $a$. The complementary region $A'_0$ (light blue) has the entanglement wedge $a'$. CC flow generates valid states, but one-sided modular flow is only defined with a UV cutoff. For example, one can consider regulated subregions $A^{(\epsilon)}$ (deep blue) and $A'^{(\epsilon)}$ (red). In the bulk, this amounts to excising an infrared region (gray) from the joint entanglement wedge (yellow).