Surface Theory: the Classical, the Quantum, and the Holographic

TL;DR

The paper develops a unified, operator-based framework to study how extremal and quantum extremal surfaces deform under boundary, metric, and state perturbations in holographic settings. By translating geometric variation into elliptic-operator language via the extremal-deviation (Jacobi) operator and introducing a covariant functional-derivative calculus for nonlocal entropy functionals, it derives a quantum extension called the equation of quantum extremal deviation. It then demonstrates how subregion/subregion duality constraints—causal wedge inclusion and entanglement wedge nesting—impose new bulk-geometry constraints, including smeared quantum energy inequalities and quantum focusing-type relations, both perturbatively near AdS and in general spacetimes. The results connect deep holographic consistency conditions to concrete geometric and energetic bounds, with explicit AdS examples and a robust mathematical toolkit for further exploration of bulk reconstruction and boundary constraints.

Abstract

Motivated by the power of subregion/subregion duality for constraining the bulk geometry in gauge/gravity duality, we pursue a comprehensive and systematic approach to the behavior of extremal surfaces under perturbations. Specifically, we consider modifications to their boundary conditions, to the bulk metric, and to bulk quantum matter fields. We present a unified framework for treating such perturbations for classical extremal surfaces, classify some of their stability properties, and develop new technology to extend our treatment to quantum extremal surfaces, culminating in an "equation of quantum extremal deviation". The power of this formalism stems from its ability to map geometric statements into the language of elliptic operators; to illustrate, we show that various a priori disparate bulk constraints all follow from basic consistency of subregion/subregion duality. These include familiar properties such as (smeared) versions of the quantum focusing conjecture and the generalized second law, as well as new constraints on (i) metric and matter perturbations in spacetimes close to vacuum and (ii) the bulk stress tensor in generic (not necessary close to vacuum) spacetimes. This latter constraint is highly reminiscent of a quantum energy inequality.

Figures (13)

• Figure 1: \ref{['subfig:CWI']}: CWI requires the causal wedge $W_C[R]$ to lie inside the entanglement wedge $W_E[R]$, and therefore the causal surface $C_R$ must be achronally separated and to the outside of the HRT surface $X_R$. \ref{['subfig:EWN']}: EWN requires the entanglement wedge $W_E[R]$ to shrink into itself as the boundary region $R$ is shrunk; this implies that the HRT surface $X_R$ must move in an achronal direction towards $R$. (The dashed lines indicate caustics and intersections of generators of $\partial W_E[R]$.)
• Figure 2: The equation of geodesic deviation, which constrains the deviation vector $\eta^a$ along a one-parameter family $\Sigma(\lambda)$ of geodesics, can be interpreted as describing the relative acceleration of nearby geodesics in a congruence or alternatively as the perturbation of a geodesic as its boundary conditions are changed.
• Figure 3: A surface in some manifold $M$ is the image of a lower-dimensional manifold $\Sigma$ under an embedding map $\psi$. The map $\psi$ can be used to push forward the tangent bundle $T\Sigma$ to a subset of $TM$ or to pull back the cotangent bundle $T_{\psi(\Sigma)}^* M$ to $T^*\Sigma$. (Here we depict $\Sigma$ with a boundary, but whether or not this is the case is immaterial to the discussion.)
• Figure 4: A one-parameter family of surfaces $\Sigma(\lambda)$ can be obtained from a starting surface $\Sigma$ by evolving it along a one-parameter family of diffeomorphisms $\phi_\lambda$. For each point $p \in \Sigma$, $\phi_\lambda(p)$ is a curve parametrized by $\lambda$ (shown as a dashed line); the tangent to such curves at all points on $\Sigma$ is a deviation vector field $\eta^a$ along this family of surfaces. There is some freedom in the choice of $\eta^a$ on $\Sigma$ (corresponding to the freedom in how each point $p \in \Sigma$ is mapped to subsequent surfaces), but the evolution of the geometry of the family $\Sigma(\lambda)$ is captured by the component $\eta^a_\perp$ normal to $\Sigma$.
• Figure 5: \ref{['subfig:active']}: in the active picture, the initial surface $\Sigma$ is evolved to a one-parameter family of surfaces $\Sigma(\lambda)$ by a one-parameter group of diffeomorphisms $\phi_\lambda$. Each $\Sigma(\lambda)$ is sensitive only to the ambient geometry $g^\mathrm{act}_{ab}$ in its neighborhood, illustrated schematically as the dark gray shading. \ref{['subfig:passive']}: equivalently, in the passive picture $\Sigma$ is left unchanged, but the ambient geometry is pulled back to $\Sigma$ by $(\phi^{-1}_\lambda)_* = \phi_{-\lambda}^*$. $\Sigma$ is then sensitive to a one-parameter family of passive metrics $g^\mathrm{pas}_{ab}(\lambda) = \phi^*_{-\lambda} g^\mathrm{act}_{ab}$.

Theorems & Definitions (11)

• Theorem 1
• Definition 1
• Theorem 2
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4