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

## TL;DR

This paper develops a unified framework to analyze extremal surfaces in gauge/gravity duality, including quantum effects, providing new insights into bulk constraints and stability properties.

## Contribution

It introduces a systematic approach to perturbations of extremal surfaces, extending classical results to quantum extremal surfaces with a novel deviation equation.

## Key findings

- Derived an equation of quantum extremal deviation.
- Unified treatment of classical and quantum extremal surface perturbations.
- Identified new bulk constraints consistent with subregion duality.

## 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.