Quantum Extremal Islands Made Easy, PartIII: Complexity on the Brane

TL;DR

This paper explores holographic complexity in doubly holographic models with quantum extremal islands, deriving leading contributions and proposing a generalized volume conjecture for higher curvature gravity theories.

Contribution

It introduces a generalized CV proposal for higher curvature gravity theories and verifies it through consistency checks with Gauss-Bonnet and f(R) gravity.

Findings

Derived leading contributions to complexity using Fefferman-Graham expansion.

Proposed a generalized volume for higher curvature gravity theories.

Validated the proposal with consistency checks in Gauss-Bonnet and f(R) gravity.

Abstract

We examine holographic complexity in the doubly holographic model introduced in [arXiv:2006.04851][arXiv:2010.00018] to study quantum extremal islands. We focus on the holographic complexity=volume (CV) proposal for boundary subregions in the island phase. Exploiting the Fefferman-Graham expansion of the metric and other geometric quantities near the brane, we derive the leading contributions to the complexity and interpret these in terms of the generalized volume of the island derived from the induced higher-curvature gravity action on the brane. Motivated by these results, we propose a generalization of the CV proposal for higher curvature theories of gravity. Further, we provide two consistency checks of our proposal by studying Gauss-Bonnet gravity and f(R) gravity in the bulk.