# Handlebody phases and the polyhedrality of the holographic entropy cone

## TL;DR

This paper investigates the structure of the holographic entropy cone, showing that when restricted to geometries dual to CFT states, its polyhedrality is not guaranteed and certain extremal geometries may not be dual to CFT states.

## Contribution

It introduces a new definition of the holographic entropy cone based on dual geometries and demonstrates that its polyhedrality is not automatic under this definition.

## Key findings

- Extremal rays correspond to subdominant bulk phases.
- Some geometries do not appear as dominant phases in the path integral.
- The polyhedrality of the cone depends on the duality condition.

## Abstract

The notion of a holographic entropy cone has recently been introduced and it has been proven that this cone is polyhedral. However, the original definition was fully geometric and did not strictly require a holographic duality. We introduce a new definition of the cone, insisting that the geometries used for its construction should be dual to states of a CFT. As a result, the polyhedrality of this holographic cone does not immediately follow. A numerical evaluation of the Euclidean action for the geometries that realize extremal rays of the original cone indicates that these are subdominant bulk phases of natural path integrals. The result challenges the expectation that such geometries are in fact dual to CFT states.