A framework for generalizing toric inequalities for holographic entanglement entropy

TL;DR

This paper proposes a multi-parameter generalization of toric inequalities related to holographic entanglement entropy, extending proof methods through geometric tiling and graph constructions, and explores their validity in specific cases.

Contribution

It introduces a new multi-parameter generalization of toric inequalities and develops novel graph-based proof techniques for their validation.

Findings

Constructed graphs via tiling Euclidean space and entanglement wedge nesting.

Built cycle graphs using graph Cartesian products to analyze inequalities.

Proved conjectures for certain parameter ranges in decomposed torus geometries.

Abstract

We conjecture a multi-parameter generalization of the toric inequalities of \cite{Czech:2023xed}. We then extend their proof methods for the generalized toric inequalities in two ways. The first extension constructs the graph corresponding to the toric inequalities and the generalized toric conjectures by tiling the Euclidean space. An entanglement wedge nesting relation then determines the geometric structure of the tiles. In the second extension, we exploit the cyclic nature of the inequalities and conjectures to construct cycle graphs. Then, the graph can be obtained using graph Cartesian products of cycle graphs. In addition, we define a set of knots on the graph by following \cite{Czech:2023xed}. These graphs with knots then imply the validity of their associated inequality. We study the case where the graph can be decomposed into disjoint unions of torii. Under the specific case, we explore and prove the conjectures for some ranges of parameters. We also discuss ways to explore the conjectured inequalities whose corresponding geometries are $d$-dimensional torii $(d>2)$