Graph duality as an instrument of Gauge-String correspondence

TL;DR

This paper investigates a mathematical identity between two branching graphs and interprets it physically within the gauge-gravity correspondence, linking probabilities to three-point functions and LLM geometries.

Contribution

It establishes a physical interpretation of a mathematical identity between branching graphs using three-point functions and LLM geometries in the gauge-gravity context.

Findings

Probabilities correspond to squares of three-point functions involving hook-shaped backgrounds.

Hook-shaped backgrounds act as domain walls interpolating different AdS spaces.

Probabilities sometimes match eigenvalues of embedding chain charges.

Abstract

We explore an identity between two branching graphs and propose a physical meaning in the context of the gauge-gravity correspondence. From the mathematical point of view, the identity equates probabilities associated with $\mathbb{GT}$, the branching graph of the unitary groups, with probabilities associated with $\mathbb{Y}$, the branching graph of the symmetric groups. In order to furnish the identity with physical meaning, we exactly reproduce these probabilities as the square of three point functions involving certain hook-shaped backgrounds. We study these backgrounds in the context of LLM geometries and discover that they are domain walls interpolating two AdS spaces with different radii. We also find that, in certain cases, the probabilities match the eigenvalues of some observables, the embedding chain charges. We finally discuss a holographic interpretation of the mathematical identity through our results.