Tensor and Matrix models: a one-night stand or a lifetime romance?

TL;DR

This paper explores the mathematical structure of tensor and matrix models, revealing a deep connection through combinatorial coefficients and proposing a duality that links their physical spectra and phases.

Contribution

It derives a novel formula relating Kronecker coefficients with Littlewood-Richardson numbers and conjectures a duality between tensor and multi-matrix model sectors.

Findings

Eigenstates of tensor models correspond to fluctuations in multi-matrix models.

A duality between tensor and matrix sectors is proposed.

High-energy phases of tensor models may be described by multi-matrix models.

Abstract

The spectra of energy eigenstates of free tensor and matrix models are organized by Kronecker coefficients and Littlewood-Richardson numbers, respectively. Exploiting recent results in combinatorics for Kronecker coefficients, we derive a formula that relates Kronecker coefficients with a hook shape with Littlewood-Richardson numbers. This formula has a natural translation into physics: the eigenstates of the hook sector of tensor models are in one-to-one correspondence with fluctuations of 1/2-BPS states in multi-matrix models. We then conjecture the duality between both sectors. Finally, we study the Hagedorn behaviour of tensor models with finite rank of the symmetry group and, using similar arguments, suggest that the second (high energy) phase could be entirely described by multi-matrix models.