Permutation invariant tensor models and partition algebras

TL;DR

This paper develops permutation invariant tensor models using symmetric group representation theory, providing explicit formulas for their two-point functions, with implications for quantum gravity and matrix ensemble analysis.

Contribution

It introduces the most general permutation invariant Gaussian tensor models in dimension D, constructed via symmetric group and partition algebra representation theory.

Findings

Explicit formula for the two-point function in tensor basis.

Diagonalization of the two-point function in a representation basis.

General construction applicable to any dimension D.

Abstract

Matrix models with continuous symmetry are powerful tools for studying quantum gravity and holography. Tensor models have also found applications in holographic quantum gravity. Matrix models with discrete permutation symmetry have been shown to satisfy large $N$ factorisation properties relevant to holography, while also having applications to the statistical analysis of ensembles of real-world matrices. Here we develop 3-index tensor models in dimension $D$ with a discrete symmetry of permutations in the symmetric group $S_D$. We construct the most general permutation invariant Gaussian tensor model using the representation theory of symmetric groups and associated partition algebras. We define a representation basis for the 3-index tensors, where the two-point function is diagonalised. Inverting the change of basis gives an explicit formula for the two-point function in the tensor basis for general $D$.