# Matrix models for classical groups and Toeplitz$\pm $Hankel minors with applications to Chern-Simons theory and fermionic models

## TL;DR

This paper develops matrix integral techniques over classical Lie groups using symmetric functions, deriving new factorizations and expansions, and applies these to compute Chern-Simons theory invariants and analyze fermionic models.

## Contribution

It introduces novel factorizations and expansions for matrix integrals over classical groups, connecting symmetric function theory with topological quantum field theory and fermionic models.

## Key findings

- Computed partition functions, Wilson loops, and Hopf links in Chern-Simons theory for various groups.
- Established identities relating observables in Chern-Simons theory.
- Evaluated spectra of fermionic models using character expansions in random matrix ensembles.

## Abstract

We study matrix integration over the classical Lie groups $U(N),Sp(2N),O(2N)$ and $O(2N+1)$, using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz$\pm$Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large $N$ the partition functions, Wilson loops and Hopf links of Chern-Simons theory on $S^{3}$ with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern-Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom.