Matrix models for classical groups and Toeplitz ± Hankel minors with applications to Chern–Simons theory and fermionic models

David García-García,Miguel Tierz

doi:10.1088/1751-8121/ab9b4d

Abstract

We study matrix integration over the classical Lie groups U(N), Sp(2N), SO(2N) and SO(2N + 1), using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz ± 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 S3 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.

Highlights

There is a well known relation between matrix integrals over the classical Lie groups and the determinants of structured matrices, such as Toeplitz and Hankel matrices
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±Hankel matrices
We establish a number of factorizations and expansions for such integrals, with insertions of irreducible characters. We compute both at finite and large N the partition functions, Wilson loops and Hopf links of Chern-Simons theory on S3 with the aforementioned symmetry groups

Summary

Introduction

There is a well known relation between matrix integrals over the classical Lie groups and the determinants of structured matrices, such as Toeplitz and Hankel matrices. Recovers the partition function of Chern-Simons theory on S3 with symmetry group G(N ), and the coefficients in the corresponding Toeplitz and Toeplitz±Hankel matrices are dk = qk2/2, according to (25).

Results

Conclusion