Free particles from Brauer algebras in complex matrix models

TL;DR

This paper reveals hidden free particle structures in complex matrix models with U(N) symmetry, using Brauer algebra projections and Schur decomposition, with detailed analysis for N=2 and implications for string duals.

Contribution

It introduces a novel use of Brauer algebra basis to project onto free particle sectors in complex matrix models, linking gauge invariants to free fermion behavior.

Findings

Brauer algebra basis projects to free fermion sector

Complete characterization of gauge invariants for N=2

Identification of five commuting Hamiltonians for free particles

Abstract

The gauge invariant degrees of freedom of matrix models based on an N x N complex matrix, with U(N) gauge symmetry, contain hidden free particle structures. These are exhibited using triangular matrix variables via the Schur decomposition. The Brauer algebra basis for complex matrix models developed earlier is useful in projecting to a sector which matches the state counting of N free fermions on a circle. The Brauer algebra projection is characterized by the vanishing of a scale invariant laplacian constructed from the complex matrix. The special case of N=2 is studied in detail: the ring of gauge invariant functions as well as a ring of scale and gauge invariant differential operators are characterized completely. The orthonormal basis of wavefunctions in this special case is completely characterized by a set of five commuting Hamiltonians, which display free particle structures. Applications to the reduced matrix quantum mechanics coming from radial quantization in N=4 SYM are described. We propose that the string dual of the complex matrix harmonic oscillator quantum mechanics has an interpretation in terms of strings and branes in 2+1 dimensions.