Symmetry and Integrability of Non-Singlet Sectors in Matrix Quantum Mechanics

TL;DR

This paper investigates the structure and spectrum of non-singlet sectors in matrix quantum mechanics, revealing their algebraic properties, conserved charges, and connections to integrable systems, with implications for bosonization.

Contribution

It introduces a nonlinear extension of the W_infinity algebra for non-singlet sectors and derives their spectrum and conserved charges using group theory.

Findings

Non-singlet spectra match singlet sectors except for multiplicities

Explicit form of conserved charges in terms of eigenvalues

Interaction terms resemble those in Calogero-Sutherland systems

Abstract

We study the non-singlet sectors of matrix quantum mechanics (MQM) through an operator algebra which generates the spectrum. The algebra is a nonlinear extension of the W_\infty algebra where the nonlinearity comes from the angular part of the matrix which can not be neglected in the non-singlet sector. The algebra contains an infinite set of commuting generators which can be regarded as the conserved currents of MQM. We derive the spectrum and the eigenfunctions of these conserved quantities by a group theoretical method. An interesting feature of the spectrum of these charges in the non-singlet sectors is that they are identical to those of the singlet sector except for the multiplicities. We also derive the explicit form of these commuting charges in terms of the eigenvalues of the matrix and show that the interaction terms which are typical in Calogero-Sutherland system appear. Finally we discuss the bosonization and rewrite the commuting charges in terms of a free boson together with a finite number of extra degrees of freedom for the non-singlet sectors.