Collective field theory of the matrix-vector model

TL;DR

This paper develops a collective field theory for matrix-vector models related to spin Calogero Moser systems, revealing an extended algebraic structure combining Virasoro and SU(r) current algebras.

Contribution

It introduces a new collective field framework for matrix-vector models, uncovering an extended algebraic structure and providing a method to construct exact eigenstates.

Findings

Derived collective field theories for matrix-vector models.

Discovered an extended algebra combining Virasoro and SU(r) currents.

Constructed exact eigenstates for the coupled field theory.

Abstract

We construct collective field theories associated with one-matrix plus $r$ vector models. Such field theories describe the continuum limit of spin Calogero Moser models. The invariant collective fields consist of a scalar density coupled to a set of fields in the adjoint representation of $U(r)$. Hermiticity conditions for the general quadratic Hamiltonians lead to a new type of extended non-linear algebra of differential operators acting on the Jacobian. It includes both Virasoro and $SU(r)$ (included in $sl(r, {\bf C}) \times sl(r, {\bf C})$) current algebras. A systematic construction of exact eigenstates for the coupled field theory is given and exemplified.