Collective field theory of the matrix-vector model

TL;DR

This work develops a collective field theory for a matrix-vector model that interpolates between one- and multi-matrix theories and connects to spin Calogero-Moser systems via Hamiltonian reduction. It introduces invariant collective fields $\phi_n^0 = Tr M^n$ and $\psi_n^{ab} = \bar{x}^a M^n x^b$, and derives a non-linear algebra of differential operators acting on the Jacobian that naturally contains the Virasoro algebra and an $SU(r)$ current algebra, enabling a systematic construction of exact eigenstates. The spectrum of the reduced matrix Laplacian $H_1$ is organized by conserved quantities ($N_V$, $N_0$) and by representation theory of $SU(N)$ and $S_n$, with explicit results in low sectors linked to Young tableaux; the formalism extends to a classical collective theory via density and current fields, clarifying the spin-Calogero-Moser connection through the Poisson structure and its quantization to a central extension of the current algebra. Overall, the paper provides a framework for exact eigenstates in matrix-vector collective field theory and illuminates the role of Virasoro and non-Abelian current algebras in the continuum limit of spin Calogero-Moser models.

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.