For U(N) vector models, primary invariants number f^2 for f≤N and N^2+2N(f−N) for f>N, with secondary invariants appearing and growing as e^{2N log 2 f}.
Collective field theory of the matrix-vector model
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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.
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Bosonic Fortuity in Vector Models
For U(N) vector models, primary invariants number f^2 for f≤N and N^2+2N(f−N) for f>N, with secondary invariants appearing and growing as e^{2N log 2 f}.