Yangian-invariant field theory of matrix-vector models

TL;DR

This work constructs a Yangian-invariant field theory for dynamical matrix–vector models by embedding a c=1 collective boson into an su(R) current algebra at level k=1 and adding precisely tuned cubic and bilinear terms to achieve ${\ m Y}(sl(R))$ symmetry. The authors derive a one-parameter family of Hamiltonians H with couplings fixed by requiring commutation with the Yangian generators, yielding a nontrivial, finite-block spectrum amenable to explicit calculation via a spinon basis. They demonstrate that the full Hamiltonian can be restricted to finite-dimensional subspaces by mode-sum and su(R) representation constraints, and they illustrate the one-spinon spectrum, showing rationality conditions on the coupling parameter. This work links matrix-model dynamics, Haldane–Shastry–Calogero–Moore–Spinon structures, and Yangian symmetry, providing an exact solvable framework for one-dimensional spinful many-body problems with potential insights into related string-theoretic limits and collective field theories.

Abstract

We extend our study of the field-theoretic description of matrix-vector models and the associated many-body problems of one dimensional particles with spin. We construct their Yangian-su(R) invariant Hamiltonian. It describes an interacting theory of a c=1 collective boson and a k=1 su(R) current algebra. When $R \geq 3$ cubic-current terms arise. Their coupling is determined by the requirement of the Yangian symmetry. The Hamiltonian can be consistently reduced to finite-dimensional subspaces of states, enabling an explicit computation of the spectrum which we illustrate in the simplest case.