A Class of Quantum Integrable Models Associated with the Infra-red Limit of Massive Chern-Simons Theory

Abstract

We study the infra-red limit of non-abelian Chern-Simons gauge theory perturbed by a non-topological, albeit gauge invariant, mass term. It is shown that, in this limit, we may construct an infinite class of integrable quantum mechanical models which, for the case of SU(2) group, are labelled by the angular momentum eigenvalue. The first non-trivial example in this class is obtained for the triplet representation and it physically describes the gauge invariant coupling of a non-abelian Chern-Simons particle with a particle moving on $S^3$ - the SU(2) group manifold. In addition to this, the model has a fascinating resemblance to the Landau problem and may be regarded as a non-abelian and a non-linear generalisation of the same defined on the three-sphere with the uniform magnetic field replaced by an angular momentum field. We explicitly solve for some eigenstates of this model in a closed form in terms of some generalised orthogonal polynomials. In the process, we unravel some startling connections with Anderson's chain models which are important in the study of disordered systems in condensed matter physics. We also sketch a method which allows us, in principle, to find the energy eigenvalues corresponding to the above eigenstates of the theory if the Lyapunov exponents of the transfer matrix of the infinite chain model involved are known.