Quantum theory of compact and non-compact σ models

We discuss quantization of SO(N + 1) σ models and CP^N models, and of certain non-compact counterparts, SO(N, 1) and QP^N respectively, of these, both in canonical operator formalism and the covariant path integral formulation, showing the equivalence of the two approaches. We discuss also a class of supersymmetric σ models formulated in d ? 3 dimensions and apply the results to the SO(N + 1) and SO(N, 1) cases. This allows us to calculate the Witten index in each case. For SO(2l + 1,1) we thereby find supersymmetry breaking. However, for SO(2l, 1), we find supersymmetry is unbroken. Moreover, there is no unique ground state, invariant under SO(2l, 1), rather an infinite multiple of zero energy states, carrying a unitary irreducible representation of the non-compact SO(2l, 1) group. We discuss also field theoretic aspects of the models in d ? 2 dimensions, stressing differences of the non-compact to the compact cases. These include infrared instead of ultraviolet asymptotic freedom, lack of an energy gap, failure (in the QP^N case) of the auxiliary vector field to become dynamical. A further conclusion that is argued concerns the absence of a consistent particle interpretation for the QP^N model in exactly two dimensions. For d > 2 the non-compact symmetry of QP^N is broken down to the compact subgroup.