TheCP N?1 1/N-action by inverse scattering transformation in angular momentum

Abstract

The effective action which generates 1/N expansion of theCPN−1 model in two dimensions is studied here by inverse-problem methods. The action contains a functional determinant, in which auxiliary scalar and vector fields are assumed to have a spherical symmetry. This leads to the introduction, as an associated linear problem, of a radial Schrodinger equation with two potentialsv and ϑ, and a potential-dependent centrifugal term {(l−rθ)2/r2−1/4r2}. The full inverse scattering formalism is developed here for this diffusion problem. It is formulated in terms of two-component Jost solutions, and leads to a matricial Gel'fand-Levitan-Marchenko equation. The scattering data associated to the potentials by this IST are then used to obtain a closed local form for the whole effective action. This is indeed possible for theCPN−1 model, owing to the classical integrability. Moreover it is found that no spherically symmetric instanton exists in this case. However the absence of supplementary informations on the 1/N series, due to the non-integrability at quantum level, does not allow safe quantitative conclusions on the general behaviour of the 1/N series at large orders.