Nahm Transform and Moduli Spaces of C P N - Models on the Torus

There is a Nahm transform for two-dimensional gauge fields which establishes a one-to-one correspondence between the orbit space of U (N) gauge fields with topological charge k defined on a torus and that of U (k) gauge fields with charge N on the dual torus. The main result of this paper is to show that a similar duality transform cannot exist for CP N instantons. This fact establishes a significative difference between 4-D gauge theories and CP N models. The result follows from the global analysis of the moduli space of instantons based on a complete and explicit parametrization of all self-dual solutions on the two-dimensional torus. The boundary of the space of regular instantons is shown to coincide with the space of singular instantons. This identification provides a new approach to analyzing the role of overlapping instantons in the infrared sector of CP N sigma models.