Nahm Transform and Moduli Spaces of CPn Models on the Torus

TL;DR

This paper explores the limitations of Nahm transform dualities in CPn models on a torus, contrasting them with known gauge theory dualities, and analyzes the structure of instanton moduli spaces to understand their singularities.

Contribution

It demonstrates that a Nahm transform duality does not exist for CPn instantons, highlighting a key difference from 4-D gauge theories, and provides a detailed analysis of the instanton moduli space.

Findings

Nahm transform duality fails for CPn instantons.

Boundary of regular instantons coincides with singular instantons.

Explicit parametrization of all self-dual solutions on a torus.

Abstract

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 CPn instantons. This fact establishes a significative difference between 4-D gauge theories and CPn 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 CPn sigma models.