Analysis of sigma models via projective structures

Abstract

This paper represents a study of projector solutions to the Euclidean sigma model in two dimensions and their associated surfaces immersed in the su(N) Lie algebra. Any solution for the sigma model defined on the extended complex plane with finite action can be written as a raising operator acting on a holomorphic one. Here the proof is formulated in terms rank-1 projectors so it is explicitly gauge invariant. We apply these results to the analysis of surfaces associated with the models defined using the generalized Weierstrass formula for immersion. We show that the surfaces are conformally parametrized by the Lagrangian density, with finite area equal to the action of the model, and express several other geometrical characteristics of the surface in terms of the physical quantities of the model. Finally, we provide necessary and sufficient conditions that a surface be related to a sigma model.