Geometry and analysis in non-linear sigma models

TL;DR

This paper reviews the geometric and analytical properties of non-linear sigma models with homogeneous space targets, exploring their topological, homotopical, and variational aspects, and proposing new invariants for manifolds.

Contribution

It introduces a unified approach to analyzing configuration spaces of sigma models using flat connections, extending topological results to Sobolev maps, and suggests novel manifold invariants.

Findings

Topological and cohomological descriptions of configuration spaces

Representation of maps by flat connections for analysis

Applications to homotopy theory and minimization problems

Abstract

The configuration space of a non-linear sigma model is the space of maps from one manifold to another. This paper reviews the authors' work on non-linear sigma models with target a homogeneous space. It begins with a description of the components, fundamental group, and cohomology of such configuration spaces together with the physical interpretations of these results. The topological arguments given generalize to Sobolev maps. The advantages of representing homogeneous space valued maps by flat connections are described, with applications to the homotopy theory of Sobolev maps, and minimization problems for the Skyrme and Faddeev functionals. The paper concludes with some speculation about the possiblility of using these techniques to define new invariants of manifolds.