# Whats the difference between a linear and non-linear sigma model?

Wikipedia says

In physics, a sigma model is a physical system that is described by a Lagrangian density of the form:

$$L(\phi_1,...,\phi_n)=g_{ij} d\phi_i \wedge d\phi_j$$

With Einsteins summation convention understood. Depending on the scalars in $$g_{ij}$$, it is either a linear sigma model or a non-linear sigma model.

Question: what exactly are the conditions on $$g$$ that distinguish a linear sigma model from a non-linear sigma model?

The fields $$\phi_i$$, in general, provide a map from a base manifold called the worldsheet to a target Riemannian manifold of the scalars linked together by internal symmetries.

This suggests that the sigma model is in fact a section of a principal bundle with a structure group that represents the internal symmetries.

Question: Is this right? Is there an exposition of that develops the properties of sigma models from this point of view?

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Although, I didn't ask this in my original question. I'm also interested in the provenance of the term sigma-model and it's importance. I know that S. Weinberg did some work on what he called a sigma-model, the sigma here denoting a particular particle.

Question: is how sigma-models used now merely an abstraction of the model that Weinberg used then with the specific model made obsolete by the advances made since then?

• Related. Nonlinear is the infinite curvature limit of the Higgs potential of the linear one. Dec 25, 2018 at 15:12

$$s: S \rightarrow M$$
Where $$S$$ is the abstract world-line and $$M$$ is spacetime. Bundles on $$M$$ represent forces acting on the particle. For example, a frame bundle would represent a metric on $$M$$ and hence the force of gravity, or a $$U(1)$$-bundle the EM force.
Weinberg modelled $$M$$ as a vector space, this was later generalised to manifold, and in particular a group manifold; hence the qualifier 'linear' as opposed to 'non-linear' distinguishes these two cases.