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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 article further adds:

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?

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

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A sigma-model is best understood as a map

$s: S \rightarrow M$

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.

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