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What exactly is a non linear $\sigma$ model? In many books one can view many different types of non linear $\sigma$ models but I don't understand what is the link between all of them and why it is called $\sigma$.

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Lubos answered the physics question, but the history is off. The origin of the term "sigma model" for a field theory where the scalar values are on a manifold is from Gell-Mann and Levy's 1960 paper "The Axial Vector Current in $\beta$-Decay" which introduced two models.

The first of these is called the "linear sigma model", and it is a renormalized Heisenberg-inspired Mexican hat model for a pion condensate. The model has four fields, $\phi^i$ where i=0,1,2,3, which have a regular Mexican hat potential, so that the vaccum values are on a sphere S_3.

This makes 3 field directions light, and these modes are the three pions, and one field direction heavy, and this mode was called the "sigma". It was a predicted particle, and I believe it was identified with the $\sigma$(600) broad resonance, except that this resonance is very strange and was delisted, and is too broad to be a real sigma, so the model is not good.

Ignoring renormalizability, the mass of the $\sigma$ is adjusted by making the wall of the Mexican-hat potential tighter oscillating, and in the limit of infinitely fast oscillations, you just end up restricting the $\pi$ fields to a sphere, and there is no finite energy $\sigma$. This limit is the non-renormalizable non-linear sigma model in the paper. It was called that, because it is the nonlinear version of the renormalizable sigma-model Gell-Mann and Levy believed, but it is a misnomer, because the nonlinear theory doesn't have a sigma, that's the whole point of going to the nonlinear version.

If you start with a microscopic nonlinear sigma model and a lattice Lagrangian, you will generate a sigma dynamically and you will get linear sigma model dynamics at long distances. Something like this was already known to Gell-Mann and Levy. But Gell-Mann wasn't sure what was going on at short distances, and was open to some sort of S-matrix thing taking over at the hadron scale, making renormalizability considerations secondary, so he left the nonlinear model as an option, even though it wasn't renormalization consistent (this is my opinion on Gell-Mann, somebody could ask the guy and get a better opinion, he never lies).

Historically, the nonlinear sigma model was the first time someone had considered a field theory where the field values were restricted to a manifold. All other such constructions have been called nonlinear sigma-models from that point on, and they historically evolved from generalizations of this construction. The term "current algebra" is also sometimes used for a special case of such constructions, when the manifold is a group. 1970s Witten says 2d current-algebra whenever he means that there are dynamical fields which take values in a Lie Group, like in the WZW models.

The modern form of the Gell-Mann Levy construction is Chiral Perturbation Theory, and it is a low-energy approximation to QCD. The pion fields are the chiral rotations of the quark condensate, while the sigma excitation is not exactly necessary, because it is not a symmetry motion. Since the SU(2) of chiral rotations is a 3-sphere topologically, it is not much different from what Gell-Mann and Levy first suggested.

The nonlinear sigma-models take on a new life due to Friedan's work, because in string theory, the space-time itself is a sigma-model on the worldsheet. The nonlinear sigma models in Friedan's paper are qualitatively more sophisticated than Gell-Mann and Levy, and really should be called by a new name. But they aren't. That's history, we deal with it.

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Oops, you're right, +1. –  Luboš Motl Apr 13 '12 at 6:26
    
@LubošMotl: I +1ed you too, the history is not as important as the physics, but it is weird usage. –  Ron Maimon Apr 13 '12 at 6:39
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I find the Wikipedia article that the OP has linked to comprehensible and self-explanatory. Nevertheless, again.

A nonlinear sigma-model is a model describing scalar fields that span a usually curved manifold equipped with a Riemannian metric. The Riemannian metric $g_{ab}(\Sigma^c)$ appears in the kinetic term as a coefficient of the $\partial^\mu \Sigma^a \partial_\mu \Sigma^b$. Such models may be interpreted as describing the motion of a particle or a higher-dimensional string/brane on the curved manifold.

The word "model" refers to a particular theory with some particular laws of physics (Lagrangian). It's "nonlinear" because the kinetic term isn't simply bilinear; it is higher-order and depends on the fields and not just their derivatives. The solutions are then nonlinear as well; they may be understood as a motion of a particle (or branes) on a curved target manifold which is not along "straight paths", so it's nonlinear.

They're called sigma models because the letter $\Sigma$ used for these scalar fields is pronounced as "sigma". Well, it was a lowercase sigma in the important Dan Friedan's 1980 thesis,

http://www.osti.gov/energycitations/servlets/purl/5001689-j6L3sY/5001689.pdf

which didn't have "sigma" in the title but it did use the sigma letter for the fields. Some letter had to be chosen, it was this one, and physicists economically avoided redundant terminology and named the model after the letter, too.

Having nonlinear coefficients of the kinetic terms is particularly natural for scalar fields where it can be linked to the Riemannian geometry. Similar things can't be done with spinning fields, at least not equally naturally. So the non-linear sigma models represent an important class. It appears in many situations in which the scalar fields are more complicated than just "parameters labeling some flat space". The relevant manifold spanned by the scalar fields may be a sphere, a quotient of groups (e.g. in supergravity theories), an arbitrary spacetime manifold if we describe string theory by a world sheet Lagrangian, the same thing for branes, and so on.

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What is the target manifold that you are speaking about? Is is the space where the fields map to? From which space is the map defined? –  ramanujan_dirac Mar 13 '13 at 12:20
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