Linear/Non-linear sigma model
The sigma model started life as a model for pions and derives its name from one of the fields in the theory (denoted $\sigma$). This, however, is another story and not the reason why I would get "excited" about sigma models.
For me the basic reason to be interested in sigma models is that they are the natural bridge between quantum field theory and geometry. One can define certain geometric notions in terms of properties of the corresponding sigma models. For example, one can characterise Kähler manifolds as those which are the target spaces for supersymmetric four-dimensional sigma models. Similarly, you can define Hodge manifolds (i.e., Kähler manifolds with integral Kähler class) as those whose associated sigma model admits a coupling to four-dimensional $N=1$ supergravity. In the same way, a hyperkähler manifold is one which is the target for an $N=2$ four-dimensional supersymmetric sigma model and a quaternionic kähler manifold as one for which the associated sigma model can be coupled to $N=2$ supergravity. These are all well-known results from the 1980s, basically. The relevant papers are those of Zumino, Bagger and Witten, Alvarez-Gaumé and Freedman, among others.
Supersymmetric sigma models are the basis for the quantum mechanical approaches to the Atiyah-Singer index theorem, based on the notion of the Witten index. Some relevant papers are those of Witten, Alvarez-Gaumé and Goodman. In fact, via the sigma model, one can understand in terms of supersymmetry much of the Hodge-Lefschetz theory of Kähler manifolds and the analogue for hyperkähler manifolds. If I may be forgiven to point to one of my papers, this is explained here.
Now one might think that this is all simply rewriting or re-interpreting geometrical notions, but the point is that there are physical techniques applicable to supersymmetric quantum field theories, which when applied to sigma models give rise to geometrically interesting constructions. The hyperkähler and quaternionic Kähler quotients were discovered in this way.
Finally, since I'm running out of time, starting from this paper of Witten's, the linear gauged sigma model became part of the toolkit of the physics approach to algebraic geometry. This is also explained in Chapter 15 of the Mirror Symmetry book (PDF file).
Added
As Igor Khavkine points out in his comment, nonlinear sigma models are such that the "coupling constants" are field dependent. In fact, for the two-dimensional nonlinear sigma model of harmonic maps $\phi: \Sigma \to M$, with action functional relative to local chart $x^i: M \cdots\to \mathbb{R}$ given by $$ \int_\Sigma g_{ij}(\phi) \partial_+ \phi^i \partial_- \phi^j d^2\sigma,$$ the metric $g_{ij}$ plays the rôle of the coupling constant. A fascinating result of Dan Friedan's PhD thesis is that the one-loop renormalisation group flow is precisely the Ricci flow of the metric. In fact, I have often speculated (admittedly from a position of ignorance) that perhaps the analytical difficulties inherent in the Ricci flow could perhaps be avoided if one were to perturb the Ricci-flow by the higher loop corrections coming from the sigma model. In other words, substitute the Ricci flow with the renormalisation group flow.
I don't know anything about the QFT side, so I'll refrain from saying things about it.
For the mathematics, one of the reasons that there aren't that many expository/introductory references for it maybe because the development of the (non-linear) theory is rather incomplete. (The linear theory is sort-of trivial: it boils down to decoupled linear wave equations.) The simplest version of the non-linear sigma model is the harmonic map/wave map systems (the former is Riemannian/elliptic, the latter is Lorentzian/hyperbolic).
Perhaps I should say a few words here to establish notation. Here sigma model generally means a Lagrangian theory of maps for $\phi: M\to N$, where $M$, endowed with a pseudo-Riemannian metric $g$, is called the source manifold, and $N$ the target. The Lagrangian density is given by $\mathcal{L} = L dvol_g$, where in index notation $L = g^{ij}k_{AB}\partial_i\phi^A\partial_j\phi^B$ where $k_{AB}$ is some symmetric tensor depending, possibly, on the map $\phi$ and its first jet.
Then the linear sigma model can be interpreted as when $N$ is some finite dimensional vector space and $k$ an inner product on $N$.
For the harmonic/wave map systems, $N$ is endowed with a Riemannian metric $h$, and $k_{AB}$ is set to be equal to the metric $h$. So we can also view $L$ to be the $g$-trace of the pull-back metric $\phi^*h$.
A lot of words have been written about harmonic maps. For an introduction, Jost's book Riemannian Geometry and Geometric Analysis has a good section on it. The notes of Helein Harmonic Maps, Conservation Laws, and Moving Frames is also quite nice. Schoen and Yau's Lectures on Harmonic Maps, as well as Eells and Lemaire's book Selected Topics in Harmonics Maps, are both very good.
One instance where the Riemannian harmonic maps have come into play is the study of stationary axisymmetric solutions to Einstein's equations in general relativity. I refer you to the works of Gilbert Weinstein or to Luc Nguyen's PhD Thesis at Rutgers University.
For the Lorentzian version, the question is much more open. In the general case, local well-posedness follows from general theory (the system of PDEs forms a semilinear hyperbolic system of equations). As far as I know, all further work (blow-up or global existence for various target manifolds) have been done only with the source manifold being the Minkowski space. A reasonably complete set of references can be found at the Dispersive Wiki. Some of the notable news recently are the blow-up results of Rodnianski-Sterbenz and Raphael-Rodnianski, and the global wellposedness results for negatively curved targets due to Tao and Krieger-Schlag (you can find all these on the arXiv).
Now, the harmonic/wave map systems can be described as the simplest of a family of nonlinear sigma models for maps between pseudo-Riemannian manifolds. Assume now $(M,g)$ and $(N,h)$ are the source and target manifolds, and $\phi: M\to N$ some map. We shall write $D^\phi$ for the (1,1)-tensor field given by $g^{-1}\circ \phi^*h$. $D^\phi$ induces at every point a linear map from $T_pM$ to itself. Note that the harmonic/wave-map Lagrangian is characterized by $L = \mathop{tr} D^\phi$, or the first invariant $\lambda_1$ of the matrix $D^\phi$. Naturally one asks whether Lagrangian field theories with the Lagrangian being (linear combinations of) other invariants $\lambda_k$ are interesting.
There are two special cases which I know that are actively studied. The case where $L = \lambda_1 + \lambda_2$ is known as the Skyrme model (there are, again, a Riemannian and a Lorentzian version depending on the signature in the source manifold). You can read a lot more about it in Manton and Sutcliffe's book Topological Solitons (for the traditional case where the target manifold is $SU(2)$ with the bi-invariant metric and the source manifold is either Minkowski space or $\mathbb{R}^3$). This model originally arose in nucleon physics. One interesting fact is that the harmonic map system from $\mathbb{R}^3\to \mathbb{S}^3$ does not admit finite energy solutions; but it looks like the Riemannian Skyrme model might (it is still an open problem). You may want to look up the works of Lev Kapitanski if you are interested in theoretical work in this direction. For the Lorentzian version not much is known (it is one of the things I am working on; a pre-print from Jared Speck and me should be available after the job application season).
The other special case which has been studied is the case where $L = (\det D^\phi)^p$ ... roughly speaking. (In the case where $M$ and $N$ are both Riemannian and have the same dimensions, this is correct; in the case where $M$ is Lorentzian and has one more dimension than $N$, the determinant should be thought of as being restricted to space-like slices [else it vanishes identically].) This is the case of non-linear elasticity and fluid dynamics (though this is not how most books in elasticity and fluids formulate their theory, the various formulations are roughly equivalent). I'm not sure who the active participants are for this model, but I vaguely remember Lars Andersson having some interest in it. (For a bit more about the mathematical set-up of this model, a good reference is, if I recall correctly, Demetrios Christodoulou's 1998 AIHP paper "On the geometry and dynamics of crystalline continua" as well as his book Action Principle and Partial Differential Equations.)
It is strange to distinguish the QFT point of view from the mathematics point of view. That's comparing apples to oranges. One can distinguish the physics (non-mathematically rigorous) point of view from that of rigorous mathematics. In this case QFT has overlaps with both. I think the distinction here is between classical sigma models, an example of geometric PDE, and quantum sigma models which involves probability on a space of fields. As Willie Wong covered classical sigma models from the mathematics point of view and José covered quantum sigma models from the physics point of view; let me add some pointers to the literature regarding quantum sigma models from the mathematics point of view.
For rigorous work on perturbative (in the sense of formal power series) renormalization of the quantum sigma model, see:
- P. K. Mitter and T. R. Ramadas, "The two-dimensional $O(N)$ nonlinear $\sigma$-model: renormalisation and effective actions", Comm. Math. Phys. 122 (1989), no. 4, 575–596.
- T. Nguyen, "Quantization of the nonlinear sigma model revisited", J. Math. Phys. 57 (2016), no. 8, 082301, 40 pp.
For rigorous nonperturbative results see:
- C. Kopper, "Mass generation in the large N-nonlinear σ-model", Comm. Math. Phys. 202 (1999), no. 1, 89–126. This proves a mass gap but in the presence of an ultraviolet cutoff.
- K. Gawędzki and A. Kupiainen, "Continuum limit of the hierarchical $O(N)$ nonlinear $\sigma$-model". Comm. Math. Phys. 106 (1986), no. 4, 533–550. This removes the ultraviolet cutoff in a finite volume but concerns a hierarchical toy model.
Also, another account from the physics point of view and related to José's answer, the Ricci flow and all that, is the article:
- M. Carfora, "Renormalization group and the Ricci flow", Milan J. Math. 78 (2010), no. 1, 319–353.