This is slightly an open-ended invitation to discuss references and reasons for excitement about the linear and non-linear sigma model. 


I gauge from some other interactions that it has considerable interest even in mathematics (geometry?) apart from QFT. 


I would like if people can type in what are the canonical explanatory/expository references about the linear and non-linear sigma model from both the QFT and mathematical points of view. It would be illuminating if experts can shed light on why is this model exciting again from both the QFT and mathematics point of view. 


{As an example of the kind of expository reference I am looking for I would like to mention [this one](http://arxiv.org/abs/hep-ph/9802419) as an example for another interesting model in QFT} 



Linear/Non-linear sigma model
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](http://www.math.uab.edu/weinstei/index.shtml) 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](http://tosio.math.utoronto.ca/wiki/index.php/Wave_maps). 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*.)


