A [nonlinear σ model (NLSM)](https://en.wikipedia.org/wiki/Non-linear_sigma_model) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold  T. 

The target manifold T is equipped with a Riemannian metric g. Σ is a differentiable map from Minkowski space M (or some other space) to T.

The Lagrangian density in chiral form is given by
$$
{\displaystyle {\mathcal {L}}={1 \over 2}g(\partial ^{\mu }\Sigma ,\partial _{\mu }\Sigma )-V(\Sigma )} \mathcal{L},
$$
One can also add the [Wess–Zumino–Witten term](https://en.wikipedia.org/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model) into this NLSM.

My question is that 

> - Are there any mathematical studies and mathematical/physics uses of nonlinear sigma model (NLSM) with non-compact groups $G$ or non-compact target space T?

In all the context that I am familiar, I always deal with a NLSM of compact (Lie) groups $G$ or compact target space T. So any comments on non-compact cases are welcome. Thanks!