TITLE: Nonlinear sigma models with non-compact groups / target spaces
QUESTION [5 upvotes]: A nonlinear σ model (NLSM) 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
$$
{\displaystyle {\mathcal {L}}={1 \over 2}g(\partial ^{\mu }\Sigma ,\partial _{\mu }\Sigma )-V(\Sigma )},
$$
One can also add the Wess–Zumino–Witten term 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?


Are these theories "unitary"?
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 and lectures on non-compact cases are welcome. Thanks!

REPLY [5 votes]: Of course there are.  A nice early example is the Wess--Zumino--Witten model based on a non-semisimple group admitting a bi-invariant lorentzian metric:
@article{Nappi:1993ie,
      author         = "Nappi, Chiara R. and Witten, Edward",
      title          = "{A WZW model based on a nonsemisimple group}",
      journal        = "Phys. Rev. Lett.",
      volume         = "71",
      year           = "1993",
      pages          = "3751-3753",
      doi            = "10.1103/PhysRevLett.71.3751",
      eprint         = "hep-th/9310112",
      archivePrefix  = "arXiv",
      primaryClass   = "hep-th",
      reportNumber   = "IASSNS-HEP-93-61",
      SLACcitation   = "%%CITATION = HEP-TH/9310112;%%"
}