# Nonlinear sigma models with non-compact groups / target spaces

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!