# 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 $${\mathcal {L}}={1 \over 2}g(\partial ^{\mu }\Sigma ,\partial _{\mu }\Sigma )-V(\Sigma )},$$ 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!

## 1 Answer

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;%%" }

• Is the theory "unitary"? many thanks +1. – wonderich Oct 29 '18 at 22:05
• It depends what you mean by unitary. The conformal field theory associated to this WZW model is a module over the loop algebra of the Nappi-Witten Lie algebra (which is solvable) and it has negative norm states. However, one can build an exact bosonic string background, say, by adding 22 free scalars and by restricting to a subclass of representations one can arrive at a no-ghost theorem. See, e.g., arxiv.org/abs/hep-th/9503222 – José Figueroa-O'Farrill Oct 29 '18 at 22:29