Summary: Nonlinear sigma models with non-compact target space and non-amenable symmetry group were introduced long ago in the study of disordered electron systems. They also occur in dimensionally reduced quantum gravity; recently they have been considered in the context of the AdS/CFT correspondence. These models show spontaneous symmetry breaking in any dimension, even one and two (superficially in contradiction with the Mermin-Wagner theorem) as a consequence of the non-amenability of their symmetry group. The low-dimensional models show other peculiarities: invariant observables remain dependent on boundary conditions in the thermodynamic limit and the Osterwalder-Schrader reconstruction yields a non-separable Hilbert space. The ground state space, however, under quite general conditions, carries a unique unitary and continuous representation. The existence of a continuum limit in 2D is an open question: while the perturbative Renormalization Group suggests triviality, other arguments hint at the existence of a conformally invariant continuum limit at least for suitable observables.
This talk gives an overview of the work done during the last several years in collaboration first of all with Max Niedermaier, some of it also with Peter Weisz and Tony Duncan [M. Niedermaier and the author, Ann. Henri Poincaré 6, No. 6, 1025–1090 (2005; Zbl 1088.82007); A. Duncan, M. Niedermaier and the author, Nucl. Phys., B 720, No. 3, 235–288 (2005; Zbl 1194.81144); M. Niedermaier and the author, Commun. Math. Phys. 270, No. 2, 373–443 (2007; Zbl 1116.22005); M. Niedermaier, the author and P. Weisz, Nucl. Phys. B 788, No. 3, 89–119 (2008), arXiv:hep-th/0703212; M. Niedermaier and the author, J. Math. Phys. 49, No. 7, Article No. 073301, 12 p. (2008; Zbl 1152.81571)].
For the entire collection see [Zbl 1207.81005].