Quantum field theory has been shown recently renormalizable on flat Moyal space and better behaved than on ordinary space-time. Some models at least should be completely finite, even beyond perturbation theory. In this article a first step is taken to extend such theories to non-flat backgrounds such as solvable symmetric spaces.
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Pierre Bieliavsky, Razvan G. Gurau, Vincent Rivasseau, Noncommutative field theory on rank one symmetric spaces. J. Noncommut. Geom. 3 (2009), no. 1, pp. 99–123DOI 10.4171/JNCG/32