Anomalies and Schwinger terms in NCG field theory models. (English) Zbl 1018.81033
Summary: This talk is based on the paper [E. Langmann, J. Mickelsson and S. Rydth, J. Math. Phys. 42, 4779-4801 (2001; Zbl 1017.58018)], which contains references to the original literature. We study the quantization of chiral fermions coupled to generalized Dirac operators arising in NCG Yang-Mills theory. The cocycles describing chiral symmetry breaking are calculated. In particular, we introduce a generalized locality principle for the cocycles. Local cocycles are by definition expressions which can be written as generalized traces of operator commutators. In the case of pseudodifferential operators, these traces lead in fact to integrals of ordinary local de Rham forms. As an application of the general ideas we take the case of noncommutative tori.
MSC:
81T10 | Model quantum field theories |
81T50 | Anomalies in quantum field theory |
58J42 | Noncommutative global analysis, noncommutative residues |
81T13 | Yang-Mills and other gauge theories in quantum field theory |
81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
81T75 | Noncommutative geometry methods in quantum field theory |
58A12 | de Rham theory in global analysis |