The Fermion Doubling Problem and Noncommutative Geometry

We propose a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its Cartesian products. The nonperturbative formulation of chiral gauge theories is a long standing programme in particle physics. It seems clear that one should regularise these theories with all symmetries intact. There are two major problems associated with conventional lattice approaches to this programme, both with roots in topological features: (1) The well-known Nielsen-Ninomiya theorem [1] states that if we want to maintain chiral symmetry, then under plausible assumptions, one cannot avoid the doubling of fermions in the usual lattice formulations . (2) It is not straightforward to understand features like anomalies with naive ultraviolet cut-off. Recently a novel approach to discrete physics has been developed. It works with quantum fields on a “fuzzy space” MF obtained by treating the underlying manifoldM as a phase space and quantising it [2–9]. The earliest contributions to topological features of the emergent fuzzy physics came from Grosse, Klimč́ık and Prešnajder [4]. They dealt with monopoles and chiral anomaly for the fuzzy two-sphere S F and took particular advantage of supersymmetry . Later Baez et al. [10] further elaborated on their monopole work and developed the fuzzy physics of σ− models and their solitons using cyclic cohomology [11, 12]. An attractive feature of this cohomological approach