Exact partition functions for gauge theories on \(\mathbb{R}_\lambda^3\). (English) Zbl 1349.81173
Summary: The noncommutative space \(\mathbb{R}_{\lambda}^3\), a deformation of \(\mathbb{R}^3\), supports a 3-parameter family of gauge theory models with gauge-invariant harmonic term, stable vacuum and which are perturbatively finite to all orders. Properties of this family are discussed. The partition function factorizes as an infinite product of reduced partition functions, each one corresponding to the reduced gauge theory on one of the fuzzy spheres entering the decomposition of \(\mathbb{R}_{\lambda}^3\). For a particular sub-family of gauge theories, each reduced partition function is exactly expressible as a ratio of determinants. A relation with integrable 2-D Toda lattice hierarchy is indicated.
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 58B34 | Noncommutative geometry (à la Connes) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
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