Abstract
In this chapter quantum noncommutative \(\Phi ^{4}\) theories on Moyal-Weyl spaces, the noncommutative fuzzy torus, and the fuzzy spheres S N 2 and S N 2 ×S N 2 are presented. This includes analytical results such as the UV-IR mixing, the stripe phase, the exact solution of the self-dual theory, as well as Monte Carlo results such as the phase structure on the fuzzy sphere, and the dispersion relation on the noncommutative fuzzy torus. Other results such as quantum noncommutative \(\Phi ^{4}\) theory on fuzzy S2 ×S2 and the Wilson renormalization group approach to noncommutative \(\Phi ^{4}\) in the Moyal-Weyl picture and in the matrix basis at the self-dual point are also briefly discussed.
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