As a perturbative check of the construction of 4D N = 4 supersymmetric Yang-Mills theory (SYM) from mass-deformed N = (8, 8) SYM on the 2D lattice, the one-loop effective action for scalar kinetic terms is computed in N = 4 U(k) SYM on R2× (fuzzy S2), which is obtained by expanding 2D N = (8, 8) U(N) SYM with mass deformation around its fuzzy-sphere classical solution. The radius of the fuzzy sphere is proportional to the inverse of the mass. We consider two successive limits: (1) decompactify the fuzzy sphere to a noncommutative (Moyal) plane and (2) turn off the noncommutativity of the Moyal plane. It is straightforward at the classical level to obtain the ordinary N = 4 SYM on R4 in the limits, while it is nontrivial at the quantum level. The one-loop effective action for the SU(k) sector of the gauge group U(k) coincides with that of the ordinary 4D N = 4 SYM in the above limits. Although a "noncommutative anomaly" appears in the overall U(1) sector of the U(k) gauge group, this can be expected to be a gauge artifact not affecting gauge-invariant observables.
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