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We study the commutative limit of the non-commutative maximallysupersymmetric Yang-Mills theory in four dimensions (N=4 SYM). The commutativelimits of non-commutative spaces are important in particular in theapplications of non-commutative spaces for regularisation of supersymmetrictheories (such as the use of non-commutative spaces as alternatives to latticesfor supersymmetric gauge theories and interpretations of some matrix models asregularised supermembrane or superstring theories), which in turn can play aprominent role in the study of quantum gravity via the gauge/gravity duality.In general, the commutative limits are known to be singular and non-smooth dueto UV/IR mixing effects. We give a direct proof that UV effects do not breakthe continuity of the commutative limit of the non-commutative N=4 SYM to allorder in perturbation theory, including non-planar contributions. This isachieved by establishing the uniform convergence (with respect to thenon-commutative parameter) of momentum integrals associated with all Feynmandiagrams appearing in the theory, using the same tools involved in the proof offiniteness of the commutative N=4 SYM.

On the continuity of the commutative limit of the 4d N=4 non-commutative super Yang-Mills theory