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Anomalies and spontaneous compactification

For a particular class of multidimensional theories (D=6 Einstein-Yang-Mills SU(3) theory compactified to M4*S2 with magnetic monopole background configuration of U(1) part of SU(3)) it is proved that the effective theory in D=4 obtained by considering only massless modes is free from all kinds of anomalies if and only if the initial theory in D=6 is also free from anomalies. The argument may be easily generalised to any SU(N) gauge group.