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In supersymmetric theories, the presence of axions usually implies the existence of a non-compact, (pseudo)moduli space.
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An example of such phenomenon is perhaps provided by the existence of so-called ( 0 , 2 ) (0,2) compactifications in the heterotic string, which imply the vanishing of infnite numbers of otherwise-allowed superpotential terms. See, for example, E. Silverstein and E. Witten, Nucl. Phys. B 444
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Axions in string theory are discussed pedagogically in M. Dine, Supersymmetry and String Theory
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Models with accidental axions as a consequence of discrete symmetries appear, for example, in A. G. Dias, V. Pleitez and M. D. Tonasse, Phys. Rev. D 67
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L. M. Carpenter, M. Dine and G. Festuccia, arXiv:0906.1273 [hep-th]
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