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We discuss the origin of the choice of global structure for six dimensional$(2,0)$ theories and their compactifications in terms of their realization fromIIB string theory on ALE spaces. We find that the ambiguity in the choice ofglobal structure on the field theory side can be traced back to a subtle effectthat needs to be taken into account when specifying boundary conditions atinfinity in the IIB orbifold, namely the known non-commutativity of RR fluxesin spaces with torsion. As an example, we show how the classification of$\mathcal{N}=4$ theories by Aharony, Seiberg and Tachikawa can be understood interms of choices of boundary conditions for RR fields in IIB. Along the way weencounter a formula for the fractional instanton number of $\mathcal{N}=4$ ADEtheories in terms of the torsional linking pairing for rational homologyspheres. We also consider six-dimensional $(1,0)$ theories, clarifying therules for determining commutators of flux operators for discrete 2-formsymmetries. Finally, we analyze the issue of global structure for fourdimensional theories in the presence of duality defects.

IIB flux non-commutativity and the global structure of field theories