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

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