  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.
