  We construct 4d superconformal field theories (SCFTs) whose Coulomb branches
have singular complex structures. This implies, in particular, that their
Coulomb branch coordinate rings are not freely generated. Our construction also
gives examples of distinct SCFTs which have identical moduli space (Coulomb,
Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting
arena in which to test the relationship between moduli space geometries and
conformal field theory data.
  We construct these SCFTs by gauging certain discrete global symmetries of
$\mathcal N=4$ superYang-Mills (sYM) theories. In the simplest cases, these
discrete symmetries are outer automorphisms of the sYM gauge group, and so
these theories have lagrangian descriptions as $\mathcal N=4$ sYM theories with
disconnected gauge groups.
