  We search for superspace Chern-Simons-like higher-derivative terms in the low
energy effective actions of supersymmetric theories in four dimensions.
Superspace Chern-Simons-like terms are those gauge-invariant terms which cannot
be written solely in terms of field strength superfields and covariant
derivatives, but in which a gauge potential superfield appears explicitly. We
find one class of such four-derivative terms with N=2 supersymmetry which,
though locally on the Coulomb branch can be written solely in terms of field
strengths, globally cannot be. These terms are classified by certain Dolbeault
cohomology classes on the moduli space. We include a discussion of other
examples of terms in the effective action involving global obstructions on the
Coulomb branch.
