  We use localization to compute the partition function of a four dimensional,
supersymmetric, abelian gauge theory on a hemisphere coupled to charged matter
on the boundary. Our theory has eight real supercharges in the bulk of which
four are broken by the presence of the boundary. The main result is that the
partition function is identical to that of ${\mathcal N}=2$ abelian
Chern-Simons theory on a three-sphere coupled to chiral multiplets, but where
the quantized Chern-Simons level is replaced by an arbitrary complexified gauge
coupling $\tau$. The localization reduces the path integral to a single
ordinary integral over a real variable. This integral in turn allows us to
calculate the scaling dimensions of certain protected operators and two-point
functions of abelian symmetry currents at arbitrary values of $\tau$. Because
the underlying theory has conformal symmetry, the current two-point functions
tell us the zero temperature conductivity of the Lorentzian versions of these
theories at any value of the coupling. We comment on S-dualities which relate
different theories of supersymmetric graphene. We identify a couple of
self-dual theories for which the complexified conductivity associated to the
U(1) gauge symmetry is $\tau/2$.
