Abstract:
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 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 τ . 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 τ . 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 τ /2.
