  A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a
$U(1)$ symmetry living on a boundary. This coupling gives rise to a continuous
family of boundary conformal field theories (BCFT) parametrized by the gauge
coupling $\tau$ in the upper-half plane and by the choice of the CFT in the
decoupling limit $\tau \to \infty$. Upon performing an $SL(2,\mathbb{Z})$
transformation in the bulk and going to the decoupling limit in the new frame,
one finds a different 3d CFT on the boundary, related to the original one by
Witten's $SL(2, \mathbb{Z})$ action [1]. In particular the cusps on the real
$\tau$ axis correspond to the 3d gauging of the original CFT. We study general
properties of this BCFT. We show how to express bulk one and two-point
functions, and the hemisphere free-energy, in terms of the two-point functions
of the boundary electric and magnetic currents. We then consider the case in
which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is
mapped to itself by a bulk $S$ transformation, and it also admits a decoupling
limit which gives the $O(2)$ model on the boundary. We compute scaling
dimensions of boundary operators and the hemisphere free-energy up to two
loops. Using an $S$-duality improved ansatz, we extrapolate the perturbative
results and find good approximations to the observables of the $O(2)$ model. We
also consider examples with other theories on the boundary, such as large-$N_f$
Dirac fermions --for which the extrapolation to strong coupling can be done
exactly order-by-order in $1/N_f$-- and a free complex scalar.
