  We present the results of a conformal bootstrap study of the presumed unitary
IR fixed point of quantum electrodynamics in three dimensions (QED$_3$) coupled
to $N_f=4$ two-component Dirac fermions. Specifically, we study the four-point
correlators of the $SU(4)$ adjoint fermion bilinear $r$ and the monopole of
lowest topological charge $\mathcal{M}_{1/2}$. Most notably, the scaling
dimensions of the fermion bilinear $r$ and the monopole $\mathcal{M}_{1/2}$ are
found to be constrained into a closed island with a combination of spectrum
assumptions inspired by the $1/N_f$ perturbative results as well as a novel
interval positivity constraint on the next-lowest-charge monopole
$\mathcal{M}_1$. Bounds in this island on the $SU(4)$ and topological $U(1)_t$
conserved current central charges $c_J$, $c_J^t$, as well as on the stress
tensor central charge $c_T$, are comfortably consistent with the perturbative
results. Together with the scaling dimensions, this suggests that a part of
estimates from the $1/N_f$ expansion -- even at $N_f=4$ -- provide a
self-consistent solution to the bootstrap crossing relations, despite some of
our assumptions not being strictly justified.

High Energy Physics - Theory;Statistical Mechanics;Strongly Correlated Electrons;High Energy Physics - Lattice