  Infrared fixed points of gauge theories provide intriguing targets for the
modern conformal bootstrap program. In this work we provide some preliminary
evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and
can potentially be solved with the conformal bootstrap. We start by considering
the bootstrap for $SO(N)$ vector 4-point functions in general dimension $D$. In
the large $N$ limit, upper bounds on the scaling dimensions of the lowest
$SO(N)$ singlet and traceless symmetric scalars interpolate between two
solutions at $\Delta =D/2-1$ and $\Delta =D-1$ via generalized free field
theory. In 3D the critical $O(N)$ vector models are known to saturate the
bootstrap bounds and correspond to the kinks approaching $\Delta =1/2$ at large
$N$. We show that the bootstrap bounds also admit another infinite family of
kinks ${\cal T}_D$, which at large $N$ approach solutions containing free
fermion bilinears at $\Delta=D-1$ from below. The kinks ${\cal T}_D$ appear in
general dimensions with a $D$-dependent critical $N^*$ below which the kink
disappears. We also study relations between the bounds obtained from the
bootstrap with $SO(N)$ vectors, $SU(N)$ fundamentals, and $SU(N)\times SU(N)$
bi-fundamentals. We provide a proof for the coincidence between bootstrap
bounds with different global symmetries. We show evidence that the proper
symmetries of the underlying theories of ${\cal T}_D$ are subgroups of $SO(N)$,
and we speculate that the kinks ${\cal T}_D$ relate to the fixed points of
gauge theories coupled to fermions.
