Searching for gauge theories with the conformal bootstrap

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 $\mathrm{\Delta }=D\mathrm{/}2-1$ and $\mathrm{\Delta }=D-1$ 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 $\mathrm{\Delta }=1\mathrm{/}2$ at large $N$. We show that the bootstrap bounds also admit another infinite family of kinks ${\mathcal{T}}_D$, which at large $N$ approach solutions containing free fermion bilinears at $\mathrm{\Delta }=D-1$ from below. The kinks ${\mathcal{T}}_D$ appear in general dimensions with a $D$-dependent critical $N^{\ast }$ 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 ${\mathcal{T}}_D$ are subgroups of $SO(N)$, and we speculate that the kinks ${\mathcal{T}}_D$ relate to the fixed points of gauge theories coupled to fermions.