  We study the electromagnetic properties of Weyl semimetals with strong
interactions. Focusing on a single Weyl cone in the band structure, we induce
strong interactions by coupling the Weyl fermion with a tunable coupling
constant $g_f$ to a quantum critical system. The critical fluctuations are
described by a conformal field theory containing fermionic operators with
scaling dimension $\Delta$. Employing the methods of the holographic
correspondence, we then derive the effective theory of the Weyl fermion in the
presence of external electric and magnetic fields. In particular, we determine
the frequency and momentum-dependent anomalous magnetic moment of the Weyl
fermions. We also determine the conductivity of the Weyl semimetal including
the vertex corrections consistent with the Ward identity. Finally, we connect
our construction to the case of Coulomb interactions in Weyl semimetals by
tuning the parameters $\Delta \rightarrow 5/2$ and $g_f^2 \rightarrow
e/\sqrt{\hbar c\epsilon_0$.
