On $\mathcal{N}=4$ supersymmetry enhancements in three dimensions

We introduce a class of 3d theories consisting of strongly-coupled $\mathcal{N}=4$ systems coupled to $\mathcal{N}=3$ Chern-Simons gauge multiplets, which exhibit $\mathcal{N}=4$ enhancements when a peculiar condition on the Chern-Simons levels is met. An example is the $SU(N{)}^3$ Chern-Simons theory coupled to the 3d $T_N$ theory, which enhances to $\mathcal{N}=4$ when $1\mathrm{/}{k}_1+1\mathrm{/}{k}_2+1\mathrm{/}{k}_3=0$. We also show that some but not all of these $\mathcal{N}=4$ enhancements can be understood by considering M5-branes on a special class of Seifert manifolds. Our construction provides a large class of $\mathcal{N}=4$ theories which have not been studied previously.