Taxonomy of coupled minimal models from finite groups

Fixed points of $N$ coupled Virasoro minimal models have recently been argued to provide large classes of compact unitary CFTs with $c>1$ and only Virasoro chiral symmetry. In this paper, we vastly increase the set of such potential irrational fixed points by considering couplings that break the maximal $G=S_N$ symmetry into various subgroups $H\subset G$. We rigorously classify all the fixed points with $N=4,5$ and do an extensive search for solutions of the beta function equations with $N\ge 6$. In particular, we find non-trivial fixed points with $H={\mathbb{Z}}_{N-1}⋊{\mathbb{Z}}_2,S_M\times S_{N-M}$ and rigorously prove that real fixed points with $H=(S_{N\mathrm{/}2}\times S_{N\mathrm{/}2})⋊{\mathbb{Z}}_2$ exist for all even $N\ge 6$. We also identify fixed points with finite Lie-type symmetry $H={\mathrm{P}\mathrm{S}\mathrm{L}}_2(N)\subset S_N$ where $N=7,11,13$ and uncover a non-unitary fixed point with $H=M_{22}\subset S_{22}$, a sporadic Mathieu group. Along the way, we encounter conformal manifolds at leading order in perturbation theory which we resolve at sub-leading order.