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 $N\geq6$. In particular, we find non-trivial fixed points with $H=\mathbb{Z}_{N-1} \rtimes \mathbb{Z}_2, \, S_{M}\times S_{N-M}$ and rigorously prove that real fixed points with $H=(S_{N/2}\times S_{N/2})\rtimes \mathbb{Z}_2$ exist for all even $N\geq6$. We also identify fixed points with finite Lie-type symmetry $H=\rm{PSL}_2(N)\subset S_N$ 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.