Taxonomy of coupled minimal models from finite groups

TL;DR

This paper classifies and discovers new fixed points in coupled Virasoro minimal models with various symmetry breakings, expanding the landscape of potential irrational conformal field theories with central charge greater than one.

Contribution

It rigorously classifies fixed points for N=4,5 and finds new solutions for N≥6 with specific symmetry subgroups, including sporadic and Lie-type groups.

Findings

Classified all fixed points for N=4,5.

Discovered fixed points with specific symmetry subgroups for N≥6.

Proved existence of fixed points with certain symmetry groups for all even N≥6.

Abstract

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.