Conjecture: A distinct fixed point of the two‑field order‑q GL theory describes M(q, 3q+1) (D‑series)

Establish that, for odd integer q, the two‑field Euclidean scalar theory with real fields φ and σ and all marginal interactions of total order q with imaginary coupling constants (including terms σ φ^{q−1}, σ^3 φ^{q−3}, …, σ^{q} consistent with discrete symmetries) possesses an RG fixed point that realizes the D_{(3q+3)/2} modular invariant of the minimal model M(q, 3q+1) in two dimensions.

Background

Building on the identification of M(3,10) and M(3,8) with fixed points of a two‑field cubic GL theory, the authors define a general two‑field action with imaginary couplings and interactions of order q, designed to be renormalizable at the upper critical dimension d_c(q)=2q/(q−2).

They argue that one fixed point matches the D_{(3q+1)/2} modular invariant of M(q, 3q−1) and conjecture that a different imaginary fixed point of the same theory matches the D_{(3q+3)/2} modular invariant of M(q, 3q+1), generalizing the known q=3 case.