Ginzburg-Landau description of a class of non-unitary minimal models

TL;DR

The paper extends the Ginzburg-Landau description of non-unitary minimal models by formulating two-field PT-symmetric theories with imaginary order-$q$ interactions to capture the D-series invariants $M(q,3q-1)$ and $M(q,3q+1)$ for odd $q$. Building on the known $M(3,8)$ and $M(3,10)$ cases, it shows that appropriate fixed points describe pairs of Yang-Lee theories and their RG flows, and provides detailed operator identifications that match twisted-sector primaries and the associated currents. The authors propose a unified GL framework for the entire odd-$q$ D-series class, analyze RG flows in $d_c(q)$-epsilon dimensions, and illustrate with explicit examples $(q=5)$ and $(q=7)$, including fixed-point structure and potential identifications with $M(5,14)$, $M(5,16)$, $M(7,20)$, and $M(7,22)$. This approach connects modular-invariant CFT data to PT-symmetric scalar field theories, enabling perturbative control in higher dimensions and offering insights into non-unitary conformal dynamics and their anomaly-matching constraints.

Abstract

It has been proposed that the Ginzburg-Landau description of the non-unitary conformal minimal model $M(3,8)$ is provided by the Euclidean theory of two real scalar fields with third-order interactions that have imaginary coefficients. The same lagrangian describes the non-unitary model $M(3,10)$, which is a product of two Yang-Lee theories $M(2,5)$, and the Renormalization Group flow from it to $M(3,8)$. This proposal has recently passed an important consistency check, due to Y. Nakayama and T. Tanaka, based on the anomaly matching for non-invertible topological lines. In this paper, we elaborate the earlier proposal and argue that the two-field theory describes the $D$ series modular invariants of both $M(3,8)$ and $M(3,10)$. We further propose the Ginzburg-Landau descriptions of the entire class of $D$ series minimal models $M(q, 3q-1)$ and $M(q, 3q+1)$, with odd integer $q$. They involve $PT$ symmetric theories of two scalar fields with interactions of order $q$ multiplied by imaginary coupling constants.

Figures (1)

• Figure 1: One-loop renormalization of $J_\mu = \sigma\partial_\mu\phi-\phi\partial_\mu\sigma$. The diagrams represent the three-point function $\langle J_\mu \phi(p)\sigma(q)\rangle$. The solid line denotes the $\phi$ propagator and the dashed line denotes the $\sigma$ propagator.