Landau-Ginzburg description of an exceptional 𝒩=1 minimal model
Abstract
The 𝒩=1 superconformal minimal model with m=12 and the exceptional modular invariant (E6,D8) is the unitary minimal model of the super-W3 algebra. We propose its Landau-Ginzburg description using two real scalar superfields with the cubic superpotential cal W=g1 XY2/2 + g2X3/6. For g1=g2, this superpotential is known to describe a product of two m=3 𝒩=1 superconformal minimal models, which is the m=10 model with the (D6,E6) modular invariant. The exceptional m=12 superconformal minimal model is realized at a different fixed point of the same theory. Testing this Landau-Ginzburg description requires the fusion ring of the minimal model, which we obtain from the modular data of the extended algebra. The fusion ring has a ℤ2 grading by chiral fermion parity that the ordinary fusion coefficients do not determine. This grading, composed with conjugation, gives the generator of the R-parity ℤ2R of the Landau-Ginzburg theory. We then treat the theory with superpotential cal W as a Gross-Neveu-Yukawa model in d=4−ε and find a weakly coupled infrared fixed point with g1/g2=3/2+𝒪(ε), at which supersymmetry emerges. We also describe the renormalization group flow from this fixed point to the decoupled fixed point with g1=g2. The operator dimensions at the coupled fixed point, continued to d=2, agree approximately with their values in the m=12 superconformal minimal model. Finally, we estimate the scaling dimensions in the new interacting d=3 𝒩=1 superconformal field theory.
Related
- Topological defect trains excited by a boundary discontinuity in the two-dimensional complex Ginzburg-Landau equation
- Adiabatic Theory Data on Strongly Chirped Dissipative Solitons of the Cubic-Quintic Nonlinear Ginzburg-Landau Equation
- Type 1.5 vortex bags in single component Ginzburg-Landau models with higher order potentials
- A Mesoscopic Ginzburg--Landau Model for Vibrational Strong Coupling Enhanced Rayleigh Scattering in Molecular Liquids
- Ginzburg-Landau Theory for Non-Invertible Symmetry-Breaking Transitions
BibTeX
@misc{nakayama2026landauginzburg,
title = {{Landau-Ginzburg description of an exceptional $\mathcal N=1$ minimal model}},
author = {Yu Nakayama and Andrei Katsevich and Igor R. Klebanov and Zimo Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10972},
eprint = {2609.10972},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10972},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Landau-Ginzburg description of an exceptional 𝒩=1 minimal model
AU - Yu Nakayama
AU - Andrei Katsevich
AU - Igor R. Klebanov
AU - Zimo Sun
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.10972
UR - https://arxiv.org/abs/2609.10972
AB - The 𝒩=1 superconformal minimal model with m=12 and the exceptional modular invariant (E_6,D_8) is the unitary minimal model of the super-W_3 algebra. We propose its Landau-Ginzburg description using two real scalar superfields with the cubic superpotential cal W=g_1 XY^2/2 + g_2X^3/6. For g_1=g_2, this superpotential is known to describe a product of two m=3 𝒩=1 superconformal minimal models, which is the m=10 model with the (D_6,E_6) modular invariant. The exceptional m=12 superconformal minimal model is realized at a different fixed point of the same theory. Testing this Landau-Ginzburg description requires the fusion ring of the minimal model, which we obtain from the modular data of the extended algebra. The fusion ring has a ℤ_2 grading by chiral fermion parity that the ordinary fusion coefficients do not determine. This grading, composed with conjugation, gives the generator of the R-parity ℤ_2^R of the Landau-Ginzburg theory. We then treat the theory with superpotential cal W as a Gross-Neveu-Yukawa model in d=4−ε and find a weakly coupled infrared fixed point with g_1/g_2=3/2+𝒪(ε), at which supersymmetry emerges. We also describe the renormalization group flow from this fixed point to the decoupled fixed point with g_1=g_2. The operator dimensions at the coupled fixed point, continued to d=2, agree approximately with their values in the m=12 superconformal minimal model. Finally, we estimate the scaling dimensions in the new interacting d=3 𝒩=1 superconformal field theory.
ER -