Landau-Ginzburg description of an exceptional $\mathcal{N}=1$ minimal model

Abstract

The $\mathcal{N}=1$ 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 $\mathcal{W}=g_{1}X{Y}^2/2+g_2{X}^3/6$. For $g_1=g_2$, this superpotential is known to describe a product of two $m=3$ $\mathcal{N}=1$ 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 ${\mathbb{Z}}_2$ 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 ${\mathbb{Z}}_2^R$ of the Landau-Ginzburg theory. We then treat the theory with superpotential $\mathcal{W}$ as a Gross-Neveu-Yukawa model in $d=4-\epsilon$ and find a weakly coupled infrared fixed point with $g_1/g_2=3/2+\mathcal{O}(\epsilon )$, 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 $d=3$ $\mathcal{N}=1$ superconformal field theory.

61 pages, 1 figure