Exceptional $N=1$ minimal model as the infrared limit of a two-field cubic Landau–Ginzburg model

Let $u=(u^1,u^2)\in {\mathbb{R}}^2$ be Euclidean spacetime coordinates. Let $\theta ,\tilde{\theta }$ be independent Grassmann-odd coordinates of mass dimension $-1\mathrm{/}2$, so ${\theta }^2={\tilde{\theta }}^2=0$ and $\theta \tilde{\theta }=-\tilde{\theta }\theta$. In the source's Euclidean $d=2$ convention, take two real $N=1$ scalar superfields with component expansions

Here ${\varphi }_X,{\varphi }_Y,F_X,F_Y:{\mathbb{R}}^2\to \mathbb{R}$ are smooth real scalar profiles. Let ${\mathrm{\Lambda }}_{\mathbb{R}}$ be a real Grassmann algebra and let $\mathrm{\Lambda }={\mathrm{\Lambda }}_{\mathbb{R}}{\otimes }_{\mathbb{R}}\mathbb{C}$, with the antilinear involution $\rho (a\otimes z)=a\otimes \bar{z}$. Its odd generators anticommute. Use the real structure $\rho$ componentwise on ${\mathbb{C}}^2\otimes {\mathrm{\Lambda }}_{\bar{1}}$ and require $\rho ({\mathrm{\Psi }}_i)={\mathrm{\Psi }}_i$, where ${\mathrm{\Psi }}_i=({\psi }_i,{\tilde{\psi }}_i{)}^T$ for $i\in \{X,Y\}$. This is the Majorana reality convention for the continued Grassmann integration variables used here; the barred field below is a charge-conjugation transpose, not an independent field or a Hermitian adjoint. Take $[{\varphi }_i]=0$, $[{\mathrm{\Psi }}_i]=1\mathrm{/}2$, $[F_i]=1$, and $g_1,g_2\in \mathbb{R}$ with mass dimension one, in units $\mathrm{\hslash }=1$. The classical action is evaluated on smooth fields with boundary conditions making its spacetime integral finite; the existence of the continuum quantum theory is not assumed as a proved fact. The cubic superpotential is

Choose ${\gamma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and ${\gamma }_2=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$, so $\{{\gamma }_{\mu },{\gamma }_{\nu }\}=2{\delta }_{\mu \nu }I$. These real matrices preserve the stated real structure. Put $C=i\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, define ${\bar{\mathrm{\Psi }}}_j:={\mathrm{\Psi }}_j^{T}C$, and set $\mathrm{\partial ̸}={\gamma }^1{\mathrm{\partial }}_1+{\gamma }^2{\mathrm{\partial }}_2$. In particular the kinetic bilinear means the ordered Grassmann expression ${\bar{\mathrm{\Psi }}}_j\mathrm{\partial ̸}{\mathrm{\Psi }}_j={\sum }_{\mu =1}^2{\mathrm{\Psi }}_j^{T}C{\gamma }_{\mu }{\mathrm{\partial }}_{\mu }{\mathrm{\Psi }}_j$, with derivatives acting on the field to their right. The scalar bilinear is ${\bar{\mathrm{\Psi }}}_j{\mathrm{\Psi }}_k={\mathrm{\Psi }}_j^{T}C{\mathrm{\Psi }}_k$. This fixes both bilinears, including their common phase, in a real Clifford basis for the Euclidean continuation. Let $h_{ijk}:={\mathrm{\partial }}_i{\mathrm{\partial }}_j{\mathrm{\partial }}_k\mathcal{W}$ for polynomial derivatives in the field variables $X,Y$. The source's canonical component action after eliminating the algebraic auxiliary fields is $S_E={\int }_{{\mathbb{R}}^2}{d}^{2}u{\mathcal{L}}_E$, where

In ${\mathcal{L}}_E$, the derivatives of $\mathcal{W}$ are evaluated at $X={\varphi }_X(u)$ and $Y={\varphi }_Y(u)$. In the source's spinor convention ${\bar{\mathrm{\Psi }}}_k{\mathrm{\Psi }}_l=i({\psi }_k{\tilde{\psi }}_l+{\psi }_l{\tilde{\psi }}_k)$. The R-parity acts as $\mathrm{\Phi }(u,\theta ,\tilde{\theta })\mapsto -\mathrm{\Phi }(u,-\theta ,\tilde{\theta })$ for $\mathrm{\Phi }=(X,Y)$; equivalently it sends $({\varphi }_i,{\psi }_i,{\tilde{\psi }}_i,F_i)$ to $(-{\varphi }_i,{\psi }_i,-{\tilde{\psi }}_i,F_i)$. Determine whether the continuum theory at the proposed coupled fixed point flows in $d=2$ to the exceptional $(E_6,D_8)$, $m=12$ $N=1$ minimal-model target defined in Source-defined exceptional $N=1$ minimal-model target data, including its extended $SW(3\mathrm{/}2,5\mathrm{/}2)$ chiral algebra, R-parity grading and operator dictionary. The $N=1$ super-Virasoro subalgebra uses $N=1$ superconformal mode algebra in NS and Ramond sectors.

The target in the question is the extended super-$W_3$ model, not a theory specified only by the $N=1$ super-Virasoro algebra.

Open as of 2026-10-26. The current arXiv record lists v2, last revised 2026-09-28, and describes the identification as a proposal. Its evidence includes modular/fusion matching and a Gross–Neveu–Yukawa analysis at $d=4-ϵ$ with a weakly coupled fixed point near $g_1\mathrm{/}{g}_2=3\mathrm{/}2+O(ϵ)$; dimensions continued to $d=2$ are reported as approximate. Neither that continuation nor the modular/fusion data prove the exact two-dimensional RG endpoint, full local OPE data, or operator dictionary. The $g_1=g_2$ decoupled fixed point is the separate $m=10$, $(D_6,E_6)$ case, not the proposed target. The scope of this status check is the cited current arXiv version and targeted search recorded in batch-29 notes; it is not an exhaustive literature review.