Fix the unresolved sign in the Ramond-sector fixed-point data

Determine which of the two opposite-sign solutions for the fixed-point differences \(d_a=U_{a,R_{9/14,+}}-U_{a,R_{9/14,-}}\) gives the Ramond-sector modular matrix of the \((E_6,D_8)\) superconformal minimal model.

Background

The two Ramond fixed-point modules R9/14,+R_{9/14,+} and R9/14,−R_{9/14,-} have identical characters. Consequently, character transformations determine only the sum of their columns in the parity-weighted modular matrix UU, leaving the column differences dad_a undetermined.

Non-negative-integrality constraints from the graded Verlinde formula reduce the possibilities to two solutions related by an overall sign. Both produce the same graded fusion coefficients, so the physical choice is not resolved by the calculations presented.

References

Which of the two solutions holds is not determined here, but for definiteness we use the one that extends eq:U to the fixed-point columns, so that U_{a\rho}=\omega_a\widetilde S_{a\rho} holds on all five columns.

— Landau-Ginzburg description of an exceptional ${\mathcal N}=1$ minimal model  (2609.10972 - Nakayama et al., 10 Sep 2026) in Section 4.4, subsection “Chiral fermion parity and the Ramond sector”

Since the pair drops out of eq:twistedbr, the sign \eta is left undetermined. It appears only in operators built with \phi\pm_{1,7}, which are not in Table~\ref{tab:web}.

eq:twistedbr:

q−16∏r∈N−12(1−qr)5(1+qr)3=∑s≠7εs(∑iCi χ^hi(q)) χs(12)(q) .q^{-\frac16}\prod_{r\in\mathbb N-\frac12}(1-q^r)^5(1+q^r)^3=\sum_{s\neq7} \varepsilon_s\left(\sum_i C_i\,\hat\chi_{h_i}(q)\right)\,\chi^{(12)}_s(q)\,.

— Landau-Ginzburg description of an exceptional ${\mathcal N}=1$ minimal model  (2609.10972 - Nakayama et al., 10 Sep 2026) in Section 5.4, subsection “The spinning primaries and their charges”