Analyze generalized S-fold theories with even trace

Construct a limiting behavior or ansatz for the Bethe vacua of generalized S-fold SCFTs that permits analysis when the trace of the monodromy is even, with the aim of extending the solvable sector to arbitrary numbers of $T[\mathrm{SU}(2)]$ segments and testing the proposed modular $S$ matrix.

Background

The detailed Bethe-vacua analysis in the paper assumes that both Chern–Simons levels k1k_1 and k2k_2 are odd, which makes the quantities p±=tr⁡φ±2p_\pm=\operatorname{tr}\varphi\pm2 odd. When the trace of the monodromy is even, the local linear ansatz produces inconsistent equations at third order in the expansion parameter.

Resolving this issue would extend the analysis beyond the odd-level cases and could enlarge the class of generalized S-fold theories for which modular data are accessible. The authors specifically connect this extension to a possible consistency check of the proposed modular matrix for the L2RnL^2R^n family.

References

After expanding the Bethe equations eq: BE without ansatz to third order in $\e$ using the $(A,D)$-type (or $(D,A)$-type) local ansatz introduced in appendix~\ref{sec: Local ansatz appendix}, we find that an inconsistency among the equations arises when ${\rm tr}\varphi$ is even. This is why we assume that both $k_1$ and $k_2$ are odd integers. It would be interesting to find a limiting behavior(i.e., an ansatz) for the Bethe vacua that allows us to analyze the cases in which ${\rm tr}\varphi$ is even.

— Non-unitary Haagerup-like TQFTs and RCFTs from generalized S-fold SCFTs  (2608.11946 - Jeong et al., 12 Aug 2026) in Discussion and Future directions, paragraph “Bethe vacua analysis when ${\rm tr}\varphi\in2\mathbb{Z}$”