Determine complete boundary RCFT characters for general monodromies

Determine the complete set of boundary rational conformal field theory characters associated with generalized S-fold SCFTs for general monodromies by identifying the full set of simple lines in the dual abelian description.

Background

The paper derives explicit characters only for generalized S-fold theories associated with particular monodromies, notably those represented by LRnL R^n and L2RnL^2R^n. For more general monodromies, the authors use Bethe-vacua methods to obtain partial modular data, but the boundary RCFT characters require knowledge of all simple lines in the corresponding TQFT.

The authors note that a Riemann–Hilbert method could help reconstruct the characters if the modular SS and TT matrices and sufficiently many initial characters are known. They further suggest that enlarging the solvable sector and identifying universal simple lines could make it possible to determine the complete character set for generic monodromies.

References

In the previous section, we proposed the central charges, conformal weights, modular $S$ matrices and characters of the Haagerup-like RCFTs only for the theories labeled by certain monodromies $\varphi$. For general monodromies $\varphi$, we cannot determine the complete set of characters because the full set of simple lines is not known.

— Non-unitary Haagerup-like TQFTs and RCFTs from generalized S-fold SCFTs  (2608.11946 - Jeong et al., 12 Aug 2026) in Section 3, opening paragraph, Section 3.1, and Discussion and Future directions, paragraph “Characters of the boundary RCFT for general ...”

In contrast, we find that the exponentiated Bethe vacua of the $\mathbb{S}(\vec{k})$ theory can be written in terms of integer powers of two principal roots of unity, $e{\frac{\pi i}{p_+}}$ and ${e{\frac{\pi i}{p_-}}$. This makes it difficult to determine which Wilson lines correspond to the decoupled TQFT, which is expected to have identical Handle-gluings based on the HF data analysis. Identifying the complete set of simple lines of the $\mathbb{S}(\vec{k})$ theory would allow us to check whether the modular matrices proposed in section~\ref{subsec: HI from S(k)} are correct by constructing the full modular $S$ matrix through the dictionary~eq: S matrix from simple lines and examining its decoupling. We leave this as an interesting direction for future work.

— 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 “Simple lines of the $\mathbb{S}(\vec{k})$ theory”