General vacuum identification

Establish whether the Galois conjugate of the positive Bethe vacuum lies in the unit class for every hyperbolic once-punctured-torus monodromy word, thereby determining the unit vacuum and extending the half-index central-charge formula beyond the cases presently justified.

Background

The paper introduces a physical input asserting that the Galois conjugate of the positive vacuum belongs to the unit class. This identification is used to derive the central charge modulo 24 and to identify the unit’s handle-gluing value. It is verified in certain families and follows under some modularity hypotheses, but the paper does not establish it for arbitrary hyperbolic words.

Resolving this question would remove a principal conditional assumption in the interpretation of the Bethe vacua as modular-data objects and would make the central-charge formula unconditional within the proposed framework.

References

For $LRn$ it agrees, through the dictionary, with $S_{00}=a_0$, the value of $|Z_{S3_b}|$ in the degenerate limit (2.11), (3.107) (Remark~\ref{rem:internal}); it follows from modularity under the hypotheses of Theorem~\ref{thm:main} and of Corollary~\ref{cor:diagonal}; and it is open in general.

— Haagerup-Izumi modular data from hyperbolic words: S-fold TQFTs, central charge, torsion and the Bethe vacua  (2610.18512 - Evans, 13 Oct 2026) in Section 1.3, subsection “What is proved and what is assumed” (Section \ref{sub:status})