Establish fixed-point resolution for Gepner-model simple-current extensions

Establish whether the representation of the Gepner-model superchiral algebra on a simple-current orbit with nontrivial 2-torsion decomposes into the expected number of distinct irreducible representations, thereby resolving the associated simple-current fixed points.

Background

The paper constructs representations of the Gepner-model superchiral algebra by extending representations of the minimal tensor-model algebra along simple-current orbits. When the relevant orbit stabilizer contains nontrivial 2-torsion, several copies of an identical minimal tensor-model representation occur in the induced vector space.

The authors expect these copies to split into distinct irreducible representations, but do not establish the decomposition. This is the classical fixed-point resolution problem for simple-current extensions and is important for obtaining the complete irreducible representation theory of the Gepner-model superchiral algebra.

References

The present authors expect that the representation of ${\cal A}{\rm Gepn.}$ on this vector space splits into $2{r_{shrt}}$ distinct irreducible representations; this question is known as the fixed-point resolution in the representation theory of simple current extensions, and there is an extensive literature (see e.g., , , , % , and references therein).

— Notes on Gepner Construction  (2608.26017 - Nishikawa et al., 26 Aug 2026) in Section 5.1, Through the Theory of Simple Current Extension, footnote following the construction of $YY^{\rm Gepn.}_{(\lambda,[\vec{m}_0+2\vec{y}_0])}$

Although the sign of \check S is a definite property of the theory, we will not attempt to determine it in this paper.

— Landau-Ginzburg description of an exceptional ${\mathcal N}=1$ minimal model  (2609.10972 - Nakayama et al., 10 Sep 2026) in Section 4.3, subsection “Modular data and the fixed-point resolution”