Complete the representation theory of the \(SU(2)/\mathbb{Z}_6\) quotient

Determine the flavored modular-linear-differential equations and classify the ordinary modules and their characters for the vertex operator algebra associated with the discretely gauged \(\mathcal{N}=3\) \(SU(2)/\mathbb{Z}_6\) theory.

Background

The paper derives closed-form Schur indices and analyzes several characters for discretely gauged SU(2)SU(2) theories, including the Z4\mathbb{Z}_4 quotient. For the Z6\mathbb{Z}_6 quotient, it obtains an unflavored modular differential equation and identifies several unflavored solutions.

The flavored equations and the classification of ordinary modules are not obtained. Resolving these issues would complete the representation-theoretic analysis of the Z6\mathbb{Z}_6 quotient and clarify its relation to the corresponding ABJM6\mathrm{ABJM}_6 reduction.

References

For n=6, the flavored equations and the classification of ordinary modules remain open.

— Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity  (2609.10685 - Guo et al., 9 Sep 2026) in Discussion

Although we have not constructed the flavored MLDEs, we conjecture that the five terms \begin{equation} \frac{\vartheta'_4(\mathfrak{a})}{\vartheta_1(2 \mathfrak{a})}, \quad \frac{\vartheta_3(\mathfrak{a} + \frac{1}{6})}{\vartheta_2(2 \mathfrak{a} + \frac{1}{6})}, \quad \frac{\vartheta_4( \mathfrak{a})}{\vartheta_2(2 \mathfrak{a} + \frac{1}{6})}, \quad \frac{\vartheta_4( \mathfrak{a})}{\vartheta_1(2 \mathfrak{a} + \frac{1}{3})}, \quad \frac{\vartheta_4(\mathfrak{a} + \frac{1}{3})}{\vartheta_1(2 \mathfrak{a} + \frac{1}{3})}, \ \end{equation} in the flavored Schur index are separately flavored solutions, in complete analogy with the $n = 4$ case.

— Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity  (2609.10685 - Guo et al., 9 Sep 2026) in Section 4, subsection “Closed-form index and non-vacuum characters”