The orbifold quantum-dimension conjecture for the Heisenberg vertex algebra

Let $\mathcal{H}(3)$ be the Heisenberg vertex operator algebra, let $S_3$ act on it by permutations, and let $M$ be an irreducible ordinary $\mathcal{H}(3)^{S_3}$-module. For $g\in S_3$, call a module a $g$-twisted module if it is a $g$-twisted $\mathcal{H}(3)$-module. The quantum dimension of a module $M$ is

$$\operatorname{qdim}[M]:=\lim_{t\to 0^+}\frac{\operatorname{ch}[M](it)}{\operatorname{ch}[\mathcal{H}(3)^{S_3}](it)}.$$

Orbifold quantum-dimension conjecture. Every irreducible ordinary $\mathcal{H}(3)^{S_3}$-module $M$ appears in the decomposition of a $g$-twisted $\mathcal{H}(3)$-module for some $g\in S_3$. Moreover,

$$\operatorname{qdim}(M)\in\{1,2,6,+\infty\}.$$

The conjecture predicts both that all irreducible ordinary modules arise from twisted sectors and that their quantum dimensions are restricted to four values. The preceding computations establish these values for several modules, while the remaining assertion was stated as to be addressed in the cited future work.