The fixed-manifold correspondence for unpunctured class-$S$ theories

Let $V_{g,0}$ be the vertex operator algebra associated with the unpunctured genus-$g$ theory, with simple modules $\{L_a\}$ and conformal dimensions $h(L_a)$. Let ${\cal M}_{g,0}$ be the moduli space of the $SU(2)/\mathbb{Z}_2$ Hitchin system, let ${\cal M}^T_{g,0}$ denote its fixed manifolds under $U(1)_r$, and let $\mu$ be the corresponding moment map. Write ${\cal M}^T_{g,0}=\{M_a\}_{0\leq a\leq g-1}$. Fixed-manifold correspondence. When $n=0$, there is a bijection between the $g$ simple modules $\{L_a\}$ of $V_{g,0}$ and the $g$ fixed manifolds ${\cal M}^T_{g,0}=\{M_a\}_{0\leq a\leq g-1}$ such that

$$h(L_a)=\mu(M_a)-g+\frac{3}{2}-\frac{1}{2}\delta_{\mu(M_a),0}.$$

Moreover, the Jordan type of the modular $T$ matrix is

$$\left[(1-\delta_{\mu(M_a),0})\dim M_a+g+1\right]_{M_a\in{\cal M}^T_{g,0}},$$

with $g$ Jordan blocks. This proposal identifies the simple modules and modular data of the chiral algebra with fixed-manifold data of the Hitchin-system moduli space; the source provides it as a proposal with supporting observations, but no resolution status is stated.