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

Let $V_{g,n}$ be the vertex operator algebra associated with the genus-$g$ theory with $n$ regular punctures, and let $\{L_a\}$ be its simple modules. Let ${\cal M}_{g,n}$ be the moduli space of the $SU(2)/\mathbb{Z}_2$ Hitchin system, with fixed manifolds ${\cal M}^T_{g,n}=\{M_a\}$ under $U(1)_r$ and moment map $\mu$. Let the punctures have parabolic parameters satisfying $0<\alpha_n<\cdots<\alpha_1<1/2$, let $\mu_{\max}$ be the maximum critical value of the moment map, let $\omega^{(i)}$ be the fundamental weight of the $i$-th flavor $\mathfrak{su}(2)$, and let the highest-weight state of $L_a$ have conformal dimension $h(L_a)$ and flavor representation $[\lambda_1,\lambda_2,\ldots,\lambda_n]$. Fixed-manifold correspondence. When $n>0$, there is a bijection between the simple modules $\{L_a\}$ of $V_{g,n}$ and the fixed manifolds ${\cal M}^T_{g,n}=\{M_a\}$ such that, for even $n$,

$$\left.\mu_{M_a}-\mu_{\max}\right|_{\alpha_i\mapsto-\omega^{(i)}}=-h(L_a)+\sum_{i=1}^n\lambda_i\omega^{(i)},$$

and, for odd $n$,

$$\left.\mu_{M_a}-\mu_{\max}\right|_{\alpha_i\mapsto\frac{1}{2}+\omega^{(i)}}=-h(L_a)+\sum_{i=1}^n\lambda_i\omega^{(i)}.$$

Moreover, the Jordan type of modular $T$ (respectively $STS$) is

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

for even (respectively odd) $n$. This proposal extends the proposed correspondence between VOA modules and fixed manifolds to regular-puncture theories, relating conformal and flavor data to moment-map values and modular Jordan types; the source gives no evidence resolving it.