The renormalized mixed-Hodge-polynomial correspondence

Let $C_{g,n}$ be the character variety of the $SU(2)/\mathbb{Z}_2$ Hitchin system on the punctured surface, and let $PH_{g,n}(q)$ be its pure mixed Hodge polynomial. Define the renormalized polynomial $P_{g,n}(q)$ by

$$P_{g,n}(q)=q^{\delta_{n,0}+g(1-\delta_{n,0})+\delta_{g,0}-\frac{d_{g,n}}{2}}PH_{g,n}(q)=\sum_i a_iq^{d_i},$$

where $d_{g,n}=-(2-2g-n)\dim\mathfrak{g}$ is the dimension of $C_{g,n}$. Let ${\cal M}_{g,n}$ be the corresponding Hitchin moduli space, ${\cal M}^T_{g,n}$ its fixed manifolds, and $V_{g,n}$ the associated vertex operator algebra. Renormalized mixed-Hodge-polynomial correspondence. The total number of fixed manifolds of ${\cal M}_{g,n}$, equivalently simple modules of $V_{g,n}$, is

$$P_{g,n}(1)=\sum_i a_i,$$

while

$$\dim\mathbb{V}^{\mathrm{mod}}_{g,n}=\left.\frac{dP_{g,n}(q)}{dq}\right|_{q=1}=\sum_i a_id_i,$$

and the Jordan type of the modular matrix is $[d_i^{a_i}]$. This proposal connects the pure mixed Hodge data of the character variety to the number of VOA simple modules, the dimension of the modular representation, and its Jordan form; it invokes a conjectural mixed-Hodge-polynomial formula and the source does not state that the resulting correspondence has been proved.