Hausel–Rodriguez-Villegas conjecture for twisted Higgs moduli

Let $X$ be a smooth projective curve of genus $g$, let $L$ be a line bundle of degree $2g-2+p$ with $p\ge0$, and let $\mathbb L$ denote the Lefschetz motive. For each partition $\lambda$ define

$$\lH_{\lambda}^{(p)}(t)=(-1)^{p|\lambda|}t^{(1-g)(2n(\lambda)+|\lambda|)+p(n(\lambda')-n(\lambda))}\mathbb L^{pn(\lambda')}\prod_{x\in d(\lambda)}Z_X(t^h\mathbb L^a)\in\mathcal V[[t]].$$

Define $H_n^{(p)}(t)$ by

$$\sum_{\lambda}\lH_{\lambda}^{(p)}(t)T^{|\lambda|}=\operatorname{Exp}\left(\sum_{n\ge1}\frac{(-1)^{pn}t^{(1-g)n^2-p\binom n2}H_n^{(p)}(t)}{(1-t)(1-t\mathbb L)}T^n\right).$$

Hausel–Rodriguez-Villegas conjecture. The functions $H_n^{(p)}(t)$ are polynomials in $t$ and, for the moduli space $\lM=\lM(L,n,d)$ of twisted Higgs bundles of coprime rank and degree,

$$[\lM]=\mathbb L^{\dim\lM/2}H_n^{(p)}(1),$$

where $\dim\lM=2((g-1)n^2+p\binom n2+1)$.

This extends the motivic Higgs-moduli formula to $L$-twisted Higgs bundles and specializes to several motivic and cohomological invariants. The supplied text gives no resolution status.