Members’ Seminar

Start with a compact Riemann surface $X$ and a complex reductive group $G$, like $\mathrm{GL}(n,\mathbb C)$. According to Hitchin-Simpson's ``non abelian Hodge theory", the pair $(X,G)$ comes with two new complex manifolds: the character variety $\mathcal M_B$ and the Higgs moduli space $\mathcal{M}_\text{Dolbeault}$. When $G= \mathbb C^*$, these manifolds are two instances of the usual first cohomology group of $X$ with coefficients in the abelian $\mathbb C^*$. For general $G$, we do not have cohomology groups, but we can study the two manifolds $\mathcal M_B$ and $\mathcal M_D$. I will present some aspects of this story and discuss a new identity --$P = W$ for $G = \mathrm{GL}(2,\mathbb C)$-- occurring inside the singular cohomology of $\mathcal M_B$ and $\mathcal M_D$, where $P$ and $W$ dwell. The question as to whether the analogous identity $P = W$ holds for other $G$'s, e.g. $\mathrm{GL}(3,\mathbb C)$, is open.