On the rational homotopy type of a moduli space of vector bundles over a curve

We study the rational homotopy of the moduli space ${\mathcal{N}}_X$ of stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface $X$ of genus $g\ge 2$. The symplectic group $Aut(H_1(X,\mathbb{Z}))=Sp(2g,\mathbb{Z})$ has a natural action on the rational homotopy groups ${\pi }_n({\mathcal{N}}_X)\otimes \mathbb{Q}$. We prove that this action extends to an action of $Sp(2g,\mathbb{C})$ on ${\pi }_n({\mathcal{N}}_X)\otimes \mathbb{C}$. We also show that $\pi_n({\mathcal N}_X) \otimes {\mathbb C}$isanon-trivial$Sp(2g,{\mathbb C})$-representation for each $n\ge 2g-1$. In particular, ${\mathcal{N}}_X$ is a rationally hyperbolic space. In the special case where $g=2$, we compute the leading $Sp(2g,\mathbb{C})$-representation occurring in $\pi_n({\mathcal N}_X) \otimes {\mathbb C}$,foreach$n$.