Opers versus nonabelian Hodge

Abstract

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point $\mathbf{u}$ of the base of Hitchin's integrable system for $(G,C)$. One family ${\nabla }_{\hslash ,\mathbf{u}}$ consists of $G$-opers, and depends on $\hslash \in {\mathbb{C}}^{\times }$. The other family ${\nabla }_{R,\hat{I}\P ,\mathbf{u}}$ is built from solutions of Hitchin's equations, and depends on $\hat{I}\P \in {\mathbb{C}}^{\times },R\in {\mathbb{R}}^{+}$. We show that in the scaling limit $R\to 0$, $\hat{I}\P =\hslash R$, we have ${\nabla }_{R,\hat{I}\P ,\mathbf{u}}\to {\nabla }_{\hslash ,\mathbf{u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.

23 pages