Opers versus nonabelian Hodge

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 $\nabla_{\hbar,{\mathbf u}}$consistsof$G$-opers,anddependson$\hbar \in {\mathbb C}^\times$. The other family ${\mathrm{\nabla }}_{R,\zeta ,\mathbf{u}}$ is built from solutions of Hitchin’s equations, and depends on $\zeta \in {\mathbb C}^\times, R \in {\mathbb R}^+$\mathrm{.}Weshowthatinthescalinglimit$R \to 0$,$\zeta = \hbar R$, we have ${\mathrm{\nabla }}_{R,\zeta ,\mathbf{u}}\to {\mathrm{\nabla }}_{\mathrm{\hslash },\mathbf{u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.