Opers versus nonabelian Hodge
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Opers versus nonabelian Hodge

  • Author(s): Dumitrescu, Olivia
  • Fredrickson, Laura
  • Kydonakis, Georgios
  • Mazzeo, Rafe
  • Mulase, Motohico
  • Neitzke, Andrew
  • et al.

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 $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The other family $\nabla_{R,\zeta,{\mathbf u}}$ is built from solutions of Hitchin's equations, and depends on $\zeta \in {\mathbb C}^\times, R \in {\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $\zeta = \hbar R$, we have $\nabla_{R,\zeta,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.

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