Opers versus nonabelian Hodge
Olivia Dumitrescu, Laura Fredrickson, Georgios Kydonakis, Rafe Mazzeo, Motohico Mulase, Andrew Neitzke
TL;DR
<3-5 sentence high-level summary>We prove that two natural families of flat G-connections—the oper family ∇_{ħ,{f u}} and a Hitchin-based family ∇_{R,ζ,{f u}}—become equivalent in a precise scaling limit, thereby extending Gaiotto's conjecture to all simple, simply connected complex groups. The approach unifies Hitchin system techniques with oper theory: one builds opers from Higgs-bundle data in the Hitchin component, analyzes harmonic metrics via Hitchin’s equations, and handles the R→0 limit using a natural metric, Kostant’s SL(2) triple, and real twistor lines. The main results establish that in the scaling limit ζ=Rħ, R→0, the Hitchin-derived connections converge to SL(N) or G-opers that are equivalent to those obtained from the oper-construction, both for SL(N) and general simple G. This creates a bridge between WKB-type asymptotics and Hitchin moduli, with potential implications for the Nekrasov-Shatashvili regime and related physics.
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 $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The other family $\nabla_{R,ζ,{\mathbf u}}$ is built from solutions of Hitchin's equations, and depends on $ζ\in {\mathbb C}^\times, R \in {\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $ζ= \hbar R$, we have $\nabla_{R,ζ,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}$. This establishes and generalizes a conjecture formulated by Gaiotto.