A remark on Gaiotto's conformal limit correspondence and the geometric Langlands correspondence
Abstract.
In this talk, I will illustrate that the Gaiotto correspondence, composed with the Hausel-Thaddeus mirror symmetry, is exactly the same as the oper case of the geometric Langlands correspondence due to Beilinson-Drinfeld. In 2014, Gaiotto conjectured a canonical correspondence between the moduli space of spectral curves for a G-Hitchin system on a curve X and the moduli of G-opers on X, where G is a simple and simply connected complex Lie group. Due to the fact that opers in this context are quantum curves to the topological recursion community, I worked with my collaborators and solved this conjecture (Dumitrescu-Fredrickson-Kydonakis-Mazzeo-Mulase-Neitzke 2016). Of course these moduli spaces are isomorphic. What was new in this work is the construction of a canonical map. When I explained this map to Gaitsgoy at IHES in 2016, he said, "I think I know the map." My talk is a 10-year-too-late report submission to his comment. Indeed, the map comes from the geometric Langlands correspondence!
