Gaiotto's Conformal Limit Construction and the Geometric Langlands Correspondence

In 2014, Gaiotto proposed a conjectural construction of the unique quantization of a spectral curve in terms of his "conformal limit" procedure of 4D N=2 SUSY gauge theory. In a geometric language, this mechanism should produce a "canonical" biholomorphic map from the moduli space of spectral curves to the moduli space of "opers." This way of interpretation of his conjecture was solved in 2016 in a joint work with Dumitrescu, Fredrickson, Kydonakis, Mazzeo and Neitzke.

A few weeks ago while sitting in A3-07, it occured to me that this map is "truly" canonical, in the sense that it is also a key step of the geometric Langlands correspondence for the case of opers. None of us who have worked along this line of geometry ever thought it had anything to do with GLC.

In this talk I will give a precise statement and key ideas of the relation between Gaiotto's map and the GLC, using elementary exposition as much as possible. The message of the talk is: "Algebra never gives you a canonical map between two isomorphic spaces." Anything "canonical" should come from either analysis, geometry, number theory, physics, or all of them together.