Algebraic geometry of the Lagrangian correspondence of Gaiotto

From a consideration of N=2 SUYM in four dimensions, Gaiotto [1] has conjectured that a certain scaling limit of twister lines in the space of G-Higgs bundles converge to an oper. For holomorphic G-Higgs bundles, the conjecture has been settled for an arbitrary simply connected algebraic group G in 2016 in a joint paper with my collaborators [2]. Later, it was noticed in [3] that the analysis method established in [2] can be applied to the scaling limit of Gaiotto for more general stable Higgs bundles when G = SL(n), and a correspondence between certain holomorphic Lagrangians in the moduli of Higgs bundles and the moduli of holomorphic connections on a curve was discovered.

 

In this talk, I will review the known results and present the Lagrangian correspondence in the language of algebraic geometry, in the spirit of Variation of Hodge Structures of Simpson [4], modeled after SL(n)-opers.

 

[1] D. Gaiotto: Opers and TBA (2014)

[2] O. Dumitrescu, L.Fredrickson, G. Kydonakis, R. Mazzeo, M. Mulase, and A. Neitzke: From the Hitchin section to opers through nonabelian Hodge, J. Diff. Geom., to appear

[3] B. Collier and R. Wentworth: Conformal limits and the Bialyncki-Birula stratification of the space of lambda-connections, Adv. Math., to appear

[4] C. Simpson: Iterated destabilizing modifications for vector bundles with connections (2010)