Caltech/USC Joint Algebra and Geometry Seminar (2/2)

The moduli spaces of Higgs bundles and flat connections on a Riemann surface are fundamental examples of holomorphic symplectic manifolds, lying at the intersection of integrable systems, geometric representation theory, and quantum field theory. In this talk, I will describe holomorphic Lagrangian subvarieties in the moduli spaces of Higgs bundles and flat connections, constructed by specifying a sub-line bundle inside a rank-n vector bundle. These Lagrangians are naturally labeled by divisors: on the Higgs side, they arise from generalized Baker–Akhiezer divisors, while on the de Rham side they correspond to apparent singularities of flat connections. I will further explain how these constructions lead to Lagrangian correspondences between the Higgs (respectively, de Rham) moduli spaces and Hilbert schemes of points on the (respectively, twisted) cotangent bundle of the Riemann surface. Connections to the geometric Langlands program and to the Kapustin–Witten framework of mirror symmetry will be briefly discussed.