Mirror symmetry for the parabolic G-Higgs bundle, from local to global

Abstract: Motivated by geometric Langlands, we initiate a program to study the mirror symmetry for the moduli space of parabolic G-Higgs bundles. This talk will focus on $G=\textrm{Sp}_{2n}$ and its Langlands dual $\textrm{SO}_{2n+1}$. Our goal is to prove the SYZ mirror symmetry and topological mirror symmetry (TMS). The parabolic structure of the parabolic Higgs bundle is related to the nilpotent orbit closure. So we need to first figure out the mirror pair for nilpotent orbits. Classically, there is a famous Springer duality between special orbits. Therefore, it is natural to speculate that the mirror symmetry we seek may coincide with Springer duality in the context of special orbits. Unfortunately, such a naive statement fails. To remedy the situation, together with Prof. Ruan and Prof. Fu (arXiv:2207.10533), we propose a conjecture which asserts the mirror symmetry for certain parabolic/induced covers of special orbits. Then, we prove the conjecture for Richardson orbits and obtain certain partial results in general. After understanding the mirror parabolic structures, together with W. He, X. Su, B. Wang, X. Wen, we are working in progress to prove the SYZ and TMS for the moduli space of parabolic $\textrm{Sp}_{2n}/\textrm{SO}_{2n+1}$-Higgs bundles with dual parabolic structures.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html