Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration

• 成果类型: Article
• 署名作者: Groechenig, Michael; Wyss, Dimitri; Ziegler, Paul
• 署名单位: University of Toronto; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Technical University of Munich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00957-8
• 发表日期: 2020
• 页码: 505-596
• 关键词: mckay correspondence; motivic integration; langlands duality; hodge numbers; VARIETIES

摘要

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type SLn and PGL(n). More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore, we prove for d prime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy-Schiffmann in the coprime case.