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We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type and . 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 -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 coprime to , that the number of rank Higgs bundles of degree over a fixed curve defined over a finite field, is independent of . This proves a conjecture by Mozgovoy--Schiffman in the coprime case.
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Zsolt Patakfalvi, Joseph Allen Waldron