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We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus forsmooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and$\operatorname{PGL}_n$. More precisely, we establish an equality of stringyHodge numbers for certain pairs of algebraic orbifolds generically fibred intodual abelian varieties. Our proof utilises p-adic integration relative to thefibres, and interprets canonical gerbes present on these moduli spaces ascharacters on the Hitchin fibres using Tate duality. Furthermore we prove for$d$ coprime 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 $d$. Thisproves a conjecture by Mozgovoy--Schiffman in the coprime case.

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