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

Michael Groechenig; Dimitri Wyss; Paul Ziegler

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arXiv · 1707.06417

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

Michael Groechenig·Dimitri Wyss·Paul Ziegler

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Abstract

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type $\operatorname{SL}_n$ and $\operatorname{PGL}_n$. 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 $d$ 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 $d$. This proves a conjecture by Mozgovoy--Schiffman in the coprime case.

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Michael Groechenig, Dimitri Wyss, Paul Ziegler. 2017-07-20. Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration. https://arxiv.org/abs/1707.06417

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