We prove the Topological Mirror Symmetry Conjecture by Hausel–Thaddeus for smooth moduli spaces of Higgs bundles of type $$SL_n$$ S L n and $$PGL_n$$ P G L 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.

中文翻译：

---

通过 p-adic 积分得到希格斯丛模空间的镜像对称

我们证明了 Hausel-Thaddeus 对 $$SL_n$$ SL n 和 $$PGL_n$$ PGL n 类型的希格斯丛的光滑模空间的拓扑镜像对称猜想。更准确地说，我们为通常纤维化为对偶阿贝尔变体的某些代数轨道对建立了弦霍奇数的等式。我们的证明利用了相对于纤维的 p-adic 积分，并使用 Tate 对偶性将这些模空间上存在的规范 Gerbes 解释为 Hitchin 纤维上的字符。此外，我们证明对于 d 到 n 的素数，在有限域上定义的固定曲线上，阶数为 d 的 n 阶希格斯丛的数量与 d 无关。这证明了 Mozgovoy-Schiffmann 在互质情况下的猜想。