Abstract
We prove the Topological Mirror Symmetry Conjecture by Hausel–Thaddeus for smooth moduli spaces of Higgs bundles of type SL_n 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.
Highlights
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
Inspired by the SYZ philosophy, Hausel–Thaddeus conjectured that the moduli spaces of SLn and PGLn-Higgs bundles are mirror partners, and predicted an agreement of appropriately defined Hodge numbers
We let MePGLn be the moduli space of families of PGLn-Higgs bundles, which admit over geometric points a reduction of structure group to a GLn-Higgs bundle of degree e
Summary
We will mostly consider stringy invariants of varieties, which admit a presentation as a global quotient Y / , where Y is a smooth variety, and a finite abstract group. We say that X (a) is a finite quotient stack, if there exists an algebraic space Y with a generically fixed-point free action of an abstract finite group such that X [Y / ]. For each eigenvalue ζi there exists a unique expression ζi = ζ ci with 0 ≤ ci < r With respect to this choice we define the fermionic shift of γ at x to be the sum of fractions. This number is locally constant on Y γ , and defines a function on π0(Y γ ). We reiterate that the definition of the fermionic shift depends on the choice of a primitive root of unity ζ of order r. Definition 2.4 Let X be a smooth finite quotient stack over a field k. In the subsection we will introduce a variant of this definition, which depends on a gerbe α ∈ H 2([Y / ], μr ) on the quotient stack
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