Endoscopic decompositions and the Hausel–Thaddeus conjecture

Abstract

We construct natural operators connecting the cohomology of the moduli spaces of stable Higgs bundles with different ranks and genera which, after numerical specialisation, recover the topological mirror symmetry conjecture of Hausel and Thaddeus concerning $\mathrm {SL}_n$ - and $\mathrm {PGL}_n$ -Higgs bundles. This provides a complete description of the cohomology of the moduli space of stable $\mathrm {SL}_n$ -Higgs bundles in terms of the tautological classes, and gives a new proof of the Hausel–Thaddeus conjecture, which was also proven recently by Gröchenig, Wyss and Ziegler via p-adic integration. Our method is to relate the decomposition theorem for the Hitchin fibration, using vanishing cycle functors, to the decomposition theorem for the twisted Hitchin fibration, whose supports are simpler.

Motivic mirror symmetry for Higgs bundles

“…Orlov's conjecture can also reasonably be extended to smooth proper Deligne-Mumford stacks, replacing Chow motives with orbifold Chow motives [27,Definition 2.5], in which case it is related to the generalized McKay correspondence and the motivic hyperkähler resolution conjecture [27,Conjecture 3.6]. One might speculate that Orlov's conjecture could be extended even further to a non-proper, twisted set-up like (1), and thus that the main result of [50] and our Theorem 1.1 is a natural prediction of an extension of Orlov's conjecture applied to (1).…”

“…The conjecture of Hausel and Thaddeus was proved by Groechenig, Wyss and Ziegler [32] using p-adic integration. Recently, Maulik and Shen [50] upgraded the agreement of (orbifold) E-polynomials to an agreement of (orbifold) Hodge structures (see (3) below) using perverse sheaves, the decomposition theorem, support theorems for Hitchin fibrations and vanishing cycles.…”

“…In this paper, we build on the ideas and techniques of [50] to prove a motivic version of mirror symmetry, Theorem 1.1, relating the (orbifold) Voevodsky motives of the SL n -Higgs and the PGL n -Higgs moduli spaces. The Voevodsky motive M (X) of a smooth k-variety X with coefficient in a Q-algebra Λ is an object of the triangulated category DM(k, Λ) of mixed motives over k with coefficients in Λ.…”

“…Γ-action on cohomology and motives. In both [32] and [50], a crucial ingredient was the Γ-action on cohomology and its isotypical decomposition. The corresponding motivic decomposition will also play a fundamental rôle in our proof.…”

“…The isotypical pieces in the decomposition associated to the Γ-action on the cohomology of the SL-Higgs moduli space M L were described in [50] in terms of isotypical pieces in the cohomology of the fixed locus M γ := (M L ) γ for elements γ ∈ Γ. This fixed locus comes with a restricted Hitchin map h γ :…”