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 specialization, recover the topological mirror symmetry conjecture of Hausel-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, proven recently by Gröchenig-Wyss-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.

v5: corrected a mistake regarding the $G_{\pi }$-equivariant structure; the essential changes are Prop 4.6, Cor 4.8, and the extra assumption that deg(D) is even in Sec 3. All the main theorems in the introduction are unchanged. v4: accepted version at Forum of Mathematics Pi

References in corpus (7)

• Mirror symmetry, Langlands duality, and the Hitchin system
• Global topology of the Hitchin system
• Unramified covers and branes on the Hitchin system
• Sur la cohomologie de certains espaces de modules de fibrés vectoriels
• Towards a Global Springer Theory III: Endoscopy and Langlands duality
• On the P=W conjecture for ${\mathrm{SL}}_n$
• Fibration de Hitchin et endoscopie

Cited by in corpus (1)

• Unramified covers and branes on the Hitchin system