GoH · The geometry of Higgs bundles
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2020-04-01 → 2022-11-28
- EU contribution
- €196,708
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
The geometry of Higgs bundles
The objects of study of GoH, Higgs bundles, appear as solutions to differential equations motivated by physics. They have become central in both mathematics and theoretical physics, both for their rich geometric properties and their applicability to crucial programmes such as mirror symmetry and geometric Langlands. This project aims at analysing some geometric properties of Higgs bundles meaningful both to improve the understanding of their geometry and towards applications. More precisely, Higgs bundles are solutions to equations. These equations have symmetry that takes one solution to an equivalent one. The space that classifies solutions up to equivalence is called the moduli space of Higgs bundles. The moduli space can be contracted to a subset called the nilpotent cone which captures a lot of its geometry. This space is not fully understood. This project addresses the study of some key characters therein: wobbly bundles. Wobbly bundles and their counterpart (very stable bundles) control some dynamical properties of the moduli space. According to Drinfeld’s conjecture, they moreover provide some interesting invariant of the moduli space (line bundles). Invariants are quantities that stay the same under equivalence. They can therefore be used to discard that two spaces be the same by showing that their invariants do not match. Finally, Donagi—Pantev’s conjecture claims the equality of wobbly and shaky bundles, the latter of which are crucial in geometric Langlands. The main goals for this project were the proof of Donagi—Pantev’s conjecture, Drinfeld’s conjecture in rank three, as well as the analysis of the nilpotent cone, and the generalisation of these notions in terms of Higgs bundles for real forms and in positive characteristic. In the course of this action, the researchers have accomplished the proof of the first two conjectures. They have moreover deepened the understanding of some generalisations of wobbliness by Hausel—Hitchin, proving a conjecture of theirs in rank three. Finally, progress has been made towards applications to mirror symmetry and in the context of real forms.s been made towards applications to mirror symmetry and in the context of real forms.
Data: CORDIS, © European Union
Project objective
Higgs bundles play a fundamental role in the current panorama of mathematics and theoretical physics through their many connections. Amongst the latter is the link with the geometric Langlands programme, a suitable generalization of the relation between a curve and its Picard variety, which moreover admits a natural quantum field theoretical interpretation. According to this, any G- local system on a curve yields a perverse sheaf on the moduli stack of G*-bundles (where G* is the Langlands dual to G). A simpler (abelianised) version of the geometric Langlands programme has been proven for Higgs bundles by Donagi and Pantev. A programme initiated by these two scientists aims at inducing the full Langlands correspondence from its abelianised version. Building on the work of the researcher and the hosts, we will fill in the gaps of this program and provide alternative tools broadening the current state of the art also beyond this action. In doing so, we will study central elements of the geometry of Higgs bundles from a new perspective. More precisely, we will give a way to understand the Bialynicki-Birula stratification via algebraic techniques, and, related to that, carefully study the irreducible components of the nilpotent cone, applying also the theory of SU(p,q)-Higgs bundles. Finally, we will explore the case of positive characteristic, with the aim to shed light on the Hecke eigenproperty in this setting.
Original text from CORDIS.
Participants
- UNIVERSITE COTE D'AZUR · NiceCoordinatorFrance
Links
Data: CORDIS, © European Union