Periodic Reporting for period 1 - GoH (The geometry of Higgs bundles)
Periodo di rendicontazione: 2020-04-01 al 2022-03-31
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.
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.
The work was performed through 3 work packages:
The first one was designed to prove Donagi-Pantev's and Drinfeld's conjectures, which was fully achieved in the smooth moduli space (the former) and for rank three bundles (the latter).
In a second work package, a generalisations of wobbliness as defined by Hausel and Hitchin is explored, as well as applications to mirror symmetry. Criteria for wobbliness for higher fixed points are proven, which solves a conjecture by Hausel and Hitchin. This is important given their ongoing programme to compute some geometric properties (multiplicities) in the nilpotent cone, to which some obstructions exist which are totally identified by the results of GoH. Regarding mirror symmetry, this is a vast programme involving some dualities in mathematics and physics. These are equivalences of objects a priori totally different in nature. Hence, mirror symmetry is a powerful engine. The results by the researcher and secondment collaborators provide some branes flowing to wobbly bundles, of which few examples are known. These are moreover specially important, as they are related to global invariants of the moduli space.
The third work package studies wobbliness relative to some meaningful subspaces in mirror symmetry. We are able to classify wobbly fixed point components for U(p,q)-Higgs bundles and show that these are also wobbly components in the whole moduli space. We compute their equivariant multiplicities, which as applications yields obstructions to existence of very stable components and emptiness of fixed points.
The results have been disseminated through conferences and seminars (a total of 16 events in 17 months) including a the conferences Quantum Fields, Geometry and Representation Theory 2021 (Bangalore, India) and Quantisation of moduli spaces (Oberwolfach, 2022), COW seminar and Oxford seminar.
The first one was designed to prove Donagi-Pantev's and Drinfeld's conjectures, which was fully achieved in the smooth moduli space (the former) and for rank three bundles (the latter).
In a second work package, a generalisations of wobbliness as defined by Hausel and Hitchin is explored, as well as applications to mirror symmetry. Criteria for wobbliness for higher fixed points are proven, which solves a conjecture by Hausel and Hitchin. This is important given their ongoing programme to compute some geometric properties (multiplicities) in the nilpotent cone, to which some obstructions exist which are totally identified by the results of GoH. Regarding mirror symmetry, this is a vast programme involving some dualities in mathematics and physics. These are equivalences of objects a priori totally different in nature. Hence, mirror symmetry is a powerful engine. The results by the researcher and secondment collaborators provide some branes flowing to wobbly bundles, of which few examples are known. These are moreover specially important, as they are related to global invariants of the moduli space.
The third work package studies wobbliness relative to some meaningful subspaces in mirror symmetry. We are able to classify wobbly fixed point components for U(p,q)-Higgs bundles and show that these are also wobbly components in the whole moduli space. We compute their equivariant multiplicities, which as applications yields obstructions to existence of very stable components and emptiness of fixed points.
The results have been disseminated through conferences and seminars (a total of 16 events in 17 months) including a the conferences Quantum Fields, Geometry and Representation Theory 2021 (Bangalore, India) and Quantisation of moduli spaces (Oberwolfach, 2022), COW seminar and Oxford seminar.
The project has reached its end, so no more results will be produced. However, a few new lines of research have been explored which will continue to be a part of the researcher’s work. In particular, in collaboration with the supervisor, they keep working on the general description of wobbly divisors. Likewise, wobbliness criteria for higher fixed points are a subject of great projection, which has already shown impact by its applications to e.g. Hausel-Hitchin’s programme. Regarding real forms, the researcher continues to study wobbliness relative to these moduli subspaces. These are all hot topics at the moment and therefore it is expected for these projects to impact the state of the art beyond the duration of this project.
Societal impact has mostly take place through popular science events (European Researcher’s Night 2021, Salon des Jeux Mathématiques 2021), many of them with a gender perspective (Celebrating women in STEM; Le Café des Matheuses Francophones). A woman’s leadership programme (Aurora Leadership Programme for Women) was also completed, which will impact the researcher's future management skills.
Societal impact has mostly take place through popular science events (European Researcher’s Night 2021, Salon des Jeux Mathématiques 2021), many of them with a gender perspective (Celebrating women in STEM; Le Café des Matheuses Francophones). A woman’s leadership programme (Aurora Leadership Programme for Women) was also completed, which will impact the researcher's future management skills.
