Intitulé du sujet: Root systems, global Lie theory and the irregular Deligne--Simpson problem

Sujet

Codirection:

Nombre de mois: 36 mois

Ecole Doctorale: ED 386 - Sciences Mathématiques de Paris-Centre

Unité de recherche et équipe:

Institut de Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG), UMR7586,
Equipe Groupes, Représentations et Géométrie

Coordonnées de l’équipe:

Bâtiment Sophie Germain, 8 Place Aurélie Nemours, 75205 Paris, France

Secteur: Sciences Physiques et Ingénierie / Physical sciences and Engineering

Langue attendue: Anglais

Niveau de langue attendu: B1

Description

Description du sujet:

The aim of this project is to better understand the link between Borcherds/Kac-Moody root systems [K] and
the (wild) nonabelian Hodge moduli spaces [BB] occurring in global Lie theory [B23].

This involves an intricate system of moduli space occurring in 2d gauge theory, that plays a central role, tying together several strands: i) finite dimensional algebraic integrable systems, ii) isomonodromy systems (such as the Painleve equations), iii) dynamics on (wild) character varieties, iv) harmonic bundles  (see [B17] for more background).
For example, a key question is to see how far is it possible to extend the celebrated work of Crawley-Boevey [CB] on the Deligne-Simpson problem
giving precise criteria, in terms of Kac-Moody roots, for the non-emptiness of these more general moduli spaces.
This irregular Deligne-Simpson problem (see [B14] section 9.4) is a much larger problem than the traditional tame problem
and involves a much larger class of Kac-Moody root systems (see [B24] and  [BY20], [D]). A specific aim is to extend existing
results beyond the "tame" world of star-shaped quivers, for example to the supernova quivers [B12], involving all the complete multipartite graphs, and to seek better proofs in existing cases.  
This leads to more general questions about 1) the role of non-affine Kac--Moody root systems/algebras in 2d gauge
theory, 2) the notion of global Weyl group controlling the isomorphisms between these spaces, and 3) the geometry and
topology of the wild nonabelian Hodge moduli spaces, providing tools towards their classification.


In more detail a key specific question appears as follows:

1) The works [B02,B09,B14,BY15] constructed a large class of complex algebraic Poisson varieties M_B(X), the Poisson wild character varieties, or "Betti moduli space", attached to objects X called wild Riemann surfaces. A wild Riemann surface is a triple X=(Sigma, \a, \Theta), where Sigma is a compact Riemann surface, \a is a finite subset of Sigma, and \Theta is the choice of some boundary data (an irregular class at each point of \a). Any meromorphic connection on a vector bundle on Sigma determines an irregular class at each of its poles and, in brief, the space M_B(X) parameterises the polystable connections with fixed irregular classes. A summary of the relation between M_B and meromorphic connections, involving deep work on the Riemann-Hilbert-Birkhoff correspondence starting in 1857, is in the paper [B20], and section 13 of that paper also summarises the construction of M_B(X).

If the irregular classes Theta are trivial then M_B(X) becomes the usual character variety, the moduli space of representations of the fundamental group of the punctured surface Sigma \ (\a). In general M_B(X) is the space of Stokes representation of the wild surface groupoid \Pi of X, as explained in [B14,BY15,B20,BY23]. The wild surface groupoid \Pi is a simple generalisation of the fundamental group of the punctured surface.

This algebraic construction gives the algebraic (Betti) counterpart to the earlier analytic approaches [B01] (at the symplectic level), and [BB04] (at the hyperkahler level). This last paper gave a generalisation of the strong form of nonabelian Hodge theory on curves, in effect showing that the symplectic leaves of M_B(X) have complete hyperkahler metrics, and they form a new class of complete hyperkahler manifolds (involving, for the first time, harmonic bundles of infinite energy, beyond the usual "instanton-type"/finite energy setting). High energy physicists had studied some simple examples,  stemming from the famous 1994 work of Seiberg-Witten (cf. the papers of Cherkis-Kapustin cited in [BB04]), although the physicists didn't have a rigorous construction or even a precise conjecture what the right general boundary conditions for the moduli problem were. After the construction [BB04], Witten linked them to the wild geometric Langlands [W08], and Gaiotto and others defined a new class of physical theories "Class S" that in some sense "explain" some of the other hyperkahler manifolds of [BB04] (as their "Coulomb branch compactified on R^3xS^1").


2) the symplectic leaves M_B(X,C) of each wild character variety M_B(X) are obtained by fixing the conjugacy classes C of formal monodromy at each marked point, and it is these leaves that support the hyperkahler metrics.
Thus from a pure mathematics viewpoint we have a quite concrete algebraic construction  (X,C) --> M_B(X,C)  attaching a symplectic variety to the choice of the data (X,C).  
The irregular Deligne-Simpson problem is the question of determining exactly which data (X,C)
lead to nonempty spaces M_B(X,C), or more precisely so that M_B(X,C) has an irreducible point, as in [B14] section 9.4.
This question is the natural generalisation of the original question (that arose in work of Deligne and Simpson [Sim91] somewhat before
the construction of the relevant tame hyperkahler metrics [Kon93, Nak96].
In [B12,B15] some important examples of the irregular Deligne-Simpson question are related to the root systems of a large class of non-affine Kac-Moody root systems, showing in particular how some of the work of Crawley-Boevey [CB] extends beyond the star-shaped case (i.e. to a large class of generalised Cartan matrices whose (generalised) Dynkin graphs are not star-shaped, e.g. as in the many drawings in [B24]). The basic aim now it prove some of the conjectures in these papers and extend to more general cases.

Compétences requises:

1) knowledge of meromorphic connections on vector bundles on smooth algebraic curves
2) basic definition related to Kac-Moody root systems and Weyl groups
3) some familiarity with the general definition of monodromy/Stokes data of linear connections and
the approach via Stokes automorphisms and wild surface groupoids would be useful.
4) some experience with geometric aspects of nonlinear algebraic differential equation such as the Painleve equations would be helpful
5) elementary complex algebraic geometry and moduli theory

Références bibliographiques:

[BB] O. Biquard, and P. Boalch
Wild non-abelian Hodge theory on curves.
Compos. Math. 140, No. 1, 179-204 (2004).

[B01] P. Boalch
Symplectic manifolds and isomonodromic deformations, Adv. in Math. 163 (2001), 137–205.

[B02] P. Boalch
Quasi-Hamiltonian geometry of meromorphic connections, arXiv:0203161 (2002), Duke Math. J. 139 (2007) no. 2, 369–405.

[B09] P. Boalch
Through the analytic halo: Fission via irregular singularities, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 7, 2669–2684.

[B12] P. Boalch
Simply-laced isomonodromy systems.
Publ. Math., Inst. Hautes Étud. Sci. 116, 1-68 (2012).

[B14] P. Boalch
Geometry and braiding of Stokes data; fission and wild character varieties.
Ann. Math. (2) 179, No. 1, 301-365 (2014).

[B15] P. Boalch
Global Weyl groups and a new theory of multiplicative quiver varieties, Geometry and Topology 19 (2015), 3467–3536.

[B17] P. Boalch
Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams.
Andersen, Jørgen Ellegaard (ed.) et al., Geometry and physics. A festschrift in honour of
Nigel Hitchin. Oxford University Press. 433-454 (2018). arXiv:1703.10376

[B20] P. Boalch
Topology of the Stokes phenomenon, Integrability, Quantization, and Geometry. I, Proc. Sympos. Pure Math., vol. 103, Amer. Math. Soc., 2021, pp. 55–100.

[B23] P. Boalch
First steps in global Lie theory: wild Riemann surfaces, their character varieties and topological symplectic structures
Talk at Collège de France 2023 (in Séminaire Ngô), https://youtu.be/NxemDWvwgOM
Slides: https://webusers.imj-prg.fr/~philip.boalch/exposes.html

[B24] P. Boalch
Counting the fission trees and nonabelian Hodge graphs
arXiv:2410.23358

[BY15] P. Boalch and D. Yamakawa
Twisted wild character varieties, arXiv:1512.08091, 2015.

[BY20] P. Boalch and D. Yamakawa
Diagrams for nonabelian Hodge spaces on the affine line
C. R., Math., Acad. Sci. Paris 358, No. 1, 59-65 (2020)

[CB] W. Crawley-Boevey
Quiver algebras, weighted projective lines, and the Deligne-Simpson problem
Proceedings of ICM 2006 Madrid, arXiv:math/0604273

[D] J. Douçot,
Diagrams and irregular connections on the Riemann sphere.
Preprint 2021, arXiv:2107.02516

[K] V. Kac,
Infinite dimensional Lie algebras, CUP 1995

[Kon93] H. Konno,
Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface, J. Math. Soc. Japan 45 (1993), no. 2, 253–275.

[Nak96] H. Nakajima,
Hyper-kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces, Moduli of vector bundles (Sanda-Kyoto 1994), 1996, pp. 199–208.

[Sim91] C. Simpson
Products of matrices, Differential geometry, global analysis, and topology (Halifax, NS, 1990), CMS Conf. Proc., vol. 12, Amer. Math. Soc., Providence, RI, 1991, pp. 157–185.

[W08] E. Witten, Gauge theory and wild ramification, Anal. Appl. (Singap.) 6 (2008), no. 4, 429–501, arXiv:0710.0631.