Research

• Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids (with P. Boalch and J. Douçot)
  To appear in PRIMS
  arXiv:2209.12695, 47pp. 

• Local wild mapping class groups and cabled braids (with J. Douçot and M. Tamiozzo)
  To appear in the Annales de l'Institut Fourier
  arXiv:2204.08188, 39pp.

• Sp(1)-symmetric hyperkähler quantisation (with J.E. Andersen and A. Malusà)
  Pacific Journal of Mathematics 329, 1 (2024)
  arXiv:2111.03584, 33pp. [DOI]

• A colourful classification of (quasi) root systems and hyperplane arrangements
  Journal of Lie theory 34, 2 (2024)
  arXiv:2206.03779, 46pp. [DOI]

• Topology of irregular isomonodromy times on a fixed pointed curve (with J. Douçout)
  Transformation Groups (2023)
  arXiv:2208.02575, 33pp. [DOI]

• Genus-one complex quantum Chern–Simons theory (with J.E. Andersen and A. Malusà)
  Journal of Symplectic Geometry 20, 6 (2022)
  arXiv:2012.15630, 27pp. [DOI]

• Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks (with G. Felder)
  Sel. Math. New Ser. 29, 15 (2023)
  arXiv:2012.14793, 42pp. [DOI]

• Symmetries of the simply-laced quantum connections, and quantisation of quiver varieties
  SIGMA 16 (2020), 103
  arXiv:1905.07713, 44pp. [DOI]

• Simply-laced quantum connections generalising KZ
  Commun. Math. Phys. 368 (2019)
  arXiv:1704.08616, 53pp. [DOI]

I defined a family of (strongly) flat connections, obtained from the deformation quantisation of Hamiltonian systems controlling isomonodromic deformations of irregular singular connections on the Riemann sphere: they include (and generalise) the Knizhnik--Zamolodchikov connection (KZ), the connection of de Concini/Millson--Toledano Laredo (DMT), and the connection of Felder--Markov--Tarasov--Varchenko (FMTV).
I also constructed a family of `quantum' symmetries of the above flat connections, obtained from the deformation quantisation of analogous symmetries of the `semiclassical' isomonodromy systems: they generalise the Howe duality, which in turn relates KZ with DMT.

With J.E. Andersen and A. Malusà, I constructed a complexified version of the Hitchin connection for the geometric quantisation of moduli spaces of holomorphic connections on principal bundles over elliptic curves, with respect to (hyper)Kähler polarisations: it has been related to the connection of Witten, for the quantisation of complex Chern--Simons theory with respect to real polarisations.
With J.E. Andersen and A. Malusà, I also introduced a framework for the geometric quantisation of hyperkähler manifold in order to construct representation of distinguished group of isometries, and applied it to some gauge-theoretic moduli spaces including Atiyah--Hitchin monopoles and instantons.

With P. Boalch, J. Douçot and M. Tamiozzo, I studied deformation spaces of irregular types/classes on a pointed Riemann surface, which provide intrinsic time-variables for the most general isomonodromic deformations: their fundamental groups generalise (generalised) Artin--Tits braid groups, as well as their associated complex reflection groups, and in type A lead to cabled braids. We have also considered the `global' picture, generalising the Riemann moduli stacks of complex projective curves with marked points and the corresponding mapping class groups.
Along the same line of work, I defined certain diagrams classifying all root subsystems of the classical simple Lie algebras, but also the `generalised' root systems obtained upon restrictions on the hyperplane intersection of a Levi subsystem, leading e.g. to a noncrystallographic arrangement in type D.
Finally, with J. Douçot and D. Yamakawa, I considered the deformations of the most general possible irregular types/classes, involving any (reductive) structure group and twisted/ramified formal normal forms.

With G. Felder, I introduced a generalisation of Verma modules for affine Lie algebras. They yield bundles of irregular vacua associated with the quantisation of genus-zero moduli spaces of generic irregular singular connections, lead to a generalisation of the `irregular' KZ connection of Reshetikhin, and contain Whittaker vector for the Gaiotto--Teschner Virasoro pair from irregular Liouville CFT.
With D. Calaque, G. Felder and R. Wentworth, I then considered a further generalisation, involving nongeneric irregular singularities, getting to the deformation quantisation of the corresponding (trucated) gauge orbits for the principal parts of meromorphic connections.