arXiv subjects

Jean Douçot

Publications and source records attributed to Jean Douçot.

12 recordsLinked to original sources

arXiv2026

Combinatorics of the Fourier transform: Stokes data, Gale duality and frieze patterns

We study the action of the Fourier transform on the Stokes data of irregular connections on the complex affine line with symmetric irregular classes at infinity, both from the point of view of Stokes filtered local systems and of Stokes local systems, and we show that it is governed by a rich combinatorial structure: (1) Observing that, in this setup, a Stokes filtration is fully determined by the data of either its recessive or subdominant solution spaces, and making the link with results of T. Mochizuki, we show that the Fourier transform amounts to exchanging recessive and subdominant solutions via the Gale transform of configurations of points in projective spaces. (2) We show that the equivalence between recessive solutions and Stokes local systems is deeply connected with the triality relating point configurations, superperiodic linear difference equations and frieze patterns obtained by Morier-Genoud-Ovsienko-Schwartz-Tabachnikov: Up to signs, the coefficients of the difference equations and friezes coincide with the nontrivial Stokes matrix entries. It follows from this Stokes-frieze correspondence that the Fourier transform of Stokes representations is given by their combinatorial Gale transform, leading to explicit closed formulas.

math.AG↗

arXiv2026

Basic Representations of Genus Zero Nonabelian Hodge Spaces

In some previous work, we defined an invariant of genus zero nonabelian Hodge spaces taking the form of a diagram. Here, enriching the diagram by fission data to obtain a refined invariant, the enriched tree, including a partition of the core diagram into $k$ subsets, we show that this invariant contains sufficient information to reconstruct $k+1$ different classes of admissible deformations of wild Riemann surfaces, that are all representations of one single nonabelian Hodge space, so that the isomonodromy systems defined by these representations are expected to be isomorphic. This partially generalises to the case of arbitrary singularity data the picture of the simply-laced case featuring a diagram with a complete $k$-partite core. We illustrate this framework by discussing different Lax representations for Painlevé equations.

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arXiv2026

Fourier transform of irregular connections on $\mathbb P^1$ and classification of Argyres-Douglas theories

We give a mathematical interpretation of the dualities between type $A$ Argyres-Douglas theories recently obtained by Beem, Martone, Sacchi, Singh and Stedman, building on work of Xie. Using the fact that, via the wild nonabelian Hodge correspondence, the data defining such a theory amount to singularity data for irregular connections on $\mathbb P^1$ of a specific form, we show that these dualities can all be realized as compositions of two types of more basic operations acting on such irregular connections: the Fourier transform and a Möbius transformation exchanging zero and infinity. The proof relies on the stationary phase formula giving explicit expressions for the singularity data of the Fourier transform. We also clarify the relation between the quivers describing the 3d mirrors of type $A$ Argyres-Douglas theories and the nonabelian Hodge diagrams defined in work of Boalch-Yamakawa and of the author: the 3d mirror corresponds to the unique nonabelian Hodge diagram with no negative edges/loops among those of singularity data in the corresponding orbit under basic operations.

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arXiv2025

Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids

Following the completion of the algebraic construction of the Poisson wild character varieties (B.--Yamakawa, 2015) one can consider their natural deformations, generalising both the mapping class group actions on the usual (tame) character varieties, and the G-braid groups already known to occur in the wild/irregular setting. Here we study these wild mapping class groups in the case of arbitrary formal structure in type A. As we will recall, this story is most naturally phrased in terms of admissible deformations of wild Riemann surfaces. The main results are: 1) the construction of configuration spaces containing all possible local deformations, 2) the definition of a combinatorial object, the ``fission forest'', of any wild Riemann surface and a proof that it gives a sharp parameterisation of all the admissible deformation classes. As an application of 1), by considering basic examples, we show that the braid groups of all the complex reflection groups known as the generalised symmetric groups appear as wild mapping class groups. As an application of 2), we compute the dimensions of all the (global) moduli spaces of type A wild Riemann surfaces (in fixed admissible deformation classes), a generalisation of the famous ``Riemann's count'' of the dimensions of the moduli spaces of compact Riemann surfaces.

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arXiv2025

A topological algorithm for the Fourier transform of Stokes data at infinity

We give a topological description of the behaviour of Stokes matrices under the Fourier transform from infinity to infinity in a large number of cases of one level. This explicit, algorithmic statement is obtained by building on a recent result of T. Mochizuki about the Fourier transform of Stokes data of irregular connections on the Riemann sphere and by using the language of Stokes local systems due to P. Boalch. In particular, this induces explicit isomorphisms between wild character varieties, in a much larger range of examples than those for which such isomorphisms have previously been written down. We conjecture that these isomorphisms are compatible with the quasi-Hamiltonian structure on the wild character varieties.

math.AG↗

arXiv2025

Twisted local G-wild mapping class groups

We consider the isomonodromic deformations of irregular-singular connections defined on principal bundles over complex curves: for any complex reductive structure group G, and any polar divisor; allowing for a twisted/ramified formal normal form at each pole, and for twists in the interior of the curve. (This covers the general case in 2-dimensional meromorphic gauge theory.) We focus on the irregular moduli of the connections, studying the fundamental groups of the spaces of admissible deformations of their irregular types/classes, i.e., the local wild mapping class groups in the title. To describe them, we first take the viewpoint of (nonsplit) reflections cosets in Springer/Lehrer--Springer theory, which yields in particular new modular interpretations of complex reflection groups -- and their braid groups. Then we introduce new `fission' trees to treat structure groups of any (simple) classical type, leading to a complete classification of the corresponding hyperplane arrangements, and singling out an infinite family of noncrystallographic examples in type D. Moreover, we reinterpret Bessis' lift of Springer theory as the study of `quasi-generic' deformations, corresponding to irregular singularities whose leading coefficient is regular semisimple upon pullback along a local cyclic covering of the base curve. Finally, we rephrase much of this material in terms of generalized root-valuation stratifications.

math.GT↗

arXiv2025

Moduli spaces of untwisted wild Riemann surfaces

We construct moduli stacks of wild Riemann surfaces in the (pure) untwisted case, for any complex reductive structure group, and we define the corresponding (pure) wild mapping class groups.

math.AG↗

arXiv2025

New nonabelian Hodge graphs from twisted irregular connections

It is known that any meromorphic connection on the Riemann sphere determines a finite diagram encoding its global Cartan matrix, and that it is invariant under the Fourier-Laplace transform. If the connection is tame at finite distance and untwisted at infinity, the diagram is actually a graph, corresponding to a symmetric generalised Cartan matrix, and it was proved by Boalch/Hiroe-Yamakawa that the corresponding nonabelian Hodge moduli space contains the Nakajima quiver variety of the graph as an open subset. In this note, we show that there exist new nonabelian Hodge diagrams that are graphs, beyond the setting of this quiver modularity theorem. The proof relies on observing that edge multiplicities in nonabelian Hodge diagrams satisfy ultrametric inequalities, which in particular gives a precise characterisation of nonabelian Hodge graphs coming from the untwisted setting.

math.AG↗

arXiv2025

Local wild mapping class groups and cabled braids

We will define and study some generalisations of pure $\mathfrak{g}$-braid groups that occur in the theory of connections on curves, for any complex reductive Lie algebra $\mathfrak{g}$. They make up local pieces of the wild mapping class groups, which are fundamental groups of (universal) deformations of wild Riemann surfaces, underlying the braiding of Stokes data and generalising the usual mapping class groups. We will establish a general product decomposition for the local wild mapping class groups, and in many cases define a fission tree controlling this decomposition. Further in type A we will show one obtains cabled versions of braid groups, related to braid operads.

math.GT↗

arXiv2025

Topology of irregular isomonodromy times on a fixed pointed curve

We will define and study (moduli) spaces of deformations of irregular classes on Riemann surfaces, which provide an intrinsic viewpoint on the `times' of irregular isomonodromy systems in general. Our aim is to study the deeper generalisation of the G-braid groups that occur as fundamental groups of such deformation spaces, with particular focus on the generalisation of the full G-braid groups.

math.AG↗

arXiv2025

Simplification of exponential factors of irregular connections on $\mathbb P^1$

We give an explicit algorithm to reduce the ramification order of any exponential factor of an irregular connection on $\mathbb P^1$, using the same types of basic operations as in the Katz-Deligne-Arinkin algorithm for rigid irregular connections. The exponential factor reached when the algorithm terminates is, up to admissible deformations, the unique factor with minimal ramification order in the orbit of the initial factor under successive applications of basic operations. Furthermore, we show that for every even integer $n\geq 0$, there is up to admissible deformations a finite number of non-simplifiable exponential factors at infinity such that the corresponding elementary wild character variety has complex dimension $n$, which conjecturally implies that there is a finite number of isomorphism classes of elementary wild character varieties in any dimension. These results can be viewed as saying that the set of all possible level data of exponential factors has the structure of a disjoint union of an infinite number of infinite rooted trees, each tree being associated to a given dimension $n$ and with a finite number of trees for each $n$. In particular, in dimension 2 there is a unique tree, corresponding to the Painlevé I moduli space.

math.AG↗

arXiv2023

Diagrams and irregular connections on the Riemann sphere

We define a diagram associated to any algebraic connection on a vector bundle on a Zariski open subset of the Riemann sphere, extending the definition of Boalch-Yamakawa to the general case featuring several irregular singularities, possibly ramified. We prove that the diagram is invariant under the symplectic automorphisms of the Weyl algebra, encompassing the Fourier-Laplace transform. As an application, we establish several new cases of the observation that different Lax representations of a given Painlevé-type equation may be read off directly from the diagram, corresponding to connections with different formal data, usually on different rank bundles.

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