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Stokes’ manifolds, also known as wild character varieties, carry a natural Poisson structure. Our goal is to provide explicit log-canonical coordinates for this Poisson structure on the Stokes’ manifolds of polynomial connections of rank 2, thus including the second Painlevé hierarchy. This construction provides the explicit linearization of the Po...
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Citations
... In this article, we are able to further prove that Euler continuants are noncommutative moment maps in the sense of Van den Bergh's noncommutative quasi-Poisson geometry-see §1. 3. This is another step towards the programme that we have initiated in [31] which aims at understanding the Poisson geometry of wild character varieties in terms of Hamiltonian double quasi-Poisson algebras attached to quivers. ...
... This is an affine cubic surface endowed with a Poisson structure; in the setting of integrable systems, it is closely related to solutions to Painlevé II equation (see also [22]). Recently, based on [33] (see also [15, §5]), Bertola and Tarricone [3] explicitly wrote the Poisson bracket on the wild character variety M ( ) (with no restriction on n); note that an explicit expression for the corresponding symplectic 2-form is well-known by experts. Whereas we originally obtained the Hamiltonian double quasi-Poisson structure of Theorem 3.4 employing entirely noncommutative arguments, in §3.4 we are able to show that it induces the Poisson bracket due to Flaschka and Newell on Rep B(Γ ), (1, 1) (i.e., before performing quasi-Hamiltonian reduction to end up with M ( )). ...
... Let us note that, since the group C × × C × acting on B +1 is abelian, the quasi-Poisson bracket is in fact a Poisson bracket. This will be important in view of the following change of variables, motivated by a standard parametrisation connected to Stokes matrices [3,15]. We set ...
It was established by Boalch that Euler continuants arise as Lie group valued moment maps for a class of wild character varieties described as moduli spaces of points on by Sibuya. Furthermore, Boalch noticed that these varieties are multiplicative analogues of certain Nakajima quiver varieties originally introduced by Calabi, which are attached to the quiver on two vertices and n equioriented arrows. In this article, we go a step further by unveiling that the Sibuya varieties can be understood using noncommutative quasi-Poisson geometry modelled on the quiver . We prove that the Poisson structure carried by these varieties is induced, via the Kontsevich–Rosenberg principle, by an explicit Hamiltonian double quasi-Poisson algebra defined at the level of the quiver such that its noncommutative multiplicative moment map is given in terms of Euler continuants. This result generalises the Hamiltonian double quasi-Poisson algebra associated with the quiver by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to admits a factorisation in terms of n copies of the algebra attached to .
... Résultats du chapitre 7 Le chapitre 7 illustre les résultats contenus dans le travail [BT21] en collaboration avec Marco Bertola. Dans ce travail on s'intéresse aux structures symplectique et de Poisson XIII de certaines variétés de monodromie, apellées variétés de Stokes. ...
... The second part contains instead the original contributions obtained in the works [Tar21,BCT21,BT21], that are distributed in the last three chapters. In particular the thesis is organised as follows : ...
... (5) Finally in Chapter 7 we explain most of the content of [BT21]. We prove that the Stokes manifold associated to a polynomial system of ODEs of generic degree K and rank 2 is indeed a symplectic manifold. ...
The Painlevé II hierarchy is a sequence of nonlinear ODEs, with the Painlevé II equation as first member. Each member of the hierarchy admits a Lax pair in terms of isomonodromic deformations of a rank 2 system of linear ODEs, with polynomial coefficient for the homogeneous case. It was recently proved that the Tracy-Widom formula for the Hastings-McLeod solution of the homogeneous PII equation can be extended to analogue solutions of the homogeneous PII hierar-chy using Fredholm determinants of operators acting through higher order Airy kernels. These integral operators are used in the theory of determinantal point processes with applications in statistical mechanics and random matrix theory. From this starting point, this PhD thesis explored the following directions. We found a formula of Tracy-Widom type connecting the Fredholm determinants of operators acting through matrix-valued analogues of the higher order Airy kernels withparticular solution of a matrix-valued PII hierarchy. The result is achieved by using a matrix-valued Riemann-Hilbert problem to study these Fredholm determinants and by deriving a block-matrix Lax pair for the relevant hierarchy. We also found another generalization of the Tracy-Widom formula, this time relating the Fredholm determinants of finite-temperature versions of higher order Airy kernels operators to particular solutions of an integro-differential PII hierarchy. In this setting, a suitable operator-valued Riemann-Hilbert problem is used to study the relevant Fredholm determinant. The study of its solution produces in the end an operator-valued Lax pair that naturally encodes an integro-differential Painlevé II hierarchy. From a more geometrical point of view, we analyzed the Poisson-symplectic structure of the monodromy manifolds associated to a system of linear ODEs with polynomial coefficient, also known as Stokes manifolds. For the rank 2 case, we found explicit log canonical coordinates for the symplectic 2-form, forming a cluster algebra of specific type. Moreover, the log-canonical coordinates constructed in this way provide a linearization of the Poisson structure on the Stokes manifolds, first introduced by Flaschka and Newell in their pioneering work of 1981
The group of type is a coideal subalgebra of the quantum group , associated with the symmetric pair . In this paper, we give a cluster realisation of the algebra . Under such a realisation, we give cluster interpretations of some fundamental constructions of , including braid group symmetries, the coideal structure and the action of a Coxeter element. Along the way, we study a (rescaled) integral form of , which is compatible with our cluster realisation. We show that this integral form is invariant under braid group symmetries, and construct the Poincare‐Birkhoff‐Witt (PBW)‐bases for the integral form.
We describe the Stokes phenomenon, its main significance, and its emergence in various landscapes. We explain how to use Stokes phenomena to enrich the classical dynamics of complex dynamical systems, defining some wild dynamics. In the same line, we describe how, adding Stokes multipliers to the classical monodromy, it is possible to classify a lot of complex dynamical systems (linear or not), up to “gauge transformations” (using some cohomological invariants), and to get generalizations of the Riemann-Hilbert correspondence, with a lot of applications. In the linear case, we describe the relations between Stokes phenomena and differential Galois theory and some important consequences. We end with a short description of many other incarnations of Stokes phenomenon (singular perturbations, resurgence, difference and q-difference equations, theoretical physics …), with some insights in the several variables cases. All along our article, we insist on the historical roots and on some simple geometric ideas. The interpretation of the divergence of some power series as expressing a form of branching for an analytic function is a leitmotiv and a red thread all along the text.
