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Stokes Manifolds and Cluster Algebras

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Marco Bertola at Concordia University Montreal and Scuola Internazionale Superiore di Studi Avanzati di Trieste
Marco Bertola
  • Concordia University Montreal and Scuola Internazionale Superiore di Studi Avanzati di Trieste
Sofia Tarricone at Institut de Physique Théorique
Sofia Tarricone
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Abstract and Figures

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 Poisson structure first discovered by Flaschka and Newell and then rediscovered and generalized by Boalch. We show that, for a connection of degree K, the Stokes’ manifold is a cluster manifold of type A2KA_{2K} with one frozen vertex. The main idea is then applied to express explicitly also the log–canonical coordinates for the Poisson bracket introduced by Ugaglia in the context of Frobenius manifolds and then also applied by Bondal in the study of the symplectic groupoid of quadratic forms.
An example of Stokes’ graph Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} used in Theorem 2.5
An example of Stokes’ graph Σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} used in Theorem 2.5
… 
The Stokes graph Σ(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Sigma ^{(2)}$$\end{document}
The Stokes graph Σ(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Sigma ^{(2)}$$\end{document}
… 
The modified graph Σ0(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Sigma ^{(2)}_{0} $$\end{document}. Here we take the triangulation T0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document} of the hexagon that connects any of its vertices to v6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v_{6} $$\end{document}
The modified graph Σ0(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Sigma ^{(2)}_{0} $$\end{document}. Here we take the triangulation T0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document} of the hexagon that connects any of its vertices to v6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v_{6} $$\end{document}
… 
The Dynkin diagram associated to the 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4 $$\end{document} matrix B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {B}}_2 $$\end{document}. This quiver can also be obtained following the construction described in the paragraph below with the triangulation of the hexagon fixed to be T0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document}
The Dynkin diagram associated to the 4×4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\times 4 $$\end{document} matrix B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {B}}_2 $$\end{document}. This quiver can also be obtained following the construction described in the paragraph below with the triangulation of the hexagon fixed to be T0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_0$$\end{document}
… 
Here the triangulation T of the hexagon and the variables yj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_j$$\end{document} assigned to the relevant edges induce the Dynkin diagram with variables κ1,κ2,κ3,κ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _1, \kappa _2, \kappa _3, \kappa _4$$\end{document} in blue
+6
Here the triangulation T of the hexagon and the variables yj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_j$$\end{document} assigned to the relevant edges induce the Dynkin diagram with variables κ1,κ2,κ3,κ4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _1, \kappa _2, \kappa _3, \kappa _4$$\end{document} in blue
… 
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Digital Object Identifier (DOI) https://doi.org/10.1007/s00220-021-04293-7
Commun. Math. Phys. 390, 1413–1457 (2022) Communications in
Mathematical
Physics
Stokes Manifolds and Cluster Algebras
M. Bertola1,2, S. Tarricone1,3
1Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W.,
Montréal, QC H3G 1M8, Canada. E-mail: Marco.Bertola@concordia.ca; Marco.Bertola@sissa.it
2SISSA/ISAS, Area of Mathematics, via Bonomea 265, 34136 Trieste, Italy
3LAREMA, UMR 6093, CNRS, SFR Math-Stic, UNIV Angers, Angers, France.
E-mail: tarricone@math.univ-angers.fr; Sofia.Tarricone@concordia.ca
Received: 29 May 2021 / Accepted: 27 November 2021
Published online: 17 January 2022 – © The Author(s), under exclusive licence to Springer-Verlag GmbH
Germany, part of Springer Nature 2022
Abstract: Stokes’ manifolds, also known as wild character varieties, carry a natural
Poisson structure. Our goal is to provide explicit log-canonical coordinates for this Pois-
son structure on the Stokes’ manifolds of polynomial connections of rank 2, thus includ-
ing the second Painlevé hierarchy. This construction provides the explicit linearization
of the Poisson structure first discovered by Flaschka and Newell and then rediscovered
and generalized by Boalch. We show that, for a connection of degree K, the Stokes’
manifold is a cluster manifold of type A2Kwith one frozen vertex. The main idea is then
applied to express explicitly also the log–canonical coordinates for the Poisson bracket
introduced by Ugaglia in the context of Frobenius manifolds and then also applied by
Bondal in the study of the symplectic groupoid of quadratic forms.
Contents
1. Introduction and Results ........................... 1414
2. Symplectic Structure on Stokes’ Matrices .................. 1416
3. Stokes Manifolds for Rank-Two Polynomial Connections ......... 1425
3.1 Computation of WK........................... 1426
4. The X-Cluster Manifold Structure ...................... 1435
4.1 Flipping the edges ............................ 1437
4.2 Example: the case K=2........................ 1442
4.3 The Amanifold ............................. 1444
4.4 Computation of the Poisson brackets for the original monodromy param-
eters ................................... 1445
5. Log Canonical Coordinates for the Ugaglia Bracket ............ 1448
6. Conclusion and Outlook ........................... 1453
A. Proof of Proposition 4.7 ........................... 1454
References ..................................... 1456
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... 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 ...
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... 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. ...
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... (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. ...
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