Wild Character Varieties Research Articles

Stokes Manifolds and Cluster Algebras

Stokes' manifolds, also known as wild character varieties, carry a natural symplectic structure. Our goal is to provide explicit log-canonical coordinates for these natural Poisson structures on the Stokes' manifolds of polynomial connections of rank $2$, thus including the second Painlev\'e\ 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 $A_{2K}$. 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.

Euler continuants in noncommutative quasi-Poisson geometry

AbstractIt 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$\mathbb {P}^1$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$\Gamma _n$on two vertices andnequioriented 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$\Gamma _n$. 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$\Gamma _n$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$\Gamma _1$by Van den Bergh. Moreover, using the method of fusion, we prove that the Hamiltonian double quasi-Poisson algebra attached to$\Gamma _n$admits a factorisation in terms ofncopies of the algebra attached to$\Gamma _1$.

Twisted spectral correspondence and torus knots

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to [Formula: see text] torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

Arithmetic and representation theory of wild character varieties

We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma–Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with the previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P=W conjecture for a suitable wild Hitchin system.

BPS States, Torus Links and Wild Character Varieties

A string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants. Explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line. For this case the string theoretic construction leads to a conjectural colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

Local curves, wild character varieties, and degenerations

Conjectural results for cohomological invariants of wild character varieties are obtained by counting curves in degenerate Calabi-Yau threefolds. A conjectural formula for E-polynomials is derived from the Gromov-Witten theory of local Calabi-Yau threefolds with normal crossing singularities. A refinement is also conjectured, generalizing existing results of Hausel, Mereb and Wong as well as recent joint work of Donagi, Pantev and the author for weighted Poincar\'e polynomials of wild character varieties.

Global Weyl groups and a new theory of multiplicative quiver varieties

In previous work a relation between a large class of Kac-Moody algebras and meromorphic connections on global curves was established---notably the Weyl group gives isomorphisms between different moduli spaces of connections, and the root system is also seen to play a role. This involved a modular interpretation of many Nakajima quiver varieties, as moduli spaces of connections, whenever the underlying graph was a complete k-partite graph (or more generally a supernova graph). However in the isomonodromy story, or wild nonabelian Hodge theory, slightly larger moduli spaces of connections are considered. This raises the question of whether the full moduli spaces admit Weyl group isomorphisms, rather than just the open parts isomorphic to quiver varieties. This question will be solved here, by developing a "multiplicative version" of the previous approach. This amounts to constructing many algebraic symplectic isomorphisms between wild character varieties. This approach also enables us to state a conjecture for certain irregular Deligne--Simpson problems and introduce some noncommutative algebras, generalising the "generalised double affine Hecke algebras".

Dynamics on Wild Character Varieties

In the present paper, we will first present briefly a general research program about the study of the "natural dynamics" on character varieties and wild character varieties. Afterwards, we will illustrate this program in the context of the Painlev\'e differential equations $P_{\rm VI}$ and $P_{\rm V}$.

Poisson varieties from Riemann surfaces

Short survey based on talk at the Poisson 2012 conference. The main aim is to describe and give some examples of wild character varieties (naturally generalising the character varieties of Riemann surfaces by allowing more complicated behaviour at the boundary), their Poisson/symplectic structures (generalising both the Atiyah–Bott approach and the quasi-Hamiltonian approach), and the wild mapping class groups.

Geometry and braiding of Stokes data; Fission and wild character varieties

A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.