Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras

Abstract

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Dualities in the q‐Askey Scheme and Degenerate DAHA

“…Interestingly, dual families (for example, the continuous dual q ‐Hahn and the big q ‐Jacobi polynomials) correspond to the same monodromy manifold in the classical limit, thus suggesting an alternative approach to spot dualities. Moreover, the limit transitions in the q ‐Askey scheme correspond to the so‐called confluence procedure of the Painlevé equations, which can be viewed geometrically as a procedure to merge holes on a Riemann sphere by which cusped holes are created . In this picture, dual families in the q ‐Askey scheme correspond to Riemann spheres with the same structure.…”

Note on Character Varieties and Cluster Algebras

“…Correspondingly the moduli space (4.1) could be degenerated, and 10 families of moduli spaces as generalized monodromy data were proposed in [50]. Their geometrical aspects was studied in [10] from a viewpoint of the Teichmüller theory of a punctured Riemann surface. Motivated by [10,11], we shall study confluences of the punctures from a cluster algebraic viewpoint of the moduli space.…”

“…Their geometrical aspects was studied in [10] from a viewpoint of the Teichmüller theory of a punctured Riemann surface. Motivated by [10,11], we shall study confluences of the punctures from a cluster algebraic viewpoint of the moduli space. In our previous studies on geometric aspects of the cluster algebra [34], the cluster y-variables may be regarded as a modulus of an ideal tetrahedron in 3dimensional hyperbolic space H 3 .…”

“…Other Painlevé equations have irregular singularities, and the associated affine cubic surfaces analogous to the Fricke surface were given in [50] as the moduli space of the generalized monodromy in the Riemann-Hilbert correspondence. In [10], those affine surfaces were studied by use of the λ-length of the Teichmüller space, and discussed was a relationship with a generalized cluster algebra [12]. In this article, we revisit degeneration processes of the character variety of the 4-punctured sphere, and reformulate them using the cluster algebra.…”

Algebras of Quantum Monodromy Data and Character Varieties

“…The decorated character variety M 2 0,1,6 has dimension 9 with one Casimir. As explained in [18], the isomonodromicity condition means that we need to restrict to a 2-dimensional sub-algebra in M 2 0,1,6 defined by the set of functions that Poisson commute with the frozen cluster variables corresponding to arcs connecting pairs of bordered cusps. On the l.h.s.…”