Shear coordinate description of the quantized versal unfolding of aD₄singularity

Abstract

In this paper by using Teichmüller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D 4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D 4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on ⊕ 3 1 sl * (2, C). We realise the action of the mapping class group by the action of the braid group on the geodesic functions . This action coincides with the procedure of analytic continuation of solutions of the sixth Painlevé equation. Finally, we produce the explicit quantisation of the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere and of the braid group action.

Note on Character Varieties and Cluster Algebras

“…We can check from tedious computations that (4.6) and (4.7) fulfill (4.1). We note that slightly different parametrizations were given in [9,15,27]. The difference is crucial for the following studies of the automorphism, the confluences of the punctures, and the quantization from the view point of the cluster algebra associated to the quiver Q oct .…”

Painlevé Monodromy Manifolds, Decorated Character Varieties, and Cluster Algebras

“…We want to remind that in the case of four-holed sphere (P V I) the quasi-Hamiltonian structure on the representation space is given by the "Korotkin-Samtleben" quadratic brackets between matrix elements of the representation matrices which form a "not closed" Poisson algebra [28]. The Jacobi identity is satisfied only modding the conjugation [10]). In our case the Poisson structure on the decorated character variety is an example of a "Trace-Poisson" quadratic structure of [30] and [4] defined on the representation space Hom (U, SL 2 (C)) [12].…”

“…The real slice of this character variety is the decorated Teichmüller space of a 4 holed Riemann sphere, and can be combinatorially described by a fat-graph and shear coordinates. By complexifying the shear coordinates, flat coordinates for the character variety of a 4 holed Riemann sphere were found in [10]. For the other Painlevé equations, the interpretation of their monodromy manifolds as "character varieties" of a Riemann sphere with boundary is still an extremely difficult problem due to the fact that the linear problems associated to the other Painlevé equations exhibit Stokes phenomenon.…”

Embedding of the rank 1 DAHA into $Mat(2,\mathbb T_q)$ and its automorphisms

“…The real slice of moduli space F (θ 1 , θ 2 , θ 3 , θ ∞ ) of rank 2 meromorphic connections over P 1 with four simple poles a 1 , a 2 , a 3 , ∞ can be obtained as a quotient of the Teichmüller space of the 4-holed Riemann sphere by the mapping class group. This fact allowed us to use the combinatorial description of the Teichmüller space of the 4-holed Riemann sphere in terms of fat-graphs to produce a good parameterisation of the monodromy manifold of PVI [4]. In this sub-section we recall the main ingredients of this construction.…”

“…[S i , S i+1 ] = iπ {s i , s i+1 } = iπ , i = 1, 2, 3, i + 3 ≡ i, while the central elements, i.e. perimeters p 1 , p 2 , p 3 and S 1 + S 2 + S 3 remain nondeformed, so that the constants ω (d) i remain non-deformed [4]. The Hermitian operators G 23 , G 31 , G 12 corresponding to G 23 , G 31 , G 12 are introduced as follows: consider the classical expressions for G 23 , G 31 , G 12 is terms of s 1 , s 2 , s 3 and p 1 , p 2 , p 3 .…”