Isomonodromic deformations: Confluence, Reduction $\&$ Quantisation

Abstract

In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the Takiff algebra (i.e. truncated current algebra). Our motivation is to produce confluent versions of the celebrated Knizhnik-Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic τ -function. In order to achieve this, we study the confluence cascade of r + 1 simple poles to give rise to a singularity of arbitrary Poincaré rank r as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians.

Local wild mapping class groups and cabled braids

“…On the contrary there exist new flat "quantum" connections corresponding to deformations of the irregular types [55,56], showing the new parameters behave as the moduli of the underlying pointed surface even after quantisation (cf. [37] for a different, related viewpoint). Our main point is the "classical" theory goes beyond these examples, and is expressed in a language that lends itself to quantisation, providing a guide to prove analogous statements in a (much) more general context.…”

A note on unfolding manifolds of meromorphic connections on the Riemann sphere with unramified singularities

“…In case of moduli spaces of connections, Inaba constructed in [8] a one-parameter deformation of moduli spaces of meromorphic connections with unramifed irregular singular points whose Hukuhara-Turrittin-Levelt normal forms have distinct eigenvalues. A similar deformation also considered by Gaiur-Mazzocco-Rubtsov in [3], in which they defined a deformation of coadjoint orbits of a Lie algebra of polynomials which they call the Takiff algebra. Furthermore, Gaiur-Mazzocco-Rubtsov and Inaba considered in their papers [3] and [8] the confluence of isomonodromic deformation equations.…”

“…A similar deformation also considered by Gaiur-Mazzocco-Rubtsov in [3], in which they defined a deformation of coadjoint orbits of a Lie algebra of polynomials which they call the Takiff algebra. Furthermore, Gaiur-Mazzocco-Rubtsov and Inaba considered in their papers [3] and [8] the confluence of isomonodromic deformation equations.…”

Topology of irregular isomonodromy times on a fixed pointed curve

“…[53] for a different construction of "quantum" simply-laced isomondromy systems, and [39,40,28] for a "confluence" viewpoint on the quantisation of irregular singularities.…”