The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

Abstract

We introduce a way of presentation of pairs (E, ∇), where E is a bundle on a Riemann surface and ∇ is a logarithmic connection in E, which is based on a presentation of the surface as a factor of the exterior of the unit disc. In this presentation we write the local equation of isomonodormic deformation of pairs (E, ∇). These conditions are written as a modified Schlesinger system on a Riemann sphere (and in the typical case just as an ordinary Schlesinger system) plus some linear system. *

Symmetries of Non-Linear ODEs: Lambda Extensions of the Ising Correlations

“…We must also mention the fact that the lambda extensions of the two-point correlation functions C(M, N) also verify quadratic discrete recursions [25][26][27] (lattice recursions in the two integers M and N), that can be seen as integrable lattice recursions. For pedagogical reasons, we restrict to C(0, 5) and C (2,5). Then, taking an example of product of two diagonal two-point correlation functions, we suggest that many more families of non-linear ODEs of the Painlevé type remain to be discovered on the two-dimensional Ising model, as well as their structures, and in particular their associated lambda extensions.…”

“…We revisit, with a pedagogical heuristic motivation, the lambda extensions [14] of some two-point correlation functions C(M, N) of the two-dimensional Ising model. For simplicity, we examine in detail the lambda extensions of a particular low-temperature diagonal correlation function, namely C(0, 5) and C (2,5), in order to make crystal clear some structures and subtleties. Note however, that similar structures and results can also be obtained on other two-point correlation functions C(M, N) for the special case ν = −k studied in [7] where Okamoto sigma-forms of Painlevé VI equations also emerge.…”

“…play a selected role, since they correspond to situations where one has "enough" (possibly an infinite number of) conserved quantities to solve the problem. We are not going to recall the techniques and tools introduced to achieve that goal (Yang-Baxter equations, Bethe Ansatz, Lax pairs, Schlesinger systems [2], etc.) but we will rather focus on the linear and non-linear differential equations emerging naturally in these problems, and on the corresponding symmetries of these ordinary differential equations.…”