Schlesinger transformations for elliptic isomonodromic deformations (Q2737998)

The authors deal with the following ordinary linear differential equation for a matrix-valued function \(\psi(\lambda) \in\text{SL}(2,\mathbb{C})\) NEWLINE\[NEWLINE{ d\psi \over d\lambda}= A(\lambda)\psi,\quad A(\lambda)= \sum^N_{j=1}{A_j \over \lambda- \lambda_j},NEWLINE\]NEWLINE where the residues \(A_j\in\text{sl}(2,\mathbb{C})\) are independent of \(\lambda\). Upon analytical continuation around \(\lambda= \lambda_j\), the function \(\psi (\lambda)\) changes by right multiplication with some monodromy matrices \(M_j\). The assumption of independence of all monodromy matrices \(M_i\) of the positions of the singularities \(\lambda_j\) is called the isomonodromy condition. Schlesinger transformations are discrete monodromy-preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere, the authors construct these transformations for isomonodromic deformations on genus one Riemann surfaces.