Discrete symmetries of isomonodromic deformations of order two Fuchsian differential equations

Abstract

In the present work we calculate the group structure of the Schlesinger transformations for isomonodromic deformations of order two Fuchsian differential equations. We perform these transformations as the isomorphisms between the moduli spaces of the logarithmic sl(2)- connections with fixed eigenvalues of residues at singular points. We give the geometrical interpretation of the Schlesinger transformations and perform the calculations using the techniques of the modifications of bundles with connections. In order to illustrate this result we present classical examples of symmetries of the hypergeometric equation, the Heun equation and the sixth Painlev\'e equation.