Mathematics

Consider a trivial holomorphic vector bundle on the Riemann sphere, equipped with a meromorphic connection. Isomodromic deformations of the connection are controlled by a system of nonlinear differential equations, as one varies the position of the poles and the (diagonal) principal part.In this talk we will consider meromorphic connections with Poincaré rank 2 at infinity plus other simple poles, recall how to encode the isomonodromy equations in the flow of a time-dependent integrable Hamiltonian system, and then quantise the Hamiltonians. The result is a flat connection on a vector bundle, and the construction generalises the derivation of the Knizhnik–Zamolodchikov connection as a quantisation of the isomonodromy equations for Fuchsian systems.