The Knizhnik--Zamolodchikov equations (KZ) express the constraints satisfied by correlation functions in 2d conformal field theory, and mathematically amount to a flat connection on a vector bundle over a configuration space: the bundle of conformal blocks. Later the KZ equations were derived from the quantisation of a Hamiltonian system controlling isomonodromic deformations of connections with simple poles, entering into the quantisation of the moduli space of gauge-classes of such connections.
In this talk we will review this story and point out that much more general moduli spaces have since been introduced, involving meromorphic connections with arbitrary singularities. Their quantisation thus lead to spaces of irregular conformal blocks, and to new flat "quantum" connections generalising KZ: we will in particular review work about connections with poles of order three, and then more recent work with G. Felder about a generalisation for poles of any order.
Thomas Strobl
