Coloured refined topological vertices and parafermion conformal field theories**To Professor Mikio Sato on his 90th birthday.

Abstract

We extend the definition of the refined topological vertex to an n-coloured refined topological vertex that depends on n free bosons, and compute the 5D strip partition function made of N pairs of vertices and conjugate vertices. Using geometric engineering and the AGT correspondence, the 4D limit of this strip partition function is identified with a (normalized) matrix element of a (primary state) vertex operator that intertwines two (arbitrary descendant) states in a (generically non-rational) 2D conformal field theory with parafermion primary states.