Knizhnik–Zamolodchikov equation

The Knizhnik–Zamolodchikov equation is a system of partial differential equations governing the parallel transport of conformal blocks in two-dimensional conformal field theory. Its solutions describe how correlation functions of primary fields vary as insertion points are moved, and the monodromy of these solutions — the transformation undergone when fields are braided around one another — is governed by representations of the braid group. The KZ equation is not a peripheral tool of string theory; it is the differential realization of the same algebraic structure that underlies topological quantum computing.

The KZ equation reveals that conformal field theory and topological quantum computing are not separate disciplines that happen to share mathematics. They are the same mathematics viewed from different laboratories — one studying the vacuum, the other engineering it.