Temperley–Lieb–Jones category and the space of conformal blocks

Abstract

We describe a relationship between homological representations of the braid groups and the monodromy representations of the KZ connection. By using this topological method we show that there is an isomorphism between the space of conformal blocks and the space of morphisms of the Temperley–Lieb–Jones category, which is equivariant under the action of the braid group. We recover the unitarity of the braid group action on the space of conformal blocks by means of the positivity of the Markov trace.