Motivic factorisation of KZ local systems and deformations of representation and fusion rings

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The KZ connection is a connection on the constant bundle associated to a set of $n$ finite dimensional irreducible representations of $\mathfrak{g}$ and a nonzero $\kappa \in \mathbb{C}$, over the configuration space of $n$-distinct points on the affine line. Via the work of Schechtman--Varchenko and Looijenga, when $\kappa$ is a rational number the associated local systems can be seen to be realisations of naturally defined motivic local systems. We prove a basic factorisation for the nearby cycles of these motivic local systems as some of the $n$ points coalesce. This leads to the construction of a family (parametrised by $\kappa$) of deformations over $\mathbb{Z}[t]$ of the representation ring of $\mathfrak{g}$--we call these enriched representation rings--which allows one to compute the ranks of the Hodge filtration of the associated variations of mixed Hodge structure; in turn, this has applications to both the local and global monodromy of the KZ connection. In the case of $\mathfrak{sl}_n$ we give an explicit algorithm for computing all products in the enriched representation rings, which we use to prove that if $1/\kappa$ is an integer then the global monodromy is finite and scalar. We also prove a similar factorisation result for motivic local systems associated to conformal blocks in genus $0$; this leads to the construction of a family of deformations of the fusion rings. Computations in these rings have potential applications to finiteness of global monodromy. Several open problems and conjectures are formulated. These include questions about motivic BGG-type resolutions and the relationship between the Hodge filtration and the filtration by conformal blocks at varying levels.