Divisor braids

Abstract

We study a novel type of braid groups on a closed orientable surface $\Sigma$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $\Sigma$; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles over $\Sigma$. The elements of these braid groups, which we call divisor braids, have coloured strands that are allowed to intersect according to rules specified by a graph $\Gamma$. In situations where there is more than one strand of each colour, we show that the corresponding braid group admits a metabelian presentation as a central extension of the free Abelian group $H_1(\Sigma;\mathbb{Z})^{\oplus r}$, where $r$ is the number of colours, and describe its Abelian commutator. This computation relies crucially on producing a link invariant (of closed divisor braids) in the three-manifold $S^1 \times \Sigma $ for each graph $\Gamma$. We also describe the von Neumann algebras associated to these groups in terms of rings that are familiar from noncommutative geometry.