Divisor braids

TL;DR

This paper introduces divisor braid groups on surfaces, exploring their algebraic structure, link invariants, and connections to noncommutative geometry, with applications in gauge theory and vortex moduli spaces.

Contribution

It defines divisor braid groups with colored intersecting strands, provides a metabelian presentation, and links these groups to von Neumann algebras in noncommutative geometry.

Findings

Metabelian presentation of divisor braid groups

Construction of link invariants in $S^1 imes \Sigma$

Description of associated von Neumann algebras

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.