Algebraic loops, braids, and tropicalization

Abstract

Let $Y$ be a smooth complex algebraic variety. We develop a tropical framework for studying algebraic loops, i.e., free homotopy classes in $Y^{\mathrm{an}}$ associated with formal loops $\mathrm{Spec}\, \mathbb{C}(\!(t)\!)\to Y$. When $Y$ is very affine, every formal loop determines an integral point of $\mathrm{Trop}(Y)$. For schön varieties, we prove that the associated algebraic loop is determined by this tropical point together with a connected component of the corresponding initial variety. Consequently, when all initial varieties are connected, integral tropical points classify algebraic loops. We apply this theory to the case when $Y$ is a hyperplane arrangement complement. In this case, we identify the initial varieties with complements of graded arrangements and obtain explicit representatives for algebraic loops as products of relative full twists. In type $A$, these representatives recover pure algebraic braids. For root arrangements, our construction gives a tropical interpretation of the split root valuation data of Goresky, Kottwitz, and MacPherson. Finally, we prove that the cohomology of braid varieties associated with positive algebraic braids depends only on their root valuation data.