Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian

Let $G$ be a complex reductive group. A folklore result asserts the existence of an $\mathbb{E}_2$-algebra structure on the Ran Grassmannian of $G$ over $\mathbb{A}^1_{\mathbb{C}}$, seen as a topological space with the complex-analytic topology. The aim of this paper is to prove this theorem, by establishing a homotopy invariance result: namely, an inclusion of open balls $D' \subset D$ in $\mathbb{C}$ induces a homotopy equivalence between the respective Beilinson--Drinfeld Grassmannians $\mathrm{Gr}_{G, {D'}^n} \hookrightarrow \mathrm{Gr}_{G, D^n}$, for any positive integer $n$. We use a purely algebraic approach, showing that automorphisms of a complex smooth algebraic curve $X$ can be lifted to automorphisms of the associated Beilinson--Drinfeld Grassmannian. As a consequence, we obtain a stronger version of the usual homotopy invariance result: namely, the homotopies can be promoted to equivariant stratified isotopies, where "equivariant" refers to the action of the arc group $\mathrm{L}^+G$ and "stratified" refers to the stratification induced by the Schubert stratification of $\mathrm{Gr}_G$ and the incidence stratification of $\mathbb{C}^n$.