Infinitesimal jet spaces of $\text{Bun}_G$ in positive characteristic

Abstract

Given a semisimple reductive group $G$ and a smooth projective curve $X$ over an algebraically closed field $k$ of arbitrary characteristic, let $\text{Bun}_G$ denote the moduli space of principal $G$-bundles over $X$. For a bundle $P\in\text{Bun}_G$ without infinitesimal symmetries, we provide a description of all divided-power infinitesimal jet spaces, $J_P^{n,PD}(\text{Bun}_G)$, of $\text{Bun}_G$ at $P$. The description is in terms of differential forms on $X^n$ with logarithmic singularities along the diagonals and with coefficients in $(\mathfrak{g}_P^*)^{\boxtimes n}$. Furthermore, we show the pullback of these differential forms to the Fulton-Macpherson compactification of the configuration space, $\hat{X}^n$, is an isomorphism. Thus, we relate the two constructions of Beilinson-Drinfeld and Beilinson-Ginzburg, and as a consequence, give a connection between divided-power infinitesimal jet spaces of $\text{Bun}_G$ and the $\mathcal{L}ie$ operad.