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 $({\mathfrak{g}}_P^{\ast }{)}^{⊠n}$. 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.

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