I am trying to understand more about geometric interpretation of vertex algebras following "Vertex Algebras and Algebraic Curves" by Ben-Zvi and Frenkel, but I am in trouble with the following constrution:

Let $X$ be a smooth curve over a algebraically closed field $k$ of characteristic zero, $p \in X$ a $k-$point and $\hat{\mathcal{O}_p}$ be the completation of the local ring, then $\hat{\mathcal{O}_p} \simeq k[[z]]$. Let $G_0(R) = \{f \in \mathrm{Aut}(R[[z]])| f(z) \in z R[[z]] \}$ be de group-scheme of automorphims of the formal disc preserving the origin and $\mathrm{Coord}_p = \{\hat{\mathcal{O}_p} \to k[[z]] \,\,\,\text{isomorphism}\}$, $\mathrm{Coord}_p$ is a $G_0(k)-$right torsor. I would like to construct a $G_0-$principal bundle $\mathrm{Coord}_X$ over $X$ with fiber $\mathrm{Coord}_p$ at $p$. Frenkel and Ben-Zvi actually gives a functor of points $F(R) = \{f: R[[z]] \to X \,\, \text{with non-vanishing differential at}\,\, z=0\}$.

I am looking for references to:

  1. Principal bundles over schemes (with Zariski opens) and associated bundle;
  2. A construction of this $\mathrm{Coord}_X$ or anything similar to understand better what is going on.

1 Answer 1


The standard reference for where this comes from originally is

I. Gelfand and D. Kazhdan, Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields, Soviet Math. Dokl. 12 (1971), no. 5, 1367–1370.

But that's in Russian and not really complete. I would recommend Section 3.1 in

Roman Bezrukavnikov and Dmitry Kaledin. Fedosov quantization in algebraic context. Mosc. Math. J, 4(3):559–592, 2004


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