Fix $n$ and a field $k$ of characteristic zero. Let $G$ be the pro-algebraic group of automorphims of $k[[x_1,...x_n]]$. Let $G_0$ be the subgroup of automorphisms preserving the closed point (note that for general $T$, $G_0(T)$ can be a proper subgroup of $G(T)$). Let $X$ be a regular variety over $k$ and let $P$ be the principal $G$ bundle of formal coordinate systems, naturally a $G$ torsor over $X$. I hear that there is a connection between $P$ and $D_X$-modules. what is this connection?
Assume X is n-dimensional and regular. Then there is a functor from G-modules V to DX-modules, given by an associated bundle construction. Take the trivial (ind-)bundle on P with fiber V, and quotient by the action of G on P and V. If you replace G with G0 and P with the canonical G0-torsor, the same construction yields an OX-module. The extra structure of a G-action lets you identify infinitesimally nearby fibers.