  Motivated by applications to the categorical and geometric local Langlands
correspondences, we establish an equivalence between the category of filtered
$\mathcal{D}$-modules on a smooth stack $X$ and the category of
$S^1$-equivariant ind-coherent sheaves on its formal loop space
$\widehat{\mathcal{L}} X$, exchanging compact $\mathcal{D}$-modules with
coherent sheaves, and coherent $\mathcal{D}$-modules with continuous
ind-coherent sheaves. The equivalence yields a sheaf of categories over
$\mathbb{A}^1/\mathbb{G}_m$ whose special fiber is a category of coherent
sheaves on stacks appearing in categorical traces, and whose generic fiber is a
category of equivariant constructible sheaves.
