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Motivated by applications to the categorical and geometric local Langlandscorrespondences, 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 withcoherent sheaves, and coherent $\mathcal{D}$-modules with continuousind-coherent sheaves. The equivalence yields a sheaf of categories over$\mathbb{A}^1/\mathbb{G}_m$ whose special fiber is a category of coherentsheaves on stacks appearing in categorical traces, and whose generic fiber is acategory of equivariant constructible sheaves.

Categorical cyclic homology and filtered $\mathcal{D}$-modules on stacks: Koszul duality