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

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.