Coherent sheaves, sheared D-modules, and Hochschild cochains

We show that the category of ind-coherent sheaves on a quasi-smooth scheme is naturally tensored over the category of sheared D-modules on its shifted cotangent bundle, commuting with its natural action of categorified Hoschschild cochains. We prove that it defines a Morita equivalence as such. We then extend these results to quasi-smooth Artin stacks. As a consequence of our formalism, we are able to articulate a precise sense in which the space of unramified automorphic functions over a function field localizes over the stack of arithmetic Arthur parameters.