Riemann–Roch theorems in monoidal 2-categories

Abstract

Abstract Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann–Roch theorems of [21, 14]. In this paper, we prove a broad generalization of these Riemann–Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to symmetric monoidal bicategories and from the Euler class to traces of non-identity maps. Our generalization also implies spectral Riemann–Roch theorems. We regard this result as an instantiation of a 2-dimensional generalized cobordism hypothesis. This perspective draws the result close to many others that generalize results about Euler characteristics and classes to bicategorical traces.