Riemann-Roch theorems in monoidal 2-categories

TL;DR

This paper generalizes Riemann-Roch theorems to monoidal 2-categories, extending classical results about Euler classes and Hochschild homology to a higher categorical framework.

Contribution

It introduces a broad generalization of Riemann-Roch theorems from dg-algebras to monoidal bicategories, incorporating traces of non-identity maps and spectral versions.

Findings

Proves generalized Riemann-Roch theorems in monoidal 2-categories

Connects the results to a 2-dimensional cobordism hypothesis

Extends classical Euler class properties to a higher categorical setting

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 familiar pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann-Roch theorems of Shklyarov and Petit. 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 monoidal bicategories and from 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.