Higher traces, noncommutative motives, and the categorified Chern character

TL;DR

This paper introduces a categorified version of the Chern character for algebraic stacks, connecting noncommutative motives with equivariant perfect complexes on derived loop stacks, and explores its implications for secondary K-theory.

Contribution

It develops a new categorification of the Chern character using noncommutative motives and analyzes trace functoriality in higher categories, extending previous work.

Findings

Categorified Chern character as a symmetric monoidal functor

Factorization of secondary Chern character through secondary K-theory

New insights into trace functoriality in symmetric monoidal (infinity,n)-categories

Abstract

We propose a categorification of the Chern character that refines earlier work of To\"en and Vezzosi and of Ganter and Kapranov. If X is an algebraic stack, our categorified Chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over X, which we introduce, to S1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that To\"en and Vezzosi's secondary Chern character factors through secondary K-theory. Our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal (infinity,n)-categories, which is of independent interest.