Secondary Traces

TL;DR

This paper introduces the secondary trace, an invariant for commuting endomorphisms in a higher category, and proves a formula revealing its symmetries, with applications to modular invariance and a generalized Lefschetz formula.

Contribution

It establishes a secondary trace formula in a symmetric monoidal 2-category, connecting it to the mapping class group and cobordism hypothesis, with concrete Morse-theoretic proofs.

Findings

Proves a secondary trace formula with symmetry properties.

Shows modular invariance of the 2-character function.

Generalizes the Atiyah-Bott-Lefschetz formula for stacks.

Abstract

We study an invariant, the secondary trace, attached to two commuting endomorphisms of a 2-dualizable object in a symmetric monoidal higher category. We establish a secondary trace formula which encodes the natural symmetries of this invariant, identifying different realizations as an iterated trace. The proof consists of elementary Morse-theoretic arguments (with many accompanying pictures included) and may be seen as a concrete realization of the cobordism hypothesis with singularities on a marked 2-torus. From this perspective, our main result identifies the secondary trace with two alternative presentations coming from the standard generators S and T of the mapping class group SL_2(Z). We include two immediate consequences of the established invariance. The first is a modular invariance property for the 2-class function on a group arising as the 2-character of a categorical representation. The second is a generalization (for coherent sheaves or D-modules) of the Atiyah-Bott-Lefschetz formula conjectured by Frenkel-Ng\^o in the case of a self-map of a smooth and proper stack over a general base.