2013/05/30 by Dani Ben‐Zvi, David Ben-Zvi, David Nadler +3
Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.AG #math.CT #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.1305.7177
Preliminary version, comments welcome! References added
openalex publication_date 2013/05/30 · arxiv created 2013/06/03 · arxiv updated 2013/06/04 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
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 SL2(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-Ngô in the case of a self-map of a smooth and proper stack over a general base.