Secondary Traces

Understand

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).