Modified traces and the Nakayama functor

TL;DR

This paper develops a framework connecting modified trace theory with the Nakayama functor in finite abelian categories, providing criteria for existence and uniqueness of traces in tensor categories.

Contribution

It introduces the notion of a $ abla$-twisted trace and establishes a correspondence with natural transformations, advancing the understanding of modified traces in tensor categories.

Findings

Unimodular pivotal finite tensor categories admit non-zero two-sided modified traces if and only if they are spherical.

Ribbon finite tensor categories admit such traces if and only if they are unimodular.

Provides existence and uniqueness criteria for modified traces based on natural transformations.

Abstract

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor $\Sigma$ on a finite abelian category $\mathcal{M}$, we introduce the notion of a $\Sigma$-twisted trace on the class $\mathrm{Proj}(\mathcal{M})$ of projective objects of $\mathcal{M}$. In our framework, there is a one-to-one correspondence between the set of $\Sigma$-twisted traces on $\mathrm{Proj}(\mathcal{M})$ and the set of natural transformations from $\Sigma$ to the Nakayama functor of $\mathcal{M}$. Non-degeneracy and compatibility with the module structure (when $\mathcal{M}$ is a module category over a finite tensor category) of a $\Sigma$-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.