The Trace Field Theory of a Finite Tensor Category

TL;DR

This paper develops a trace field theory for finite tensor categories, linking modified traces, Hochschild complexes, and topological conformal field theories, revealing new algebraic structures and invariants.

Contribution

It introduces a novel trace field theory framework for finite tensor categories, connecting modified traces with topological and algebraic invariants.

Findings

Constructs a chain complex valued topological conformal field theory from a finite tensor category.

Defines a non-unital homotopy commutative product on Hochschild chains in degree zero.

Recovers the Cartan matrix of the category via the trace of handle elements.

Abstract

Given a finite tensor category $\mathcal{C}$, we prove that a modified trace on the tensor ideal of projective objects can be obtained from a suitable trivialization of the Nakayama functor as right $\mathcal{C}$-module functor. Using a result of Costello, this allows us to associate to any finite tensor category equipped with such a trivialization of the Nakayama functor a chain complex valued topological conformal field theory, the trace field theory. The trace field theory topologically describes the modified trace, the Hattori-Stallings trace, and also the structures induced by them on the Hochschild complex of $\mathcal{C}$. In this article, we focus on implications in the linear (as opposed to differential graded) setting: We use the trace field theory to define a non-unital homotopy commutative product on the Hochschild chains in degree zero. This product is block diagonal and can be described through the handle elements of the trace field theory. Taking the modified trace of the handle elements recovers the Cartan matrix of $\mathcal{C}$.