  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
$\mathcal{C}$.
