The Trace Field Theory of a Finite Tensor Category
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Given a finite tensor category 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 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 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 C.
Original language | English |
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Journal | Algebras and Representation Theory |
Volume | 26 |
Issue number | 5 |
Pages (from-to) | 1931-1949 |
ISSN | 1386-923X |
DOIs | |
Publication status | Published - 2023 |
Bibliographical note
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.
- Finite tensor category, Modified trace, Nakayama functor, Topological conformal field theory
Research areas
ID: 342928620