The Trace Field Theory of a Finite Tensor Category
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The Trace Field Theory of a Finite Tensor Category. / Schweigert, Christoph; Woike, Lukas.
In: Algebras and Representation Theory, Vol. 26, No. 5, 2023, p. 1931-1949.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - The Trace Field Theory of a Finite Tensor Category
AU - Schweigert, Christoph
AU - Woike, Lukas
N1 - Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Nature B.V.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - Finite tensor category
KW - Modified trace
KW - Nakayama functor
KW - Topological conformal field theory
UR - http://www.scopus.com/inward/record.url?scp=85137422281&partnerID=8YFLogxK
U2 - 10.1007/s10468-022-10147-0
DO - 10.1007/s10468-022-10147-0
M3 - Journal article
AN - SCOPUS:85137422281
VL - 26
SP - 1931
EP - 1949
JO - Algebras and Representation Theory
JF - Algebras and Representation Theory
SN - 1386-923X
IS - 5
ER -
ID: 342928620