The Trace Field Theory of a Finite Tensor Category

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Standard

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

Harvard

Schweigert, C & Woike, L 2023, 'The Trace Field Theory of a Finite Tensor Category', Algebras and Representation Theory, vol. 26, no. 5, pp. 1931-1949. https://doi.org/10.1007/s10468-022-10147-0

APA

Schweigert, C., & Woike, L. (2023). The Trace Field Theory of a Finite Tensor Category. Algebras and Representation Theory, 26(5), 1931-1949. https://doi.org/10.1007/s10468-022-10147-0

Vancouver

Schweigert C, Woike L. The Trace Field Theory of a Finite Tensor Category. Algebras and Representation Theory. 2023;26(5):1931-1949. https://doi.org/10.1007/s10468-022-10147-0

Author

Schweigert, Christoph ; Woike, Lukas. / The Trace Field Theory of a Finite Tensor Category. In: Algebras and Representation Theory. 2023 ; Vol. 26, No. 5. pp. 1931-1949.

Bibtex

@article{af17fa410403442e8a51177824a473d7,
title = "The Trace Field Theory of a Finite Tensor Category",
abstract = "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.",
keywords = "Finite tensor category, Modified trace, Nakayama functor, Topological conformal field theory",
author = "Christoph Schweigert and Lukas Woike",
note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer Nature B.V.",
year = "2023",
doi = "10.1007/s10468-022-10147-0",
language = "English",
volume = "26",
pages = "1931--1949",
journal = "Algebras and Representation Theory",
issn = "1386-923X",
publisher = "Springer",
number = "5",

}

RIS

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