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Publications (10 of 91) Show all publications
Fuchs, J. & Schweigert, C. (2023). Internal Natural Transformations and Frobenius Algebras in the Drinfeld Center. Transformation groups, 28, 733-768
Open this publication in new window or tab >>Internal Natural Transformations and Frobenius Algebras in the Drinfeld Center
2023 (English)In: Transformation groups, ISSN 1083-4362, E-ISSN 1531-586X, Vol. 28, p. 733-768Article in journal (Refereed) Published
Abstract [en]

For M and N finite module categories over a finite tensor category C, the category Rex(C)(M, N) of right exact module functors is a finite module category over the Drinfeld center Z(C). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C-C-bimodule functors to objects of Z(C), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if M and N are exact C-modules and C is pivotal, then the Z(C)-module Rex(C)(M,N) is exact. We compute its relative Serre functor and show that if M and N are even pivotal module categories, then Rex(C)(M, N) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z(C).

Place, publisher, year, edition, pages
Springer, 2023
National Category
Mathematics Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-88004 (URN)10.1007/s00031-021-09678-5 (DOI)000733868600001 ()2-s2.0-85121613594 (Scopus ID)
Available from: 2022-01-07 Created: 2022-01-07 Last updated: 2023-07-04Bibliographically approved
Fuchs, J. & Grøsfjeld, T. (2023). Tetrahedral symmetry of 6j-symbols in fusion categories. Journal of Pure and Applied Algebra, 227(1), Article ID 107112.
Open this publication in new window or tab >>Tetrahedral symmetry of 6j-symbols in fusion categories
2023 (English)In: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 227, no 1, article id 107112Article in journal (Refereed) Published
Abstract [en]

We establish tetrahedral symmetries of 6j-symbols for arbitrary fusion categories under minimal assumptions. As a convenient tool for our calculations we introduce the notion of a veined fusion category, which is generated by a finite set of simple objects but is larger than its skeleton. Every fusion category C contains veined fusion subcategories that are monoidally equivalent to C and which suffice to compute many categorical properties for C. The notion of a veined fusion category does not assume the presence of a pivotal structure, and thus in particular does not assume unitarity. We also exhibit the geometric origin of the algebraic statements for the 6j-symbols.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
6j-symbols, Fusion categories, Monoidal categories, Tetrahedral symmetry
National Category
Fusion, Plasma and Space Physics
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-91284 (URN)10.1016/j.jpaa.2022.107112 (DOI)000888220400001 ()2-s2.0-85130414419 (Scopus ID)
Funder
Swedish Research Council, 2017-03836
Available from: 2022-07-08 Created: 2022-07-08 Last updated: 2022-12-15Bibliographically approved
Fuchs, J., Schaumann, G. & Schweigert, C. (2022). A Modular Functor From State Sums For Finite Tensor Categories And Their Bimodules. Theory and Applications of Categories, 38, 436-594
Open this publication in new window or tab >>A Modular Functor From State Sums For Finite Tensor Categories And Their Bimodules
2022 (English)In: Theory and Applications of Categories, ISSN 1201-561X, Vol. 38, p. 436-594Article in journal (Refereed) Published
Abstract [en]

We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need to require that the finite categories appearing in our construction are semisimple, nor that the finite tensor categories that are assigned to two-dimensional strata are endowed with a pivotal structure. Our prescription can be understood as a state-sum construction. The state-sum variables are assigned to one-dimensional strata and take values in bimodule categories over finite tensor categories, whereby we also account for the presence of boundaries and defects. Our construction allows us to explicitly compute functors associated to surfaces and representations of mapping class groups acting on them.

Place, publisher, year, edition, pages
MOUNT ALLISON UNIV, 2022
Keywords
modular functor, state-sum construction, finite tensor category, monoidal bicategory, mapping class group, factorization, topological defect
National Category
Algebra and Logic Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-89452 (URN)000773400500001 ()2-s2.0-85135314248 (Scopus ID)
Available from: 2022-04-11 Created: 2022-04-11 Last updated: 2024-01-23Bibliographically approved
Fuchs, J., Schweigert, C. & Yang, Y. (2022). Correlators from String Nets (1ed.). In: Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki (Ed.), String-Net Construction of RCFT Correlators: (pp. 35-59). Springer, 45
Open this publication in new window or tab >>Correlators from String Nets
2022 (English)In: String-Net Construction of RCFT Correlators / [ed] Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki, Springer, 2022, 1, Vol. 45, p. 35-59Chapter in book (Refereed)
Abstract [en]

This chapter contains our central construction, the one of string-net correlators, as defined in Definition 3.26. The chief result—Theorem 3.28—states that the construction does provide a consistent system of correlators. 

Place, publisher, year, edition, pages
Springer, 2022 Edition: 1
Series
SpringerBriefs in Mathematical Physics, ISSN 2197-1757, E-ISSN 2197-1765 ; 45
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-93605 (URN)10.1007/978-3-031-14682-4_3 (DOI)2-s2.0-85146218098 (Scopus ID)978-3-031-14681-7 (ISBN)978-3-031-14682-4 (ISBN)
Available from: 2023-02-13 Created: 2023-02-13 Last updated: 2023-02-13Bibliographically approved
Fuchs, J., Schweigert, C. & Yang, Y. (2022). Correlators in Rational Conformal Field Theory (1ed.). In: Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki (Ed.), String-Net Construction of RCFT Correlators: (pp. 9-34). Springer, 45
Open this publication in new window or tab >>Correlators in Rational Conformal Field Theory
2022 (English)In: String-Net Construction of RCFT Correlators / [ed] Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki, Springer, 2022, 1, Vol. 45, p. 9-34Chapter in book (Refereed)
Abstract [en]

In this chapter we turn the physics task of finding correlation functions into the mathematically precise problem of determining specific elements in spaces of conformal blocks. These spaces are described in terms of an open-closed modular functor.

Place, publisher, year, edition, pages
Springer, 2022 Edition: 1
Series
SpringerBriefs in Mathematical Physics, ISSN 2197-1757, E-ISSN 2197-1765 ; 45
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-93600 (URN)10.1007/978-3-031-14682-4_2 (DOI)2-s2.0-85146253335 (Scopus ID)978-3-031-14681-7 (ISBN)978-3-031-14682-4 (ISBN)
Available from: 2023-02-13 Created: 2023-02-13 Last updated: 2023-02-13Bibliographically approved
Fuchs, J., Schweigert, C. & Yang, Y. (2022). Correlators of Particular Interest. In: Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki (Ed.), String-Net Construction of RCFT Correlators: (pp. 61-84). Springer, 45
Open this publication in new window or tab >>Correlators of Particular Interest
2022 (English)In: String-Net Construction of RCFT Correlators / [ed] Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki, Springer, 2022, Vol. 45, p. 61-84Chapter in book (Refereed)
Abstract [en]

The prescription presented in the previous chapter applies to all correlators of the theory. However, a few specific correlators are of particular interest; concretely, partition functions on the one hand, and correlators which determine operator products, i.e. composition morphisms on the field objects, on the other hand. The present chapter provides detailed information about such correlators. 

Place, publisher, year, edition, pages
Springer, 2022
Series
SpringerBriefs in Mathematical Physics, ISSN 2197-1757, E-ISSN 2197-1765 ; 45
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-93599 (URN)10.1007/978-3-031-14682-4_4 (DOI)2-s2.0-85146257635 (Scopus ID)978-3-031-14681-7 (ISBN)978-3-031-14682-4 (ISBN)
Available from: 2023-02-13 Created: 2023-02-13 Last updated: 2023-02-13Bibliographically approved
Fuchs, J., Schweigert, C. & Yang, Y. (2022). Internal Eckmann–Hilton Relation (1ed.). In: Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki (Ed.), String-Net Construction of RCFT Correlators: (pp. 85-93). Springer, 45
Open this publication in new window or tab >>Internal Eckmann–Hilton Relation
2022 (English)In: String-Net Construction of RCFT Correlators / [ed] Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki, Springer, 2022, 1, Vol. 45, p. 85-93Chapter in book (Refereed)
Abstract [en]

The exchange law for ordinary natural transformations turns out to have an analogue for internal natural transformations. To obtain this analogue, which we call the internal Eckmann–Hilton relation, we introduce three braided colored operads. The string-net correlator defined in Sect. 3.5 provides a morphism from the braided colored operad of world sheets to the one of string nets.

Place, publisher, year, edition, pages
Springer, 2022 Edition: 1
Series
SpringerBriefs in Mathematical Physics, ISSN 2197-1757, E-ISSN 2197-1765 ; 45
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-93603 (URN)10.1007/978-3-031-14682-4_5 (DOI)2-s2.0-85146237116 (Scopus ID)978-3-031-14681-7 (ISBN)978-3-031-14682-4 (ISBN)
Available from: 2023-02-13 Created: 2023-02-13 Last updated: 2023-02-13Bibliographically approved
Fuchs, J., Schweigert, C. & Yang, Y. (2022). Introduction. In: Nathanaël Berestycki; Mihalis Dafermosl; Atsuo Kuniba; Matilde Marcolli; Bruno Nachtergaele; Hal Tasaki (Ed.), String-Net Construction of RCFT Correlators: (pp. 1-7). Springer, 45
Open this publication in new window or tab >>Introduction
2022 (English)In: String-Net Construction of RCFT Correlators / [ed] Nathanaël Berestycki; Mihalis Dafermosl; Atsuo Kuniba; Matilde Marcolli; Bruno Nachtergaele; Hal Tasaki, Springer, 2022, Vol. 45, p. 1-7Chapter in book (Refereed)
Abstract [en]

The main topic of this book is a novel construction of the correlators of two-dimensional rational conformal field theories on arbitrary world sheets, allowing in particular for physical boundaries as well as for defects and general defect junctions.

Place, publisher, year, edition, pages
Springer, 2022
Series
SpringerBriefs in Mathematical Physics, ISSN 2197-1757, E-ISSN 2197-1765 ; 45
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-93602 (URN)10.1007/978-3-031-14682-4_1 (DOI)2-s2.0-85146240803 (Scopus ID)978-3-031-14681-7 (ISBN)978-3-031-14682-4 (ISBN)
Available from: 2023-02-13 Created: 2023-02-13 Last updated: 2023-06-19Bibliographically approved
Fuchs, J., Schweigert, C. & Yang, Y. (2022). Outlook: Universal Correlators (1ed.). In: Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki (Ed.), String-Net Construction of RCFT Correlators: (pp. 95-100). Springer, 45
Open this publication in new window or tab >>Outlook: Universal Correlators
2022 (English)In: String-Net Construction of RCFT Correlators / [ed] Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki, Springer, 2022, 1, Vol. 45, p. 95-100Chapter in book (Refereed)
Abstract [en]

It can happen that two world sheets with different defect networks and thus different defect patterns—that is, assignments of labels to the cells of the world sheet—turn out to have the same correlator. Existing approaches to correlators, including the string-net construction considered in the previous chapters, treat such world sheets as different even though they cannot be distinguished by their correlators

Place, publisher, year, edition, pages
Springer, 2022 Edition: 1
Series
SpringerBriefs in Mathematical Physics, ISSN 2197-1757, E-ISSN 2197-1765 ; 45
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-93601 (URN)10.1007/978-3-031-14682-4_6 (DOI)2-s2.0-85146243025 (Scopus ID)978-3-031-14681-7 (ISBN)978-3-031-14682-4 (ISBN)
Available from: 2023-02-13 Created: 2023-02-13 Last updated: 2023-02-13Bibliographically approved
Fuchs, J., Schweigert, C. & Yang, Y. (2022). Preface (1ed.). In: Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki (Ed.), String-Net Construction of RCFT Correlators: (pp. v-vii). Springer, 45
Open this publication in new window or tab >>Preface
2022 (English)In: String-Net Construction of RCFT Correlators / [ed] Nathanaël Berestycki, Mihalis Dafermosl, Atsuo Kuniba, Matilde Marcolli, Bruno Nachtergaele and Hal Tasaki, Springer, 2022, 1, Vol. 45, p. v-viiChapter in book (Refereed)
Place, publisher, year, edition, pages
Springer, 2022 Edition: 1
Series
SpringerBriefs in Mathematical Physics, ISSN 2197-1757, E-ISSN 2197-1765 ; 45
National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-93604 (URN)10.1007/978-3-031-14682-4 (DOI)2-s2.0-85146226202 (Scopus ID)978-3-031-14681-7 (ISBN)978-3-031-14682-4 (ISBN)
Funder
German Research Foundation (DFG), 390833306, SCHW1162/6-1Swedish Research Council, 2017-03836
Available from: 2023-02-13 Created: 2023-02-13 Last updated: 2023-02-13Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-4081-6234

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