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Fuchs, J., Schaumann, G. & Schweigert, C. (2019). Eilenberg-Watts calculus for finite categories and a bimodule Radford S4 theorem. Transactions of the American Mathematical Society
Open this publication in new window or tab >>Eilenberg-Watts calculus for finite categories and a bimodule Radford S4 theorem
2019 (English)In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Article in journal (Refereed) Epub ahead of print
Abstract [en]

We obtain Morita invariant versions of Eilenberg-Watts type the-orems, relating Deligne products of finite linear categories to categories of leftexact as well as of right exact functors. This makes it possible to switch be-tween different functor categories as well as Deligne products, which is oftenvery convenient. For instance, we can show that applying the equivalence fromleft exact to right exact functors to the identity functor, regarded as a left exactfunctor, gives a Nakayama functor. The equivalences of categories we exhibitare compatible with the structure of module categories over finite tensor cat-egories. This leads to a generalization of Radford’sS4-theorem to bimodulecategories. We also explain the relation of our construction to relative Serrefunctors on module categories that are constructed via inner Hom functors.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2019
National Category
Physical Sciences
Research subject
Mathematics; Physics
Identifiers
urn:nbn:se:kau:diva-74944 (URN)10.1090/tran/7838 (DOI)
Funder
Swedish Research Council, 621-2013-4207
Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-17Bibliographically approved
Fuchs, J. & Schweigert, C. (2019). Full Logarithmic Conformal Field theory — an Attempt at a Status Report: LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. Fortschritte der Physik, Article ID 1910018.
Open this publication in new window or tab >>Full Logarithmic Conformal Field theory — an Attempt at a Status Report: LMS/EPSRC Durham Symposium on Higher Structures in M-Theory
2019 (English)In: Fortschritte der Physik, ISSN 0015-8208, E-ISSN 1521-3978, article id 1910018Article in journal (Refereed) Published
Abstract [en]

Logarithmic conformal field theories are based on vertex algebras with non-semisimple representation categories. While examples of such theories have been known for more than 25 years, some crucial aspects of local logarithmic CFTs have been understood only recently, with the help of a description of conformal blocks by modular functors. We present some of these results, both about bulk fields and about boundary fields and boundary states. We also describe some recent progress towards a derived modular functor. This is a summary of work with Terry Gannon, Simon Lentner, Svea Mierach, Gregor Schaumann and Yorck Sommerhäuser.

Place, publisher, year, edition, pages
Wiley-VCH Verlag, 2019
Keywords
Lego-Teichmüller game, logarithmic conformal field theory, modular Frobenius algebra, modular functor, modular tensor category
National Category
Physical Sciences
Identifiers
urn:nbn:se:kau:diva-72524 (URN)10.1002/prop.201910018 (DOI)000486266200019 ()2-s2.0-85066011661 (Scopus ID)
Available from: 2019-06-13 Created: 2019-06-13 Last updated: 2019-10-10Bibliographically approved
Fuchs, J., Gannon, T., Schaumann, G. & Schweigert, C. (2018). The logarithmic Cardy case: Boundary states and annuli. Nuclear Physics B, 930, 287-327
Open this publication in new window or tab >>The logarithmic Cardy case: Boundary states and annuli
2018 (English)In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 930, p. 287-327Article in journal (Refereed) Published
Abstract [en]

We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.

Place, publisher, year, edition, pages
Elsevier, 2018
National Category
Mathematics Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-66940 (URN)10.1016/j.nuclphysb.2018.03.005 (DOI)000435647100012 ()2-s2.0-85044165224 (Scopus ID)
Available from: 2018-04-06 Created: 2018-04-06 Last updated: 2018-09-05Bibliographically approved
Fuchs, J., Schaumann, G. & Schweigert, C. (2017). A trace for bimodule categories. Applied Categorical Structures, 25(2), 227-268
Open this publication in new window or tab >>A trace for bimodule categories
2017 (English)In: Applied Categorical Structures, ISSN 0927-2852, E-ISSN 1572-9095, Vol. 25, no 2, p. 227-268Article in journal (Refereed) Published
Abstract [en]

We study a 2-functor that assigns to a bimodule category over a finite k-linear tensor category a k-linear abelian category. This 2-functor can be regarded as a category-valued tracefor 1-morphisms in the tricategory of finite tensor categories. It is defined by a universalproperty that is a categorification of Hochschild homology of bimodules over an algebra.We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance.Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Wittentopological field theories.

National Category
Other Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-45611 (URN)10.1007/s10485-016-9425-3 (DOI)000399877600004 ()
Funder
Swedish Research Council, 621-2013-4207
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2019-06-17Bibliographically approved
Fuchs, J. & Schweigert, C. (2017). Coends in conformal field theory. In: Katrina Barron, Elizabeth Jurisich, Antun Milas, Kailash Misra (Ed.), Lie Algebras, Vertex Operator Algebras, and Related Topics: (pp. 65-81). American Mathematical Society (AMS)
Open this publication in new window or tab >>Coends in conformal field theory
2017 (English)In: Lie Algebras, Vertex Operator Algebras, and Related Topics / [ed] Katrina Barron, Elizabeth Jurisich, Antun Milas, Kailash Misra, American Mathematical Society (AMS), 2017, p. 65-81Chapter in book (Refereed)
Abstract [en]

The idea of "summing over all intermediate states" that is central for implementing locality in quantum systems can be realized by coend constructions. In the concrete case of systems of conformal blocks for a certain class of conformal vertex algebras, one deals with coends in functor categories. Working with these coends involves quite a few subtleties which, even though they have in principle already been understood twenty years ago, have not been sufficiently appreciated by the conformal field theory community.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2017
Series
Contemporary Mathematics ; 695
National Category
Subatomic Physics
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-63716 (URN)10.1090/conm/695/13996 (DOI)000431843300006 ()978-1-4704-2666-8 (ISBN)978-1-4704-4196-8 (ISBN)
Available from: 2017-09-14 Created: 2017-09-14 Last updated: 2019-08-02Bibliographically approved
Fuchs, J. & Schweigert, C. (2017). Consistent systems of correlators in non-semisimple conformal field theory. Advances in Mathematics, 307, 598-639
Open this publication in new window or tab >>Consistent systems of correlators in non-semisimple conformal field theory
2017 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 307, p. 598-639Article in journal (Refereed) Published
Abstract [en]

Based on the modular functor associated with a -- not necessarily semisimple -- finite non-degenerate ribbon category D, we present a definition of a consistent system of bulk field correlators for a conformal field theory which comprises invariance under mapping class group actions and compatibility with the sewing of surfaces. We show that when restricting to surfaces of genus zero such systems are in bijection with commutative symmetric Frobenius algebras in D, while for surfaces of any genus they are in bijection with modular Frobenius algebras in D. This provides additional insight into structures familiar from rational conformal field theories and extends them to rigid logarithmic conformal field theories.

Place, publisher, year, edition, pages
Academic Press, 2017
National Category
Subatomic Physics
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-63715 (URN)10.1016/j.aim.2016.11.020 (DOI)000409285300015 ()
Funder
Swedish Research Council, 621-2013-4207
Available from: 2017-09-14 Created: 2017-09-14 Last updated: 2019-06-17Bibliographically approved
Fuchs, J. & Schweigert, C. (2017). Low-dimensional topology, low-dimensional field theory and representation theory. In: Krause, H Littelmann, P Malle, G Neeb, KH Schweigert, C (Ed.), Representation theory: Current trends and perspectives. Paper presented at Research Priority Programme SPP 1388 "Representation theory", Hamburg Universität (pp. 255-267). European Mathematical Society Publishing House
Open this publication in new window or tab >>Low-dimensional topology, low-dimensional field theory and representation theory
2017 (English)In: Representation theory: Current trends and perspectives / [ed] Krause, H Littelmann, P Malle, G Neeb, KH Schweigert, C, European Mathematical Society Publishing House, 2017, p. 255-267Conference paper, Published paper (Refereed)
Abstract [en]

Structures in low-dimensional topology and low-dimensional geometry often combined with ideas from (quantum) field theory can explain and inspire concepts in algebra and in representation theory and their categorified versions. We present a personal view on some of these instances which have appeared within the Research Priority Programme SPP 1388 "Representation theory".

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2017
Series
Hamburger Beiträge zur Mathematik ; 571
Keywords
Topological field theory, tensor categories, categorification
National Category
Mathematics
Identifiers
urn:nbn:se:kau:diva-65475 (URN)000398985900010 ()
Conference
Research Priority Programme SPP 1388 "Representation theory", Hamburg Universität
Available from: 2017-12-29 Created: 2017-12-29 Last updated: 2018-06-26Bibliographically approved
Fuchs, J. & Schweigert, C. (2016). Categorical tools for state sum constructions. Paper presented at Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker‐Vereinigung (DMV). Proceedings in Applied Mathematics and Mechanics: PAMM, 16(1), 911-912
Open this publication in new window or tab >>Categorical tools for state sum constructions
2016 (English)In: Proceedings in Applied Mathematics and Mechanics: PAMM, ISSN 1617-7061, E-ISSN 1617-7061, Vol. 16, no 1, p. 911-912Article in journal (Refereed) Published
Abstract [en]

Surface defects in (extended) three‐dimensional topological field theories have important applications, ranging from solid state physics to computations of Brauer‐Picard groups of representation categories. We present some categorical tools that are needed in such a construction, including in particular a category‐valued trace.

Place, publisher, year, edition, pages
Wiley-Blackwell, 2016
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-74946 (URN)10.1002/pamm.201610444 (DOI)
Conference
Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker‐Vereinigung (DMV)
Funder
Swedish Research Council, 621-2013-4207
Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-16Bibliographically approved
Fuchs, J. & Schweigert, C. (2016). Symmetries and defects in three-dimensional topological field theory. In: Vincent Bouchard, Charles Doran, Stefan Méndez-Diez, Callum Quigley (Ed.), Proceedings of Symposia in Pure Mathematics: . Paper presented at Conference on String Math 2014 (pp. 21-40). Providence: American Mathematical Society (AMS), 93
Open this publication in new window or tab >>Symmetries and defects in three-dimensional topological field theory
2016 (English)In: Proceedings of Symposia in Pure Mathematics, Providence: American Mathematical Society (AMS), 2016, Vol. 93, p. 21-40Conference paper, Published paper (Refereed)
Abstract [en]

Boundary conditions and defects of any codimension are natural parts of any quantum field theory. Surface defects in three-dimensionaltopological field theories of Turaev-Reshetikhin type have applications to two-dimensional conformal field theories, in solid state physics and in quantumcomputing. We explain an obstruction to the existence of surface defects thattakes values in a Witt group. We then turn to surface defects in Dijkgraaf-Witten theories and their construction in terms of relative bundles; this allowsone to exhibit Brauer-Picard groups as symmetry groups of three-dimensionaltopological field theories.

Place, publisher, year, edition, pages
Providence: American Mathematical Society (AMS), 2016
Series
Proceedings of Symposia in Pure Mathematics, ISSN 2324-707X
Keywords
Topological field theory; tensor categories; topological defects; Brauer-Picard group
National Category
Subatomic Physics
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-45610 (URN)000379639000002 ()978-1-4704-1992-9 (ISBN)
Conference
Conference on String Math 2014
Funder
Swedish Research Council, 621-2013-4207
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2019-06-17Bibliographically approved
Fuchs, J., Priel, J., Schweigert, C. & Valentino, A. (2015). On the Brauer Groups of Symmetries of Abelian Dijkgraaf-Witten Theories. Communications in Mathematical Physics, 339(2), 385-405
Open this publication in new window or tab >>On the Brauer Groups of Symmetries of Abelian Dijkgraaf-Witten Theories
2015 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 339, no 2, p. 385-405Article in journal (Refereed) Published
Abstract [en]

Symmetries of three-dimensional topological field theories are naturally defined in terms of invertible topological surface defects. Symmetry groups are thus Brauer-Picard groups. We present a gauge theoretic realization of all symmetries of abelian Dijkgraaf-Witten theories. The symmetry group for a Dijkgraaf-Witten theory with gauge group a finite abelian group A, and with vanishing 3-cocycle, is generated by group automorphisms of A, by automorphisms of the trivial Chern-Simons 2-gerbe on the stack of A-bundles, and by partial e-m dualities. We show that transmission functors naturally extracted from extended topological field theories with surface defects give a physical realization of the bijection between invertible bimodule categories of a fusion category and braided auto-equivalences of its Drinfeld center . The latter provides the labels for bulk Wilson lines; it follows that a symmetry is completely characterized by its action on bulk Wilson lines.

National Category
Physical Sciences
Research subject
Physics
Identifiers
urn:nbn:se:kau:diva-41597 (URN)10.1007/s00220-015-2420-y (DOI)000358130400003 ()
Available from: 2016-04-11 Created: 2016-04-11 Last updated: 2017-11-30Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-4081-6234

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