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Fuchs, Jürgenorcid.org/0000-0003-4081-6234

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### Fuchs, Jürgen

### Schweigert, C.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Fortschritte der Physik, ISSN 0015-8208, E-ISSN 1521-3978, article id 1910018Article in journal (Refereed) Published
##### Abstract [en]

##### 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)2-s2.0-85066011661 (Scopus ID)
#####

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Available from: 2019-06-13 Created: 2019-06-13 Last updated: 2019-06-13Bibliographically approved

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).

Universität Hamburg.

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.

Open this publication in new window or tab >>The logarithmic Cardy case: Boundary states and annuli### Fuchs, Jürgen

### Gannon, Terry

### Schaumann, Gregor

### Schweigert, Christoph

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 930, p. 287-327Article in journal (Refereed) Published
##### Abstract [en]

##### 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)
#####

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Available from: 2018-04-06 Created: 2018-04-06 Last updated: 2018-09-05Bibliographically approved

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).

University of Alberta, Edmonton, Canada.

Universität Wien, Austria.

Universität Hamburg, Germany.

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.

Open this publication in new window or tab >>A trace for bimodule categories### Fuchs, Jürgen

### Schaumann, Gregor

### Schweigert, Christoph

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 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]

##### National Category

Other Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-45611 (URN)10.1007/s10485-016-9425-3 (DOI)000399877600004 ()
#####

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##### Funder

Swedish Research Council, 621-2013-4207
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2019-06-17Bibliographically approved

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).

Max-Planck-Institut für Mathematik, Bonn.

Hamburg University.

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.

Open this publication in new window or tab >>Coends in conformal field theory### Fuchs, Jürgen

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).### Schweigert, Christoph

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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Available from: 2017-09-14 Created: 2017-09-14 Last updated: 2019-08-02Bibliographically approved

Hamburg University.

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.

Open this publication in new window or tab >>Consistent systems of correlators in non-semisimple conformal field theory### Fuchs, Jürgen

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).### Schweigert, Christoph

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 307, p. 598-639Article in journal (Refereed) Published
##### Abstract [en]

##### 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 ()
#####

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##### Funder

Swedish Research Council, 621-2013-4207
Available from: 2017-09-14 Created: 2017-09-14 Last updated: 2019-06-17Bibliographically approved

Hamburg University.

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.

Open this publication in new window or tab >>Low-dimensional topology, low-dimensional field theory and representation theory### Fuchs, Jürgen

### Schweigert, Christoph

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 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]

##### 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
#####

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Available from: 2017-12-29 Created: 2017-12-29 Last updated: 2018-06-26Bibliographically approved

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013). Karlstads Univ, Teoret Fys, Univ Gatan 21, S-65188 Karlstad, Sweden..

Univ Hamburg, Fachbereich Math, Algebra & Zahlentheorie, Bundesstr 55, D-20148 Hamburg, Germany..

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".

Open this publication in new window or tab >>Symmetries and defects in three-dimensional topological field theory### Fuchs, Jürgen

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics (from 2013).### Schweigert, Christoph

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 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]

##### 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
#####

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##### Funder

Swedish Research Council, 621-2013-4207
Available from: 2016-08-31 Created: 2016-08-31 Last updated: 2019-06-17Bibliographically approved

Hamburg University.

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.

Open this publication in new window or tab >>On the Brauer Groups of Symmetries of Abelian Dijkgraaf-Witten Theories### Fuchs, Jürgen

### Priel, Jan

### Schweigert, Christoph

### Valentino, Alessandro

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 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]

##### National Category

Physical Sciences
##### Research subject

Physics
##### Identifiers

urn:nbn:se:kau:diva-41597 (URN)10.1007/s00220-015-2420-y (DOI)000358130400003 ()
#####

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Available from: 2016-04-11 Created: 2016-04-11 Last updated: 2017-11-30Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.

Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany..

Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany..

Univ Hamburg, Fachbereich Math, D-20146 Hamburg, Germany..

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.

Open this publication in new window or tab >>On the Killing form of Lie Algebras in Symmetric Ribbon Categories### Buchberger, Igor

### Fuchs, Jürgen

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 11, article id 017Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

NATL ACAD SCI UKRAINE, 2015
##### Keywords

Lie algebra, monoidal category, Killing form, Lie superalgebra
##### National Category

Physical Sciences
##### Research subject

Physics
##### Identifiers

urn:nbn:se:kau:diva-41659 (URN)10.3842/SIGMA.2015.017 (DOI)000350561700001 ()
#####

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Available from: 2016-04-11 Created: 2016-04-11 Last updated: 2017-11-30Bibliographically approved

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics. Karlstads Univ, Teoret Fys, S-65188 Karlstad, Sweden..

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.

As a step towards the structure theory of Lie algebras in symmetric monoidal categories we establish results involving the Killing form. The proper categorical setting for discussing these issues are symmetric ribbon categories.

Open this publication in new window or tab >>Surface defects and symmetries### Fuchs, Jürgen

### Schweigert, Christoph

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: XXXTH International Colloquium on Group Theoretical Methods in Physics (ICGTMP) (GROUP30), Institute of Physics (IOP), 2015, article id 012002Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Institute of Physics (IOP), 2015
##### Series

Journal of Physics Conference Series, ISSN 1742-6588 ; 597
##### National Category

Physical Sciences
##### Research subject

Physics
##### Identifiers

urn:nbn:se:kau:diva-41650 (URN)10.1088/1742-6596/597/1/012002 (DOI)000354929400002 ()
##### Conference

30th International Colloquium on Group Theoretical Methods in Physics (ICGTMP), Jul 14-18, 2014, Ghent, Belgium
#####

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Available from: 2016-04-11 Created: 2016-04-11 Last updated: 2016-05-31Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.

Univ Hamburg, Fachbereich Math, Algebra & Zahlentheorie, D-20148 Hamburg, Germany..

In quantum field theory, defects of various codimensions are natural ingredients and carry a lot of interesting information. In this contribution we concentrate on topological quantum field theories in three dimensions, with a particular focus on Dijkgraaf-Witten theories with abelian gauge group. Surface defects in Dijkgraaf-Witten theories have applications in solid state physics, topological quantum computing and conformal field theory. We explain that symmetries in these topological field theories are naturally defined in terms of invertible topological surface defects and are thus Brauer-Picard groups.