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• 1.
CTQM, Department of Mathematical Sciences, Aarhus University, Aarhus, Denmark.
On Reducibility of Mapping Class Group Representations: The SU(N) case2010In: Noncommutative structures in mathematics and physics / [ed] Stefaan Caenepeel, Jürgen Fuchs, Simone Gutt, Christophe Schweigert, Alexander Stolin, Freddy Van Oystaeyen, Brussels: Koninklijke vlaamse academie van Belgie voor Wetenschappen en kunsten, 2010, p. 27-45Conference paper (Refereed)

We review and extend the results of [1] that gives a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. This criterion is derived using methods developed to describe rational conformal field theories, making use of Frobenius algebras and their representations in modular categories. Given a modular category C, a rational conformal field theory can be constructed from a Frobenius algebra A in C. We show that if C contains a symmetric special Frobenius algebra A such that the torus partition function Z(A) of the corresponding conformal field theory is non-trivial, implying reducibility of the genus 1 representation of the modular group, then the representation of the genus g mapping class group constructed from C is reducible for every g\geq 1. We also extend the number of examples where we can show reducibility significantly by establishing the existence of algebras with the required properties using methods developed by Fuchs, Runkel and Schweigert. As a result we show that the quantum representations are reducible in the SU(N) case, N>2, for all levels k\in \mathbb{N}. The SU(2) case was treated explicitly in [1], showing reducibility for even levels k\geq 4.

• 2.
Center for Quantum Geometry of Moduli Spaces, University of Aarhus, Århus, Denmark.
Reducibility of quantum representations of mapping class groups2010In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 91, no 3, p. 215-239Article in journal (Refereed)

In this paper we provide a general condition for the reducibility of the Reshetikhin–Turaev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a non-trivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are reducible. In particular, for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) Reshetikhin–Turaev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete decomposition into irreducible representations for high enough levels.

• 3.
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK.
Department of Physics, Nanjing University, Nanjing, China.
Modular invariant partition functions for noncompact G/Ad(H) models2011In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 83, no 8, article id 086007Article in journal (Refereed)

We propose a spectrum for a class of gauged non-compact G/Ad(H) WZNW models, including spectrally flowed images of highest, lowest, and mixed extremal weight modules. These are combined into blocks whose characters, due to the Lorentzian signature of the target space, are divergent and treated as formal expressions in need of regularisation. Assuming that this is possible, we show that these extended characters transform linearly under modular transformations, and can be used to write down modular invariant partition functions.

• 4.
Department of Physics, Nanjing University, Nanjing, China.
Department of Physics, Nanjing University, Nanjing, China. Department of Physics, Nanjing University, Nanjing, China. Faculty of Technology and Science, Karlstad University, Karlstad, Sweden. Faculty of Technology and Science, Karlstad University, Karlstad, Sweden.
No-ghost theorem for the bosonic Nappi-Witten string2009In: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 80, no 8, article id 086003Article in journal (Refereed)

We prove a no-ghost theorem for a bosonic string propagating in Nappi-Witten spacetime. This is achieved in two steps. We first demonstrate unitarity for a class of NW/U(1) modules: the norm of any state which is primary with respect to a chosen timelike U(1) is non-negative. We then show that physical states - states satisfying the Virasoro constraints - in a class of modules of an affinisation of the Nappi-Witten algebra are contained in the NW/U(1) modules. Similar to the case of strings on AdS3, in order to saturate the spectrum obtained in light-cone quantization we are led to include modules with energy not bounded from below, which are related to modules with energy bounded from below by spectral flow automorphisms.

• 5.
Örebro University, School of Science and Technology.
Conformal Field Theory and the Holgoraphic Principle2002Doctoral thesis, monograph (Other academic)

This thesis is based on five research papers with string theory as the least common denominator. The text is divided in two parts. The first part is an introduction to the constructions used in the papers; string theory, conformal field theory, topological field theory and three-dimensional gravity. The second part is a review of a concept believed to be important in any theory of quantum gravity, the so called holographic principle.A relation between three-dimensional topological field theory and two-dimensional conformal field theory is re-established in a classical formulation. This result is then used to investigate holography in a toy model, gravity in three dimensions. A construction is proposed which yields the Bekenstein-Hawking entropy of a three-dimensional black hole. The same mechanism is shown not to be responsible for the entropy of Kerr-de Sitter spacetime.Similar methods yields a relation between topological field theories in five and seven dimensions and chiral conformal field theories in four and six dimensions.There presumably exists a class of conformal field theories with novel properties, so called logarthmic conformal field theories. By starting from a conventional theory, a systematic way to deform this to a logarithmic theory is developed.

• 6.
On 7-D TFT and 6-D chiral CFT2002Manuscript (preprint) (Other academic)

We perform a canonical and BRST analysis of a seven-dimensional Chern-Simons theory on a manifold with boundary. The main result is that the 7D theory induces for consistency a chiral two-form on the 6D boundary. We also comment on similar behaviour in a five-dimensional Chern-Simons theory relevant for $\N=4$ supersymmetric Yang-Mills theory in four dimensions.

• 7.
Department of Physics, Nanjing University, Nanjing, China.
On duality and extended chiral symmetry in the SL(2,R) WZW model2011In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 44, no 23, article id 235404Article in journal (Refereed)

Two chiral aspects of the SL(2,R) WZW model in an operator formalism are investigated. First, the meaning of duality, or conjugation, of primary fields is clarified. On a class of modules obtained from the discrete series it is shown, by looking at spaces of two-point conformal blocks, that a natural definition of contragredient module provides a suitable notion of conjugation of primary fields, consistent with known two-point functions. We find strong indications that an apparent contradiction with the Clebsch-Gordan series of SL(2,R), and proposed fusion rules, is explained by nonsemisimplicity of a certain category. Second, results indicating an infinite cyclic simple current group, corresponding to spectral flow automorphisms, are presented. In particular, the subgroup corresponding to even spectral flow provides part of a hypothetical extended chiral algebra resulting in proposed modular invariant bulk spectra.

• 8.
• 9.
Logarithmic conformal field theories via logarithmic deformations2002In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 633, no 3, p. 379-413Article in journal (Refereed)

We construct logarithmic conformal field theories starting from an ordinary conformal field theory -- with a chiral algebra C and the corresponding space of states V -- via a two-step construction: i) deforming the chiral algebra representation on V\tensor End K[[z,1/z]], where K is an auxiliary finite-dimensional vector space, and ii) extending C by operators corresponding to the endomorphisms End K. For K=C^2, with End K being the two-dimensional Clifford algebra, our construction results in extending C by an operator that can be thought of as \partial^{-1}E, where \oint E is a fermionic screening. This covers the (2,p) Virasoro minimal models as well as the sl(2) WZW theory.

• 10.
Institutionen för Fysik, Karlstads Universitet, Karlstad, Sweden. Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Golm, Germany. Fachbereich Mathematik, Universität Hamburg Schwerpunkt Algebra und Zahlentheorie, Zentrum für Mathematische Physik, Hamburg, Germany.
TFT construction of RCFT correlators V: Proof of modular invariance and factorisation2006In: Theory and Applications of Categories, ISSN 1201-561X, E-ISSN 1201-561X, Vol. 16, no 16, p. 342-433Article in journal (Refereed)

The correlators of two-dimensional rational conformal field theories that are obtained in the TFT construction of [FRSI,FRSII,FRSIV] are shown to be invariant under the action of the relative modular group and to obey bulk and boundary factorisation constraints. We present results both for conformal field theories defined on oriented surfaces and for theories defined on unoriented surfaces. In the latter case, in particular the so-called cross cap constraint is included.

• 11.
Department of Physics, Nanjing University, Nanjing, China.
RCFT with defects: Factorization and fundamental world sheets2012In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 863, no 1, p. 213-259Article in journal (Refereed)

It is known that for any full rational conformal field theory, the correlation functions that are obtained by the TFT construction satisfy all locality, modular invariance and factorization conditions, and that there is a small set of fundamental correlators to which all others are related via factorization - provided that the world sheets considered do not contain any non-trivial defect lines. In this paper we generalize both results to oriented world sheets with an arbitrary network of topological defect lines.

• 12.
Department of Mathematics, Uppsala University, Uppsala.
Mapping class group representations from Drinfeld doubles of finite groupsManuscript (preprint) (Other academic)

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular we show that they have finite images, and that for surfaces of genus at least 3 their restriction to the Torelli group is non-trivial iff G is non-abelian.

• 13.
Cardiff School of Mathematics, Cardiff University, United Kingdom; Avdelning Fysik, Karlstads Universitet, Karlstad, Sweden.
Avdelning Fysik, Karlstads Universitet, Karlstad, Sweden. Department of Mathematics, King's College London, United Kingdom. Organisationseinheit Mathematik, Universität Hamburg, Schwerpunkt Algebra und Zahlentheorie, Hamburg, Germany.
Uniqueness of open/closed rational CFT with given algebra of open states2008In: Advances in Theoretical and Mathematical Physics, ISSN 1095-0761, E-ISSN 1095-0753, Vol. 12, no 6, p. 1283-1375Article in journal (Refereed)

We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly non-empty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: For a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two,- and three-point correlators on the disk.

Thus for any such theory every consistent collection of correlators can be obtained by the TFT approach of hep-th/0204148, hep-th/0503194. As morphisms of the category of world sheets we include not only homeomorphisms, but also sewings; interpreting the correlators as a natural transformation then encodes covariance both under homeomorphisms and under sewings of world sheets.

• 14.
Department of Physics, Nanjing University, Nanjing, China.
Partition functions, mapping class groups and Drinfeld doubles2012In: Symmetries and Groups in Contemporary Physics: Proceedings of the XXIX International Colloquium on Group-Theoretical Methods in PhysicsTianjin, China, 20 – 26 August 2012, Shanghai: World Scientific, 2012, p. 405-410Conference paper (Refereed)

Higher genus partition functions of two-dimensional conformal field theories have to be invariants under linear actions of mapping class groups. We illustrate recent results [4,6] on the construction of such invariants by concrete expressions obtained for the case of Drinfeld doubles of finite groups. The results for doubles are independent of the characteristic of the underlying field, and the general results do not require any assumptions of semisimplicity.

• 15.
Equivalence of Chern-Simons gauge theory and WZNW model using a BRST symmetry1999In: Physics Letters B, ISSN 0370-2693, E-ISSN 1873-2445, Vol. 466, no 2-4, p. 227-233Article in journal (Refereed)

The equivalence between the Chern-Simons gauge theory on a three-dimensional manifold with boundary and the WZNW model on the boundary is established in a simple and general way using the BRST symmetry. Our approach is based on restoring gauge invariance of the Chern-Simons theory in the presence of a boundary. This gives a correspondence to the WZNW model that does not require solving any constraints, fixing the gauge or specifying boundary conditions.

• 16.
Sectors of solutions in three-dimensional gravity and black holes2002In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 628, no 1-2, p. 331-360Article in journal (Refereed)

We examine the connection between three dimensional gravity with negative cosmological constant and two-dimensional CFT via the Chern-Simons formulation. A set of generalized spectral flow transformations are shown to yield new sectors of solutions. One implication is that the microscopic calculation of the entropy of the Banados-Teitelboim-Zanelli (BTZ) black hole is corrected by a multiplicative factor with the result that it saturates the Bekenstein-Hawking expression.

• 17.
Department of Physics, Karlstad University, Karlstad, Sweden. Institute of Theoretical Physics, Stockholm, Sweden.
CFT description of three-dimensional Kerr-de Sitter space-time2002In: Nuclear Physics B, ISSN 0550-3213, E-ISSN 1873-1562, Vol. 641, no 1-2, p. 376-392Article in journal (Refereed)

We describe three-dimensional Kerr-de Sitter space using similar methods as recently applied to the BTZ black hole. A rigorous form of the classical connection between gravity in three dimensions and two-dimensional conformal field theory is employed, where the fundamental degrees of freedom are described in terms of two dependent SL(2,C) currents. In contrast to the BTZ case, however, quantization does not give the Bekenstein-Hawking entropy connected to the cosmological horizon of Kerr-de Sitter space.

• 18.
Department of Physics, Nanjing University, Nanjing, China.
Department of Theoretical Physics, School of Engineering Sciences, Royal Institute of Technology (KTH), Stockholm, Sweden.
New symmetries of the chiral Potts model2012In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 45, no 15, article id 155208Article in journal (Refereed)

In this paper a hithertho unknown symmetry of the three-state chiral Potts model is found consisting of two coupled Temperley-Lieb algebras. From these we can construct new superintegrable models. One realisation is in terms of a staggered isotropic XY spin chain. Further we investigate the importance of the algebra for the existence of mutually commuting charges. This leads us to a natural generalisation of the boost-operator, which generates the charges.

• 19.
Dept. of Mathematics, King's College, London, UK.
Cardiff School of Math., Cardiff Univ., Cardiff, UK. Institutionen för fysik, Karlstads universitet, Karlstad, Sweden. Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany.
Topological and conformal field theory as Frobenius algebras2007In: Categories in Algebra, Geometry and Mathematical Physics / [ed] Alexei Davydov, Michael Batanin, Michael Johnson, Stephen Lack and Amnon Neeman, American Mathematical Society (AMS), 2007, p. 225-248Chapter in book (Refereed)

Two-dimensional conformal field theory (CFT) can be defined through its correlation functions. These must satisfy certain consistency conditions which arise from the cutting of world sheets along circles or intervals. The construction of a (rational) CFT can be divided into two steps, of which one is complex-analytic and one purely algebraic. We realise the algebraic part of the construction with the help of three-dimensional topological field theory and show that any symmetric special Frobenius algebra in the appropriate braided monoidal category gives rise to a solution. A special class of examples is provided by two-dimensional topological field theories, for which the relevant monoidal category is the category of vector spaces

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