# Publications

The following is a list of everything with scientific content, which I have written. Thus, in additon to the listings on the arXiv or INSPIRE-HEP, you will also find some unpublished texts. Some of the older texts are only provided as scanned version, as their original TeX sources are no longer available.
Further below, you find a list of publications and theses of members of my group, which were written under my supervision.

## Papers et cetera

• Integrable anyon chains:
from fusion rules to face models to effective field theories

Authors: Peter Finch, Michael Flohr, Holger Frahm
Sources: Abstract Starting from the fusion rules for the algebra $$SO(5)_2$$ we construct one-dimensional lattice models of interacting anyons with commuting transfer matrices of interactions round the face' (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. Two of the anyon models are critical at zero temperature. From the analysis of the finite size spectrum we find that they are effectively described by rational conformal field theories invariant under extensions of the Virasoro algebra, namely $$\mathcal{W}B_2$$ and $$\mathcal{W}D_5$$, respectively. The latter contains primaries with half and quarter spin. The modular partition function and fusion rules are derived and found to be consistent with the results for the lattice model.   ::   pdf   ::   arXiv:1408.1282 [cond-mat.str-el]
Journal: Nucl. Phys. B889 (2014) 299-332

• What the characters of irreducible subrepresentations of Jordan cells can tell us about LCFT
Authors: Michael Flohr, Michael Köhn
Sources: Abstract In this article, we review some aspects of logarithmic conformal field theories which can be inferred from the characters of irreducible submodules of indecomposable modules. We will mainly consider the $$\mathcal{W}(2,2p-1,2p-1,2p-1)$$ series of triplet algebras and a bit logarithmic extensions of the minimal Virasoro models. Since in all known examples of logarithmic conformal field theories the vacuum representation of the maximally extended chiral symmetry algebra is an irreducible submodule of a larger, indecomposable module, its character provides a lot of non-trivial information about the theory such as a set of functions which spans the space of all torus amplitudes. Despite such characters being modular forms of inhomogeneous weight, they fit in the $$ADET$$-classification of fermionic sum representations. Thus, they show that logarithmic conformal field theories naturally have to be taken into account when attempting to classify rational conformal field theories.   ::   pdf   ::   arXiv:1307.5844 [hep-th]
Journal: J. Phys. A49 (2013) 494007

• Correlation Functions in $$\boldsymbol{\mathrm{N}=3}$$ Superconformal Theory
Authors: Dmitriy Drichel, Michael Flohr
Sources: Abstract Using a superspace representation of the $$\mathrm{N}=3$$ Neveau-Schwarz super Virasoro algebra, we find solutions of super Ward identities. Global transformations generated by the non-abelian supercurrent require not only superfields, but also functions of Grassmann variables (in particular correlation functions and their linear combinations) to be $$\mathfrak{su}(2)$$ representations. As a consequence of this, the only admissible fields in the theory are isospin singlets and doublets. We show how to compute the generic form of any $$\mathrm{N}=3\;\ n$$-point function and demonstrate a construction of all $$\mathfrak{su}(2)$$ representations on the space of $$\mathrm{N}=3$$ superfunctions.   ::   pdf   ::   arXiv:1006.3346 [hep-th]
Journal: submitted to Phys.Lett.B   ::   ITP-UH-11/10, June 2010, 10pp.

• On a Logarithmic Deformation of the Supersymmetric $$\boldsymbol{bc}$$-system on Curved Manifolds
Authors: Michael Flohr, Kirsten Vogeler
Sources: Abstract E. Frenkel, A. Losev and N. Nekrasov claim that a certain class of theories on compact Kähler manifolds and in particular the “gauged” supersymmetric $$bc$$-system on $$\mathbb{CP}^1$$ are logarithmic conformal field theories. We discuss that proposition on a classical level for the $$bc$$-system on $$\mathbb{CP}^1$$. The outcome of our investigation conforms to their conjecture. The property of being a logarithmic CFT thus can be interpreted as an effect of gravity.   ::   pdf   ::   arXiv:0902.0729 [hep-th]
Journal: J. Math. Phys. 50 (2009) 112305

• On Factorization Constraints for Branes in the $$\boldsymbol{\mathrm{H}_3^+}$$ Model
Authors: Hendrik Adorf, Michael Flohr
Sources: Abstract We comment on the brane solutions for the boundary $$\mathrm{H}_3^+$$ model that have been proposed so far and point out that they should be distinguished according to the patterns regular/irregular and discrete/continuous. In the literature, mostly irregular branes have been studied, while results on the regular ones are rare. For all types of branes, there are questions about how a second factorization constraint in the form of a $$b^{-2}/2$$-shift equation can be derived. Here, we assume analyticity of the boundary two point function for $$AdS_2$$ branes, which means that the Cardy-Lewellen constraints remain unweakened. This enables us to derive unambiguously the desired $$b^{-2}/2$$-shift equations in all cases of $$AdS_2$$ branes. They serve as important additional consistency conditions. For the regular $$AdS_2$$ branes, we also derive $$1/2$$-shift equations that were not known previously. Case by case, we discuss possible solutions to the enlarged system of constraints. We find that the wellknown irregular continuous $$AdS_2$$ branes are consistent with our new factorization constraint. Furthermore, we establish the existence of a new type of brane: The shift equations in the regular discrete case possess a no–trivial solution that we write down explicitly. All other types are found to be inconsistent when using our second constraint. We discuss these results in view of the Hosomichi-Ribault proposal and some of our earlier results on the derivation of $$b^{-2}/2$$-shift equations.   ::   pdf   ::   arXiv:0801.2711 [hep-th]
Journal: Int. J. Mod. Phys. A23 (2008) 3963-4010

• Towards the Construction of Local Logarithmic Conformal Field Theories
Authors: Anne-Lý Do, Michael Flohr
Sources: Abstract Although logarithmic conformal field theories (LCFTs) are known not to factorise many previous findings have only been formulated on their chiral halves. Making only mild and rather general assumptions on the structure of an chiral LCFT we deduce statements about its local non-chiral equivalent. Two methods are presented how to construct local representations as subrepresentations of the tensor product of chiral and anti-chiral Jordan cells. Furthermore we explore the assembly of generic non-chiral correlation functions from generic chiral and anti-chiral correlators. The constraint of locality is studied and the generality of our method is discussed.   ::   pdf   ::   arXiv:0710.1783 [hep-th]
Journal: Nucl. Phys. B802 [FS] (2008) 475-493

• Continuously crossing $$\boldsymbol{u=z}$$ in the $$\boldsymbol{\mathrm{H}_3^+}$$ boundary CFT
Authors: Hendrik Adorf, Michael Flohr
Sources: Abstract For $$AdS$$ boundary conditions, we give a solution of the $$\mathrm{H}_3^+$$ two point function involving degenerate field with $$\mathrm{SL}(2)$$-label $$b^{-2}/2$$, which is defined on the full $$(u,z)$$ unit square. It consists of two patches, one for $$z < u$$ and one for $$u < z$$. Along the $$u = z$$ “singularity”, the solutions from both patches are shown to have finite limits and are merged continuously as suggested by the work of Hosomichi and Ribault. From this two point function, we can derive $$b^{-2}/2$$-shift equations for $$AdS_2$$ D-branes. We show that discrete as well as continuous $$AdS_2$$ branes are consistent with our novel shift equations without any new restrictions.   ::   pdf   ::   arXiv:0707.1463 [hep-th]
Journal: JHEP 11 (2007) 024

• On Verlinde-Like Formulas in $$\boldsymbol{c_{p,1}}$$ Logarithmic Conformal Field Theories
Authors: Michael Flohr, Holger Knuth
Sources: Abstract Two different approaches to calculate the fusion rules of the $$c_{p,1}$$ series of logarithmic conformal field theories are discussed. Both are based on the modular transformation properties of a basis of chiral vacuum torus amplitudes, which contains the characters of the irreducible representations.
One of these is an extension, which we develop here for a non-semisimple generalisation of the Verlinde formula introduced by Fuchs et al., to include fusion products with indecomposable representations. The other uses the Verlinde formula in its usual form and gets the fusion coefficients in the limit, in which the basis of torus amplitudes degenerates to the linear dependent set of characters of irreducible and indecomposable representations. We discuss the effects, which this linear dependence has on any result for fusion rules, which are calculated from these character's modular transformation properties.
We show that the two presented methods are equivalent. Furthermore we calculate explicit BPZ-like expressions for the resulting fusion rules for all $$p$$ larger than $$2$$.
::   pdf   ::   arXiv:0705.0545 [math-ph]
Journal: submitted to Commun.Math.Phys.   ::   ITP-UH-10/07, May 2007, 38pp.

• On the Various Types of D-Branes in the Boundary $$\boldsymbol{\mathrm{H}_3^+}$$ Model
Authors: Hendrik Adorf, Michael Flohr
Sources: Abstract We comment on the D-brane solutions for the boundary $$\mathrm{H}_3^+$$ model that have been proposed so far and point out that many more types of D-branes should be considered possible. We start a systematic derivation of the $$1/2$$- and $$b^{-2}/2$$-shift equations corresponding to each type. These equations serve as consistency conditions and we discuss their possible solutions. On this basis, we show for the known $$AdS_2^{(d)}$$ branes, that only strings transforming in finite dimensional $$\mathrm{SL}(2)$$ representations can couple to them. Moreover, we also demonstrate that strings in the infinite dimensional continuous $$\mathrm{SL}(2)$$ representations do not couple consistently to the known $$AdS_2$$ branes. For some other types, we show that no consistent solutions seem to exist at all.   ::   pdf   ::   arXiv:hep-th/0702158
Journal: submitted to Nucl.Phys.B   ::   ITP-UH-01/07, February 2007, 28pp.

• Fermionic expressions for the characters of $$\boldsymbol{c_{p,1}}$$ logarithmic conformal field theories
Authors: Michael Flohr, Carsten Grabow, Michael Köhn
Sources: Abstract We present fermionic quasi-particle sum representations consisting of a single fundamental fermionic form for all characters of the logarithmic conformal field theory models with central charge $$c_{p,1}$$, $$p\geq 2$$, and suggest a physical interpretation. We also show that it is possible to correctly extract dilogarithm identities.   ::   pdf   ::   arXiv:hep-th/0611241
Journal: Nucl. Phys. B768 (2007) 263-276

• Pure Gauge $$\boldsymbol{\mathrm{SU}(2)}$$ Seiberg-Witten Theory and Modular Forms
Authors: Michael Flohr, Kirsten Vogeler
Sources: Abstract We identify the spectral curve of pure gauge $$\mathrm{SU}(2)$$ Seiberg-Witten theory with the Weierstrass curve $\mathbb{C}/L \ni z \mapsto (1,\wp(z),\wp(z)')$ and thereby obtain explicitly a modular form from which the moduli space parameter $$u$$ and lattice parameters $$a$$, $$a_D$$ can be derived in terms of modular respectively theta functions. We further discuss its relationship with the $$c=-2$$ triplet model conformal field theory.   ::   pdf   ::   arXiv:hep-th/0607142
Journal: J. Math. Phys. 48 (2007) 083505

• Virasoro representations and fusion for general augmented minimal models
Authors: Holger Eberle, Michael Flohr
Sources: Abstract In this paper we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models, the augmented $$c_{2,3} = 0$$ model as well as the augmented Yang-Lee model at $$c_{2,5} = -22/5$$. We analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra. We also describe the first few higher rank representations in detail. In particular, we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure.
Knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented $$c_{p,q}$$ models.
::   pdf   ::   arXiv:hep-th/0604097
Journal: J. Phys. A39 (2006) 15245-15286

• Indecomposable Representations in $$\boldsymbol{\mathbb{Z}_n}$$ Symmetric $$\boldsymbol{b,c}$$ Ghost Systems via Deformations of the Virasoro Field
Authors: Michael Flohr, Julia Voelskow
Sources: Abstract The Virasoro field associated to $$b,c$$ ghost systems with arbitrary integer spin $$\lambda$$ on an $$n$$-sheeted branched covering of the Riemann sphere is deformed. This leads to reducible but indecomposable representations, if the new Virasoro field acts on the space of states, enlarged by taking the tensor product over the different sheets of the surface. For $$\lambda = 1$$, proven LCFT structures are made explicit through this deformation. In the other cases, the existence of Jordan cells is ruled out in favour of a novel kind of indecomposable representations.   ::   pdf   ::   arXiv:hep-th/0602056
Journal: Nucl. Phys. B792 [FS] (2008) 207-227

• Notes on generalised nullvectors in logarithmic CFT
Authors: Holger Eberle, Michael Flohr
Sources: Abstract In these notes we discuss the procedure how to calculate nullvectors in general indecomposable representations which are encountered in logarithmic conformal field theories. In particular, we do not make use of any of the restrictions which have been imposed in logarithmic nullvector calculations up to now, especially the quasi-primarity of all Jordan cell fields.
For the quite well-studied $$c_{p,1}$$ models we calculate examples of logarithmic nullvectors which have not been accessible to the older methods and recover the known representation structure. Furthermore, we calculate logarithmic nullvectors in the up to now almost unexplored general augmented $$c_{p,q}$$ models and use these to find bounds on their possible representation structures.
::   pdf   ::   arXiv:hep-th/0512254
Journal: Nucl. Phys. B741 (2006) 441-466

• Notes on non-trivial and logarithmic CFTs with $$\boldsymbol{c = 0}$$
Authors: Michael Flohr, Annekathrin Müller-Lohmann
Sources: Abstract We examine the properties of two-dimensional conformal field theories (CFTs) with vanishing central charge based on the extended Kac-table for $$c_{(9,6)} = 0$$ using a general ansatz for the stress energy tensor residing in a Jordan cell of rank two.
Within this setup we will derive the OPEs and two point functions of the stress energy tensor $$T(z)$$ and its logarithmic partner field $$t(z)$$ and illustrate this by a bosonic field realization.
We will show why our approach may be more promising than those chosen in the literature so far, including a discussion on properties of the augmented minimal model with vanishing central charge such as full conformal invariance of the vacuum as a state in an irreducible representation.
Furthermore we will present a more general solution of another solution to the $$c \rightarrow 0$$ catastrophe based on a logarithmic CFT tensor model. As an example, we consider a tensor product of the well-known $$c = -2$$ logarithmic CFT with a fourfold Ising model.
We give an overview of the possible configurations and various consequences on the two point functions and the OPEs of the stress energy tensor $$T(z)$$ and its logarithmic partner field $$t(z)$$. We will motivate that due to the full conformal invariance of the vacuum at $$c = 0$$, we should assume a Jordan cell for the identity since $$t(z)$$ is presumably a descendant of a new $$h = 0$$ field.
::   pdf   ::   arXiv:hep-th/0510096
Journal: J. Stat. Mech. 0604:P002 (2006)

• Logarithmic torus amplitudes
Authors: Michael Flohr, Matthias Gaberdiel
Sources: Abstract For the example of the logarithmic triplet theory at $$c = -2$$ the chiral vacuum torus amplitudes are analysed. It is found that the space of these torus amplitudes is spanned by the characters of the irreducible representations, as well as a function that can be associated to the logarithmic extension of the vacuum representation. A few implications and generalisations of this result are discussed.   ::   pdf   ::   arXiv:hep-th/0509075
Journal: J. Phys. A: Math. Gen. 39 (2006) 1955-1967

• Nonmeromorphic operator product expansion and $$\boldsymbol{C_2}$$-cofiniteness for a family of $$\boldsymbol{{\cal W}}$$-algebras
Authors: Nils Carqueville, Michael Flohr
Sources: Abstract We prove the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras, the triplet $${\cal W}$$-algebras, using results from $$P(z)$$-tensor product theory. While doing this, we also show that all these vertex operator algebras are $$C_2$$-cofinite.   ::   pdf   ::   arXiv:math-ph/0508015
Journal: J. Phys. A: Math. Gen. 39 (2006) 951-966

• Proposal for a CFT interpretation of Watts' differential equation for percolation
Authors: Michael Flohr, Annekathrin Müller-Lohmann
Sources: Abstract G.M.T. Watts derived in his paper arXiv:cond-mat/9603167 that in two dimensional critical percolation the crossing probability $$\Pi_{h v}$$ satisfies a fifth order differential equation which includes another one of third order whose independent solutions describe the physically relevant quantities $$1$$, $$\Pi_h$$, $$\Pi_{h v}$$.
We will show that this differential equation can be derived from a level three null vector condition of a rational $$c = -24$$ CFT and suggest a new interpretation of the generally known CFT properties of percolation.
::   pdf   ::   arXiv:hep-th/0507211
Journal: J. Stat. Mech. 0512:P004 (2005)

• Four-Point Functions in Logarithmic Conformal Field Theories
Authors: Michael Flohr, Marco Krohn
Sources: Abstract The generic structure of 4-point functions of fields residing in indecomposable representations of arbitrary rank is given. The used algorithm is described and we present all results for Jordan-rank $$r = 2$$ and $$r = 3$$ where we make use of permutation symmetry and use a graphical representation for the results. A number of remaining degrees of freedom which can show up in the correlator are discussed in detail. Finally we present the results for two logarithmic fields for arbitrary Jordan-rank.   ::   pdf   ::   arXiv:hep-th/0504211
Journal: Nucl. Phys. B743 (2006) 276-306

• A Note on Four-Point Functions in Logarithmic Conformal Field Theory
Authors: Michael Flohr, Marco Krohn
Sources: Abstract The generic structure of 4-point functions of fields residing in indecomposable representations of arbitrary rank is given. The presented algorithm is illustrated with some non-trivial examples and permutation symmetries are exploited to reduce the number of free structure-functions, which cannot be fixed by global conformal invariance alone.   ::   pdf   ::   arXiv:hep-th/0501144
Journal: Fortschr. Phys. 53 (2005) 456-462

• Explicit Formulas for the Scalar Modes in Seiberg-Witten Theory with an Application to the Argyres-Douglas Point
Authors: Nikolas Akerblom, Michael Flohr
Sources: Abstract General formulas for the scalar modes $$a_i$$ and $$a_D^{\ \ i}$$ in the Seiberg-Witten $$\mathrm{SU}(N)$$ setting are derived in the cases with and without massive hypermultiplets. Subsequently these formulas are applied in a study of the $$\mathrm{SU}(3)$$ Argyres-Douglas point. We use this example to study the question, whether the scalar modes admit an interpretation in terms of BPS mass states everywhere in moduli space. The paper collects, in an appendix, various facts on the function Lauricella $$F_D^{(n)}$$ which naturally appears in the derived formulas.   ::   pdf   ::   arXiv:hep-th/0409253
Journal: JHEP 02 (2005) 057

• Logarithmic Conformal Field Theory — or — How to Compute a Torus Amplitude on the Sphere
Authors: Michael Flohr
Sources: Abstract We review some aspects of logarithmic conformal field theories which might shed some light on the geometrical meaning of logarithmic operators. We consider an approach, put forward by V. Knizhnik, where computation of correlation functions on higher genus Riemann surfaces can be replaced by computations on the sphere under certain circumstances. We show that this proposal naturally leads to logarithmic conformal field theories, when the additional vertex operator insertions, which simulate the branch points of a ramified covering of the sphere, are viewed as dynamical objects in the theory.
We study the Seiberg-Witten solution of supersymmetric low energy effective field theory as an example where physically interesting quantities, the periods of a meromorphic one-form, can effectively computed within this conformal field theory setting. We comment on the relation between correlation functions computed on the plane, but with insertions of twist fields, and torus vacuum amplitudes.

In memoriam Ian Kogan.

::   pdf   ::   arXiv:hep-th/0407003
Journal: In: From Fields to Strings: Circumnavigating Theoretical Physics Ian Kogan memorial collection (edited by Misha Shifman, Arkady Vainshtein, John Wheater), World Scientific, Singapore (2005), see also Ian Kogan memorial Volume

• Operator Product Expansion and Zero Mode Structure in Logarithmic CFT
Authors: Michael Flohr, Marco Krohn
Sources: Abstract The generic structure of 1-, 2- and 3-point functions of fields residing in indecomposable representations of arbitrary rank are given. These in turn determine the structure of the operator product expansion in logarithmic conformal field theory. The crucial role of zero modes is discussed in some detail.   ::   pdf   ::   arXiv:hep-th/0312185
Journal: Fortschr. Phys. 52 (2004) 503-508

• Foundations and Applications of Logarithmic Conformal Field Theory
Authors: Michael Flohr
Sources: Abstract Diese Schrift versammelt meine hauptsächlichen Arbeiten zum Gebiet der sogenannte logarithmischen konformen Feldtheorien. Dies sind zwei-dimensionale konforme Quantenfeldtheorien, bei denen Korrelationsfunktionen logarithmische Divergenzen aufweisen können. Aus heutiger Sichtweise ist es sinnvoller, solche konforme Feldtheorien dadurch zu charakterisieren, dass einige der Darstellungen ihrer chiralen Symmetrie-Algebra unzerlegbar aber reduzibel sind.
Das Hauptgewicht meiner Arbeiten liegt auf der Begründung der logarithmischen konformen Feldtheorien als bona fide Theorien, bei denen sich alle Strukturen und Methoden, die von den gewöhnlichen konformen Feldtheorien bekannt sind und als nützlich geschätzt werden, entsprechend übertragen lassen. Dies gilt insbesondere für die Strukturen, die für rationale konforme Feldtheorien besonders wichtig sind. Dies sind vor allem die Invarianz unter der Modulgruppe, die Fusionsalgebra und ihr Zusammenhang mit der Modulgruppe via der Verlinde-Formel; die explizite Form von Nullvektoren und die daraus resultierenden Differentialgleichungen, welche die Form der Korrelationsfunktionen festzulegen vermögen; die Operatorproduktentwicklung; und das explizite Spektrum der Randzustände. Mein Zugang zur logarithmischen konformen Feldtheorie konzentriert sich dabei vor allem auf die darstellungstheoretischen Aspekte.
Die Verallgemeinerung dieser Strukturen auf den logarithmischen Fall hat zum Ergebnis, dass logarithmischen konformen Feldtheorien ähnlich exakt lösbar sind, wie gewöhnliche konforme Feldtheorien, dass also im Prinzip alle Korrelatoren explizit und exakt berechnet werden können. Es gibt aber auch einige überraschende Aspekte, die kein Pendant bei den gewöhnlichen konformen Feldtheorien haben, zum Beispiel die Erweiterung des Satzes von Charakteren um Torusamplituden, die keine Interpretation als Spur über einen Darstellungsmodul zulassen, oder eine interessante Faktorisierungseigenschaft logarithmische Nullvektoren. Ein Beispiel für logarithmische konforem Feldtheorie zieht sich durch die meisten meiner Arbeiten (und auch durch die Arbeiten der meisten anderen Autoren), das Spin $$(1,0)$$ Geistsystem mit zentraler Ladung $$c = -2$$, das die am besten verstandene logarithmische Theorie darstellt.
Drei Anwendungen werden kurz vorgestellt, um das Potential der logarithmischen Feltheorie zur erfolgreichen Beschreibung von einer weiten Klasse von Phänomenen, besonders aus der Welt der zwei-dimensionalen Festkörperphysik, zu demonstrieren. Ich diskutiere, in wie weit konforme Feldtheorie zwei-dimensionale Turbulenz beschreibt, und komme zu der Schlußfolgerung, dass dies nur mit logarithmischen Theorien möglich ist. Ich untersuche die Möglichkeit, die Bulk-Wellenfunktionen der Eigenzustände des fraktionalen Quanten-Hall-Effektes als Korrelatoren von konformen Feldtheorien auszudrücken. Für den speziellen Zustand mit Füllfaktor $$\nu = 5/2$$ geht dies nur mit Hilfe der logarithmischen $$c = -2$$ Theorie. Ebenso zeigen ich, dass Jain's Idee des Anheftens von Flußquanten zur Erzeugung von komponierten Fermionen eine Beschreibung in Form von Operatorproduktentwicklungen bzw. Fusionsprodukten in der konformen Feldtheorie hat, wenn dabei Tensorprodukte mit geeigneten logarithmischen konformen Theorien betrachtet werden. Schließlich verwende ich die $$c = -2$$ Theorie, um die Perioden des Seiberg-Witten-Differentials auf einfache und elegante Weise berechnen zu können. Diese Perioden bestimmen ja den Modulraum der exakten Lösungen effektiver niederenergetischer Feldtheorien von supersymmetrischen Yang-Mills Theorien. Das BPS-Spetkrum der masselosen Zustände kann dann besonders einfach durch Operatorproduktentwicklungen gefunden werden, da letztere automatisch degenerierte bzw. Rand-Bereiche des Modulraumes sondieren.
Die hier zusammengefassten Arbeiten ergeben damit einen umfassenden Überblick der Grundlagen der logarithmischen konformen Feldtheorie, sowie einen kleinen Ausblick auf ihr Anwendungspotential.
[in German]   ::   pdf   [German introduction only, 12+24pp]
Journal: Habilitation thesis, December 2002, 333pp.

• Proceedings of School & Workshop on Logarithmic Conformal Field Theory and its Applications   IPM Tehran, Iran, September 4-18, 2001
Authors: Michael Flohr, Shahin Rouhani (eds.)
Sources: Abstract TABLE OF CONTENTS
International Journal of Modern Physics A (IJMPA)   ::   Volume 18, Issue 25 (10 October 2003, Special Issue)

 1 Organizing Committee v 2 Preface [SPIRES:irn/5764017] vii - viii 3 Bits and Pieces in Logarithmic Conformal Field Theory Michael A.I. Flohr [arXiv:hep-th/0111228] 4497 - 4591 4 An Algebraic Approach to Logarithmic Conformal Field Theory Matthias R. Gaberdiel [arXiv:hep-th/0111260] 4593 - 4638 5 Boundary States in Boundary LCFT: An Algebraic Approach Yukitaka Ishimoto [SPIRES:irn/5756740] 4639 - 4654 6 Logarithmic Conformal Field Theory with Boundary Shinsuke Kawai [arXiv:hep-th/0204169] 4655 - 4684 7 The $$\boldsymbol{\mathrm{SU}(2)_0}$$ WZNW Model Alexander Nichols [SPIRES:irn/5756758] 4685 - 4702 8 Disordered Systems and Logarithmic Conformal Field Theory M. Reza Rahimi Tabar [arXiv:cond-mat/0111327] 4703 - 4745 9 Use of Nilpotent Weights in Logarithmic Conformal Field Theories S. Moghimi-Araghi, Shahin Rouhani, M. Saadat [arXiv:hep-th/0201099] 4747 - 4770 10 Stress Energy Tensor in LCFT and Logarithmic Sugawara Construction Ian I. Kogan, Alexander Nichols [arXiv:hep-th/0112008] 4771 - 4788 11 Some Photo Impressions from the 1st International School & Workshop on Logarithmic Conformal Field Theory and its Applications 4789 - 4796

::   pdf   [Preface only]
Journal: Special Issue Int. J. Mod. Phys. A18 Vol. 25 (10 October 2003)

• Ghost Systems Revisited: Modified Virasoro Generators and Logarithmic Conformal Field Theories
Authors: Marco Krohn, Michael Flohr
Sources: Abstract We study the possibility of extending ghost systems with higher spin to a logarithmic conformal field theory. In particular we are interested in $$c = -26$$ which turns out to behave very differently to the already known $$c = -2$$ case. The energy momentum tensor cannot be built anymore by a combination of derivatives of generalized symplectic fermion fields. Moreover, the logarithmically extended theory is only consistent when considered on nontrivial Riemann surfaces. This results in a LCFT with some unexpected properties. For instance the Virasoro mode $$L_0$$ is diagonal and for certain values of the deformation parameters even the whole global conformal group is non-logarithmic.   ::   pdf   ::   arXiv:hep-th/0212016
Journal: JHEP 01 (2003) 020

• A Novel CFT Approach to Bulk Wave Functions in the Fractional Quantum Hall Effect
Authors: Michael Flohr, Klaus Osterloh
Sources: Abstract We propose to describe bulk wave functions of fractional quantum Hall states in terms of correlators of non-unitary $$b/c$$-spin systems. These yield a promising conformal field theory analogon of the composite fermion picture of Jain. Fractional statistics is described by twist fields which naturally appear in the $$b/c$$-spin systems. We provide a geometrical interpretation of our approach in which bulk wave functions are seen as holomorphic functions over a ramified covering of the complex plane, where the ramification precisely resembles the fractional statistics of the quasi-particle excitations in terms of branch points on the complex plane. To extend Jain's main series, we use the concept of composite fermions pairing to spin singlets, which enjoys a natural description in terms of the particular $$c = -2\$$ $$\ b/c$$-spin system as known from the Haldane-Rezayi state. In this way we derive conformal field theory proposals for lowest Landau level bulk wave functions for more general filling fractions. We obtain a natural classification of the experimentally confirmed filling fractions which does not contain prominent unobserved fillings. Furthermore, our scheme fits together with classifications in terms of $$K$$-matrices of effective multilayer theories leading to striking restrictions of these coupling matrices.

PACS: 73.43.Cd, 11.25.Hf
::   pdf   ::   arXiv:cond-mat/0208429
Journal: Phys. Rev. B67 (2003) 235316

• Boundary States in $$\boldsymbol{c = -2}$$ Logarithmic Conformal Field Theory
Authors: Andreas Bredthauer, Michael Flohr
Sources: Abstract Starting from first principles, a constructive method is presented to obtain boundary states in conformal field theory. It is demonstrated that this method is well suited to compute the boundary states of logarithmic conformal field theories. By studying the logarithmic conformal field theory with central charge $$c = -2$$ in detail, we show that our method leads to consistent results. In particular, it allows to define boundary states corresponding to both, indecomposable representations as well as their irreducible subrepresentations.   ::   pdf   ::   arXiv:hep-th/0204154
Journal: Nucl. Phys. B639 (2002) 450-470

• Bits and Pieces in Logarithmic Conformal Field Theory
Authors: Michael Flohr
Sources: Abstract These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran.
These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories.
These two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic $$c = -2$$ theory.
::   pdf   ::   arXiv:hep-th/0111228
Journal: Int. J. Mod. Phys. A18 (2003) 4497-4591

• Operator Product Expansion in Logarithmic Conformal Field Theory
Authors: Michael Flohr
Sources: Abstract In logarithmic conformal field theory, primary fields come together with logarithmic partner fields on which the stress-energy tensor acts non-diagonally. Exploiting this fact and global conformal invariance of two- and three-point functions, operator product expansions of logarithmic operators in arbitrary rank logarithmic conformal field theory are investigated. Since the precise relationship between logarithmic operators and their primary partners is not yet sufficiently understood in all cases, the derivation of operator product expansion formulæ is only possible under certain assumptions. The easiest cases are studied in this paper: firstly, where operator product expansions of two primaries only contain primary fields, secondly, where the primary fields are pre-logarithmic operators. Some comments on generalization towards more relaxed assumptions are made, in particular towards the case where logarithmic fields are not quasi-primary. We identify an algebraic structure generated by the zero modes of the fields, which proves useful in determining settings in which our approach can be successfully applied.   ::   pdf   ::   arXiv:hep-th/0107242
Journal: Nucl. Phys. B634 (2002) 511-545

• Conformal Field Theory Survival Kit
Authors: Michael Flohr
Sources: Abstract The bare necessities of conformal field theory are provided such that chances to survive the string theory crash course, held fall 2000 at Hannover University within the German String Network, are increased. This is not a comprehensive introduction to the subject, and provides emergency treatment only.
Although there exist many introductions to conformal field theory, this one might in particular be useful for students without too much knowledge in general quantum field theory.
::   pdf
Journal: Notes of lectures delivered on the First String Theory Crash Course within the special research focus String theory in the context of particle physics, quantum field theory, quantum gravitation, cosmology and mathematics of the German Research Council (DFG), Hannover University, October 2000, 52pp.

• Null Vectors in Logarithmic Conformal Field Theory
Authors: Michael Flohr
Sources: Abstract The representation theory of the Virasoro algebra in the case of a logarithmic conformal field theory is considered. Here, indecomposable representations have to be taken into account, which has many interesting consequences. We study the generalization of null vectors towards the case of indecomposable representation modules and, in particular, how such logarithmic null vectors can be used to derive differential equations for correlation functions. We show that differential equations for correlation functions with logarithmic fields become inhomogeneous.   ::   pdf   ::   arXiv:hep-th/0009137
Journal: JHEP Proc. Sect. PoS(tmr2000)044   ::   Proceedings of the IV. Annual European TMR Conference on Integrability, Non-perturbative Effects and Symmetry in Quantum Field Theory, Paris, September 2000, (edited by D. Bernard, L. Bonora, E. Corrigan, C. Gomez, B. Julia, G. Mussardo, W. Nahm)

• On a Conformal Field Theory Approach to Seiberg-Witten Models
Authors: Michael Flohr
Sources: Abstract The use of conformal field theory tools in string theory is, of course, not new. However, with the advent of logarithmic conformal field theory (LCFT, see [1] and references therein) we may have a better machinery to describe physics whose underlying structure is determined by the geometry of Riemannian manifolds, in particular Riemannian surfaces. This is especially the case in modern string theory with its new family members such as branes, M-theory and the resulting exactly solvable low-energy effective field theories.   ::   pdf
Journal: Poster presented at Strings '99 at the Albert-Einstein-Institut (Max-Planck-Institut für Gravitationsphysik) Potsdam, July 1999, 4pp.

• Logarithmic Conformal Field Theory on Riemann Surfaces & Applications to Strings, Branes, and Seiberg-Witten Models
Authors: Michael Flohr
Sources: Abstract This is a brief resumé of my talk.   ::   pdf
Journal: Contribution to the workshop Mathematical Aspects of String Theory at the Mathematisches Forschungsinstitut Oberwolfach, April 1999, 2pp.   ::   See also the Proceedings 16/1999, Mathematisches Forschungsinstitut Oberwolfach, April 1999, 19pp.

• Fun with Quantum Dots
Authors: Michael Flohr
Sources: Abstract We consider quantum dots with a parabolic confining potential. The qualitative features of such mesoscopic systems as functions of the total number of electrons $$N$$ and their total angular momentum $$J$$, e.g. magic numbers, overall symmetries etc., are derived solely from combinatoric principles. The key is one simple hypothesis about such quantum dots yielding a basis of states (different from the usual single electron states one starts with) which is extremely easy to handle. Within this basis all qualitative features are already present without the need of any perturbation theory.

PACS: 73.20.Dx, 73.23.-b, 73.61.-r, 71.10.-w
::   pdf   ::   arXiv:cond-mat/9811288
Journal: Submitted to Phys.Rev.B   ::   KCL-MTH-98-56, November 1998, 9pp.

• Logarithmic Conformal Field Theory & Seiberg-Witten Models
Authors: Michael Flohr
Sources: Abstract The periods of arbitrary abelian forms on hyperelliptic Riemann surfaces, in particular the periods of the meromorphic Seiberg-Witten differential λSW, are shown to be in one-to-one correspondence with the conformal blocks of correlation functions of the rational logarithmic conformal field theory with central charge $c = c_{2,1} = -2$. The fields of this theory precisely simulate the branched double covering picture of a hyperelliptic curve, such that generic periods can be expressed in terms of certain generalised hypergeometric functions, namely the Lauricella functions of type $$F_D$$.   ::   pdf   ::   arXiv:hep-th/9808169
Journal: Phys. Lett. B444 (1998) 179-189

• On the Classification and Moduli Spaces of (supersymmetric) Conformal Field Theories
Authors: Michael Flohr
Sources: Abstract Über die Klassifikation und Modul-Räume von (supersymmetrischen) konformen Quantenfeldtheorien.
Abschlußbericht über mein von der DFG unter der Nummer FL 259/1-2 gefördertes Forschungsvorhaben am Institute for Advanced Study, Princeton, USA. Verfaßt September 1997.
[in German]   ::   pdf   [in German]
Journal: Research report DFG-Fl-259-1/2, September 1997, 28pp.

• Singular Vectors in Logarithmic Conformal Field Theories
Authors: Michael Flohr
Sources: Abstract Null vectors are generalized to the case of indecomposable representations which are one of the main features of logarithmic conformal field theories. This is done by developing a compact formalism with the particular advantage that the stress energy tensor acting on Jordan cells of primary fields and their logarithmic partners can still be represented in form of linear differential operators. Since the existence of singular vectors is subject to much stronger constraints than in regular conformal field theory, they also provide a powerful tool for the classification of logarithmic conformal field theories.   ::   pdf   ::   arXiv:hep-th/9707090
Journal: Nucl. Phys. B514 (1998) 523-552

• Aspects of (0,2) Orbifolds and Mirror Symmetry
Authors: Ralph Blumenhagen, Michael Flohr
Sources: Abstract We study orbifolds of (0,2) models and their relation to (0,2) mirror symmetry. In the Landau-Ginzburg phase of a (0,2) model the superpotential features a whole bunch of discrete symmetries, which by quotient action lead to a variety of consistent (0,2) vacua. We study a few examples in very much detail. Furthermore, we comment on the application of (0,2) mirror symmetry to the calculation of Yukawa couplings in the space-time superpotential.   ::   pdf   ::   arXiv:hep-th/9702199
Journal: Phys. Lett. B404 (1997) 41-48

• The Haldane-Rezayi Quantum Hall State and Conformal Field Theory
Authors: Michael Flohr, Victor Gurarie, Chetan Nayak
Sources: Abstract We propose field theories for the bulk and edge of a quantum Hall state in the universality class of the Haldane-Rezayi wavefunction. The bulk theory is associated with the $$c = -2$$ conformal field theory. The topological properties of the state, such as the quasiparticle braiding statistics and ground state degeneracy on a torus, may be deduced from this conformal field theory. The 10-fold degeneracy on a torus is explained by the existence of a logarithmic operator in the $$c = -2$$ theory; this operator corresponds to a novel bulk excitation in the quantum Hall state. We argue that the edge theory is the $$c = 1$$ chiral Dirac fermion, which is related in a simple way to the $$c = -2$$ theory of the bulk. This theory is reformulated as a truncated version of a doublet of Dirac fermions in which the $$\mathrm{SU}(2)$$ symmetry — which corresponds to the spin-rotational symmetry of the quantum Hall system — is manifest and non-local. We make predictions for the current-voltage characteristics for transport through point contacts.   ::   pdf   ::   arXiv:cond-mat/9701212
Journal: Nucl. Phys. B498 (1997) 513-538

• 2-Dimensional Turbulence: A Novel Approach via Logarithmic Conformal Field Theory
Authors: Michael Flohr
Sources: Abstract A new conformal field theory description of two-dimensional turbulence is proposed. The recently established class of rational logarithmic conformal field theories provides a unique candidate solution which resolves many of the drawbacks of former approaches via minimal models. This new model automatically includes magneto-hydrodynamic turbulence and the Alf'ven effect.   ::   pdf   ::   arXiv:hep-th/9606130
Journal: Nucl. Phys. B482 (1996) 567-578

• On Fusion Rules in Logarithmic Conformal Field Theory
Authors: Michael Flohr
Sources: Abstract We find the fusion rules for the $$c_{p,1}$$ series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via $c = c_{3p,3}$. This leads to the conjecture that (almost) all minimal models with $$c = c_{p,q}$$, $$(p,q) > 1$$, belong to the class of rational logarithmic conformal field theories.   ::   pdf   ::   arXiv:hep-th/9605151
Journal: Int. J. Mod. Phys. A12 (1997) 1943-1958

• Fusion and Tensoring of Conformal Field Theory and Composite Fermion Picture of Fractional Quantum Hall Effect
Authors: Michael Flohr
Sources: Abstract We propose a new way for describing the transition between two quantum Hall effect states with different filling factors within the framework of rational conformal field theory. Using a particular class of non-unitary theories, we explicitly recover Jain's picture of attaching flux quanta by the fusion rules of primary fields. Filling higher Landau levels of composite fermions can be described by taking tensor products of conformal theories. The usual projection to the lowest Landau level corresponds then to a simple coset of these tensor products with several $$\mathrm{U}(1)$$-theories divided out. These two operations — the fusion map and the tensor map — can explain the Jain series and all other observed fractions as exceptional cases. Within our scheme of transitions we naturally find a field with the experimentally observed universal critical exponent $$7/3$$.   ::   pdf   ::   arXiv:hep-th/9605152
Journal: Mod. Phys. Lett. A11 (1996) 55-68

• On Modular Invariant Partition Functions of Conformal Field Theories with Logarithmic Operators
Authors: Michael Flohr
Sources: Abstract We extend the definitions of characters and partition functions to the case of conformal field theories which contain operators with logarithmic correlation functions. As an example we consider the theories with central charge $$c = c_{p,1} = 13 - 6(p + p^{-1})$$, the “border” of the discrete minimal series. We show that there is a slightly generalized form of the property of rationality for such logarithmic theories. In particular, we obtain a classification of theories with $c = c_{p,1}$$which is similar to the$$A$$-$$D$$-$$E$ classification of $$c = 1$$ models.
Extended version, accepted for publication in Int. J. Mod. Phys. A11 (1996).
::   pdf   ::   arXiv:hep-th/9509166
Journal: Int. J. Mod. Phys. A11 (1996) 4147-4172

• On a New Universal Class of Phase Transitions and Quantum Hall Effect
Authors: Michael Flohr
Sources: Abstract We study the possible phase transitions between (2+1)-dimensional abelian Chern-Simons theories. We show that they may be described by non-unitary rational conformal field theories with ceff = 1. As an example we choose the fractional quantum Hall effect and derive all its main features from the discrete fractal structure of the moduli space of these non-unitary transition conformal field theories and some large scale principles. Rationality of these theories is intimately related to the modular group yielding severe conditions on the possible phase transitions. This gives a natural explanation for both, the values and the widths, of the observed fractional Hall plateaux.   ::   pdf   ::   arXiv:hep-th/9412053
Journal: CSIC-IMAFF-137-1994, November 1994, 37pp.

• The Rational Conformal Quantum Field Theories in Two Dimensions with Effective Central Charge $$\boldsymbol{c_{\mathrm{eff}} \leq 1}$$
Authors: Michael Flohr
Sources: Abstract We classify all rational conformal quantum field theories (RCFTs) in two dimensions with effective central charge $$c_{\mathrm{eff}}\leq 1$$. Besides the well known minimal models and the Gauss models with $$c = 1$$ we study two new series of non-unitary RCFTs with $$c_{\mathrm{eff}} = 1$$.
The first has $$c = 1 - 12k$$, $$k \in \mathbb{N}$$, and maximally extended chiral symmetry algebra $${\cal W}(2,3k/2)$$ with $$\mathbb{Z}_2$$-orbifold $${\cal W}(2,4k)$$. We prove rationality by studying their representation theory.
The second series represents the “border” of the discrete series $$c_{p,q}$$ with $$q = 1$$. These models involve fields with logarithmic correlation functions. We extend the concepts of representation theory and modular invariance to the logarithmic case and show that the maximally extended symmetry algebras $${\cal W}(2,2p-1,2p-1,2p-1)$$ yield a rational structure within our scheme.
With these results we can show that the classification of RCFTs with $$c_{\mathrm{eff}} \leq 1$$ is now complete. This generalizes to $$N = 1$$ supersymmetric theories wit $c_{\mathrm{eff}} \leq 3/2$,
Moreover, we analyse the moduli space of all RCFTs with $$c_{\mathrm{eff}}= 1$$. The non-unitary part appears to be a multi-fractal dense in $$\mathbb{R}^2$$ with a structure coming from an action of the modular group $$\mathrm{PSL}(2,\mathbb{Z})$$ on the parameters.
Finally, we apply our results to the fractional quantum Hall effect (FQHE). The assumption of non-unitarity enables us, to recover the Laughlin wave functions from correlators of chiral groundstates of our first series of non-unitary RCFTs with $$c_{\mathrm{eff}} = 1$$. The strange modular structure of the moduli space yields a natural explanation for the existence of non-vanishing plateau widths of the observed filling factors and also for the absence of the non-observed filling factors.
::   pdf   [in German]
Journal: PhD-Thesis, BONN-IR-94-11, September 1994, 130pp.

• More Curiosities at Effective $$\boldsymbol{c = 1}$$
Authors: Michael Flohr
Sources: Abstract The moduli space of all rational conformal quantum field theories with effective central charge $$c_{\mathrm{eff}} = 1$$ is considered. Whereas the space of unitary theories essentially forms a manifold, the non unitary ones form a fractal which lies dense in the parameter plane. Moreover, the points of this set are shown to be in one-to-one correspondence with the elements of the modular group for which an action on this set is defined.   ::   pdf   ::   arXiv:hep-th/9312097
Journal: Mod. Phys. Lett. A9 (1994) 1071-1082

• Infinite Symmetry in the Fractional Quantum Hall Effect
Authors: Michael Flohr, Raimund Varnhagen
Sources: Abstract We have generalized recent results on the integer quantum Hall effect constructing explicitly a $${\cal W}_{1+\infty}$$ for the fractional quantum Hall effect such that the negative modes annihilate the Laughlin wave functions. This generalization has a nice interpretation in Jain's composite fermion theory. Furthermore, for these models we have calculated the wave functions of the edge excitations viewing them as area preserving deformations of an incompressible quantum droplet, and have shown that the $${\cal W}_{1+\infty}$$ is the underlying symmetry of the edge excitations in the fractional quantum Hall effect. Finally, we have applied this method to more general wave functions.   ::   pdf   ::   arXiv:hep-th/9309083
Journal: J. Phys. A: Math.Gen. 27 (1994) 3999-4010

• $$\boldsymbol{{\cal W}}$$-Algebras in Conformal Field Theory
Authors: W. Eholzer, M. Flohr, A. Honecker, R. Hübel, R. Varnhagen
Sources: Abstract Quantum $${\cal W}$$-algebras are defined and their relevance for conformal field theories is outlined. We describe direct constructions of $${\cal W}$$-algebras using associativity requirements. With this approach one explicitly obtains the first members of series of $${\cal W}$$-algebras, including in particular ‘Casimir algebras’ (related to simple Lie algebras) and extended symmetry algebras corresponding to special Virasoro-minimal models. We also describe methods for the study of highest weight representations of $${\cal W}$$-algebras. In some cases consistency of the corresponding quantum field theory already imposes severe restrictions on the admitted representations, i.e. allows to determine the field content. We conclude by reviewing known results on $${\cal W}$$-algebras and RCFTs and show that most known rational conformal fields theories can be described in terms of Casimir algebras although on the level of $${\cal W}$$-algebras exotic phenomena occur.   ::   pdf
Journal: In Proceedings of the workshop Strings, Gravity, and Related Topics, Trieste, July 1993 (edited by E. Gava, A. Masiero, K.S. Narain, S. Randjbar-Daemi, Q. Shafi), World Scientific 1994, pp 435-445

• Modular Chaos
Authors: Michael Flohr
Sources: Abstract Short summary of my talk, presented at the Workshop on Two-Dimensional Quantum Field Theory at the Erwin-Schrödinger International Institute, Vienna, March 1993. In particular, a high-resolution image of the mulit-fractal moduli space of certain non-unitary rational conformal field theories is presented, which I discovered while working on the classification of all rational conformal quantum field theories with effective central charge $$c_{\mathrm{eff}}\leq 1$$.
The interesting point is that this is also an image of the modular group $$\mathrm{PSL}(2,\mathbb{Z})$$. [The image to the right is a very low-resolution version to give you a rough idea.]
::   pdf
Journal: In Proceedings of the workshop Two-Dimensional Quantum Field Theory at the Erwin-Schrödinger International Institute, Vienna, March 1993, 2pp.

• $$\boldsymbol{{\cal W}}$$-Algebras, New Rational Models and Completeness of the $$\boldsymbol{c = 1}$$ Classification
Authors: Michael Flohr
Sources: Abstract Two series of $${\cal W}$$-algebras with two generators are constructed from chiral vertex operators of a free field representation. If $$c = 1 - 24k$$, there exists a $${\cal W}(2,3k)$$ algebra for $$k \in \mathbb{Z}_+/2$$ and a $${\cal W}(2,8k)$$ algebra for $$k \in \mathbb{Z}_+/4$$. All possible lowest-weight representations, their characters and fusion rules are calculated proving that these theories are rational. It is shown, that these non-unitary theories complete the classification of all rational theories with effective central charge $$c_{\mathrm{eff}}= 1$$. The results are generalized to the case of extended supersymmetric conformal algebras.   ::   pdf   ::   arXiv:hep-th/9207019
Journal: Commun. Math. Phys. 157 (1993) 179-212

• Representations of $$\boldsymbol{{\cal W}}$$-Algebras with Two Generators and New Rational Models
Authors: W. Eholzer, M. Flohr, A. Honecker, R. Hübel, W. Nahm, R. Varnhagen
Sources: Abstract In conformal field theory we investigate the representations of recently discovered $${\cal W}$$-algebras with a single generator in addition to the Virasoro field. We show that many of these $${\cal W}$$-algebras have only a finite number of highest weight representations. We describe methods for the classification and give complete lists. In a sporadic case we determine characters and fusion rules. Different methods are used for $${\cal W}$$-algebras with continously variable and for those with fixed central extension.   ::   pdf
Journal: Nucl. Phys. B383 (1992) 249-290

• $$\boldsymbol{{\cal W}}$$-Algebras, Quasiprimary Fields and Nonminimal Models
Authors: M. Flohr
Sources: Abstract W-algebras are extensions of the Virasoro-algebra by primary fields. In this work an explicit construction of a basis for the quasi-primary fields contained in a $${\cal W}$$-algebra is given. In particular, the number of independent fields of given conformal dimension is determined. An efficient algorithm for explicitly constructing such algebras is presented. As an application, all $${\cal W}$$-algebras with one further primary field up to conformal dimension 10 and a few examples for $${\cal W}$$-algebras with two additional fields are calculated.
Some new series of $${\cal W}$$-algebras which are not related to any minimal model are established using a free field construction with BRST-structure. It is shown that for any $$c = 1 - 24\delta$$, $$\delta\in \mathbb{Z}_+/2$$, there exists an infinitely generated $${\cal W}$$-algebra with generators of dimension ($$r^2 - 1)\delta$$, $r \in \mathbb{Z}_+$, which has finitely generated closed sub-algebras. Finally, first results concerning the representation theory are obtained for the simplest example, the $${\cal W}(2,9/2)$$ for $$c = -35$$.
::   pdf   [in German]
Journal: Diploma Thesis, BONN-IR-91-30, June 1991, 75pp.

• Connections between Mathematical Physics (Quantum Groups, von Neumann Algebras) and the Theory of Knots
Authors: M. Flohr, M. Terhoeven, R. Varnhagen
Sources: Abstract Tagungsbericht 15/1991 of the “Arbeitsgemeinschaft Geyer-Harder” on the topic Connections Between Mathematical Physics (Quantum Groups,von Neumann Algebras) and the Theory of Knots, from 31 March to 6 April 1991. Organizer: Prof. V. G. Turaev, Université Louis-Pasteur, Strasbourg.   ::   pdf
Journal: Proceedings 15/1991, Mathematisches Forschungsinstitut Oberwolfach, April 1991, 7pp.

• $$\boldsymbol{{\cal W}}$$-Algebras with Two and Three Generators
Authors: R. Blumenhagen, M. Flohr, A. Kliem, W. Nahm, A. Recknagel, R. Varnhagen
Sources: Abstract We construct all $${\cal W}$$-algebras of chiral fields which in addition to the energy-momentum density have a single generator of conformal dimension up to 8. Some of them were unexpected, which indicates that present conjectures concerning the classification of conformally invariant quantum field theories in two dimensions are rather incomplete. We also explicitly construct the $${\cal W}A_3$$-algebra with generators of dimensions 2, 3, 4.   ::   pdf
Journal: Nucl. Phys. B361 (1991) 255-289   ::   also published in $${\cal W}$$-Symmetry (edited by P. Bouwknegt, K. Shoutens), Adv. Series in Math. Phys., World Scientific, Singapore (1995).

• Beam Test Results from a Prototype for the DELPHI Microvertex Detector
Authors: V. Chaubaud, H. Dijkstra, M. Flohr, M. Grone, R. Horisberger, L. Hubbeling, G. Mahlum, A. Peisert, A. Sandvik, P. Weilhammer (CERN), A. Czermak, P. Jalocha, P. Kapusta, M. Turala, A. Zalewska (Cracow, INP-Exp), E. Sundell (Abo Akademi), T. Tuuva (Helsinki U.), M. Battaglia, M. Caccia, W. Kucewicz, C. Meroni, N. Redaelli, R. Turchetta, A. Stocchi, C. Troncon, G. Vegni (INFN, Milan), G. Baricello, M. Mazzucato, M. Pegoraro, F. Simonetto (INFN, Padua), P. Allport, M. Tyndel (Rutherford), H.J. Seebrunner (Heilbronn, Fachhochschule)
Sources: Abstract Results are presented from a test in the CERN SPS North Area of a prototype of the DELPHI Microvertex Detector. Full-sized modules built up from prototype AC-coupled detectors and VLSI readout electronics were used. The spatial resolution of the detectors equipped with the prototype VLSI chips was measured to be 6.5 μm. The system aspects, including the readout, were found to work well. Extrapolating to the final components we expect to achieve a measurement precision of 5 μm with the DELPHI Microvertex Detector.   ::   pdf   [scanned version from the KEK Library]
Journal: Nucl. Instrum. Meth. A292 (1990) 75-80

• Beamtest and Beyond: Præludium and Tripel Fuga for Student and Delphi Microvertex Detector
Authors: Michael Flohr
Sources: Abstract At the beginning of this report a brief general description of the whole DELPHI Detector and in particular the Mocrovertex Detector is given. Then some aspects of the August 1988 beamtest of the Microvertex Detector are explained as the hardware and the data acquisition system with the online software used there. In the last part the off-line analysis and its first results are presented. Of course, in this report I will describe these topics rather short and incomplete. I will put the emphasis to the tasks I worked at during my stay at CERN as a summer student, July 5th — September 30th 1988.
N.B.: This report will also be used as documentation for some soft- and hardware. Therefore some parts of the report have to be rather detailed and are intended to be read by an insider, who has some experience with the described subject.
::   pdf   [scanned version]
Journal: CERN, DELPHI Microvertex Detector Group, internal report, September 1988, 38pp.

## Publications of members of my group

• Boundary Conformal Field Theory Analysis of the $$\boldsymbol{\mathrm{H}_3^+}$$ Model
Authors: Hendrik Adorf
Sources: Abstract The central topic of this thesis is the study of consistency conditions for the maximally symmetric branes of the $$\mathrm{H}_3^+$$ model. It is carried out by deriving constraints in the form of so-called shift equations and analysing their solutions. This results in explicit expressions for the one point functions in the various brane backgrounds. The brane spectrum becomes organized in certain continuous and discrete series.
In the first part, we give an introduction to two dimensional conformal field theory (CFT) in the framework of vertex operator algebras and their modules. As this approach has been developed along with rational CFT, we pay attention to adapt it to the special needs of the nonrational $$\mathrm{H}_3^+$$ model.
Part two deals with boundary CFT only. We start with a review of some basic techniques of boundary CFT and the Cardy-Lewellen sewing relations that will be at the heart of all following constructions. Afterwards, we introduce the systematics of brane solutions that we are going to follow. With the distinction between regular and irregular one point functions, we propose a new additional pattern according to which the brane solutions must be organized. We argue that all isospin dependencies must be subjected to the sewing constraints. At this point, the programme to be carried out is established and we are ready to derive the missing 1/2-shift equations for the various types of $$AdS_2$$ branes in order to make the list of this kind of equation complete.
Then we address the $$b^{-2}/2$$-shift equations. It turns out that their derivation is not straightforward: One needs to extend the initial region of definition of a certain (boundary CFT) two point function to a suitable patch. Therefore, a continuation prescription has to be assumed. The most natural candidate is analytic continuation. We show that it can be carried out, although it is rather technical and involves the use of certain generalized hypergeometric functions in two variables. In this way, we derive a complete set of $$b^{-2}/2$$-shift equations for $$AdS_2$$ branes, study their solutions and extract the resulting brane spectrum.
In a following interlude we review the $$\mathrm{H}_3^+$$ / Liouville correspondence and explain the Hosomichi-Ribault continuity proposal, which motivates our next construction. Its purpose is the explicit realization of the Hosomichi-Ribault proposal within the $$\mathrm{H}_3^+$$ model. As this proposal suggests a continuation prescription that differs from the above, one needs to study its implications for the brane solutions. Based on our explicit realization, we show that sensible $$b^{-2}/2$$-shift equations can be extracted from the Hosomichi-Ribault proposal and we study their solutions and the corresponding brane spectrum. The two approaches are finally compared. We outline a possible demarcation criterion that still has to be worked out and speculate about how our results might generalize to a certain class of noncompact nonrational CFTs.

Keywords: Nonrational Conformal Field Theory, H3+ Model, D-Branes.
::   pdf
Journal: PhD Thesis (May 2008)

• Explicit Formulas for the Scalar Modes in Seiberg-Witten Theory
Authors: Nikolas Akerblom
Sources: Abstract In this thesis, general formulas for the scalar modes $$a_i$$ and $$a_D^{\ \ \ i}$$ in the Seibgerg-Witten $$\mathrm{SU}(N)$$ setting are derived in the cases with and without massive hypermultiplets.
These formulas are applied in two cases: (i) the asymptotic behavior of the scalar modes in the asymptotically free region of moduli space is derived in the $$\mathrm{SU}(2)$$ case, (ii) the $$\mathrm{SU}(3)$$ Argyres-Douglas point is studied. In the latter case the approach via the explicit formulas derived raises, for example, the question whether the scalar modes admit an interpretation in terms of BPS mass states everywhere in moduli space.
There is also an appendix giving details on the class of special functions (Lauricella functions of type $$D$$) which naturally appear in the formulas for the scalar modes.
::   pdf
Journal: Diploma Thesis (October 2004)

• Boundary States and Symplectic Fermions
Authors: Andreas Bredthauer
Sources: Abstract We investigate the set of boundary states in the symplectic fermion description of the logarithmic conformal field theory with central charge $$c = -2$$. We show that the thus constructed states correspond exactly to those derived under the restrictions of the maximal chiral symmetry algebra for this model, $${\cal W}(2,3,3,3)$$. This connects our previous work to the coherent state approach of Kawai and Wheater.   ::   pdf   ::   arXiv:hep-th/0207181
Journal: Phys. Lett. B551 (2003) 378-386

• Boundary States in Logarithmic Conformal Field Theory — A novel Approach
Authors: Andreas Bredthauer
Sources: Abstract In this thesis, a constructive method is presented to obtain boundary states in conformal field theory. It is compatible to the usual approach via Ishibashi states in ordinary conformal field theories but extendible to cases that have a more complicated structure, such as rank-2 indecomposable Jordan cells as in logarithmic conformal field theories. In particular, it allows to study boundary states keeping the structure of the underlying bulk theory visible. Using this method the logarithmic conformal field theory with central charge $$c = -2$$ is studied in detail, deriving the maximal set of boundary states in this case. The analysis shows the existence of states corresponding to indecomposable representations as well as their irreducible subrepresentations. Furthermore, a new kind of boundary states emerges. So-called mixed boundary states glue together the two different irreducible representations of the $$c = -2$$ theory at the boundary. A relation between the boundary states is deduced that implies a deeper connection to the unique local logarithmic conformal field theory studied by Gaberdiel and Kausch. Both, the three-dimensional and the five-dimensional representation of the modular group are found when calculating the cylinder amplitudes, the latter one by introducing additional states that serve as duals to certain boundary states. Finally, the symplectic fermion model is studied to link the results to the coherent state approach of Kawai and Wheater.   ::   pdf
Journal: Diploma Thesis (June 2002)

• Towards logarithmic mode algebras
Authors: Nils Carqueville
Sources: Abstract It is an open question whether or not it is possible to generalize the definition of a vertex operator algebra to treat logarithmic conformal field theory already at the vacuum sector. With this task in mind, universal bracket relations for logarithmic mode algebras are given.   ::   pdf
Journal: Czech. J. Phys. 56 (2006) 11

• Aspects of Indecomposable Vertex Operator Algebras
Authors: Nils Carqueville
Sources: Abstract I started my studies in the framework of this thesis with the aim to gain a better understanding of the algebraic structure of (logarithmic) conformal field theory. The theory of vertex operator algebras soon was identified as the most promising and interesting setting, and most of the outcome presented here is formulated in this language. One main result is a proof of the existence and associativity of the nonmeromorphic operator product expansion for an infinite family of vertex operator algebras that are also known to be the vacuum sectors of logarithmic conformal field theories. Furthermore, all these vertex operator algebras are shown to be $$C_2$$-cofinite, and this important finiteness property is related to another one called rationality. Finally, I try to explore the algebraic roots of logarithmic conformal field theory in general and obtain some new but limited insight in logarithmic commutation relations.
Before several aspects of the theory of vertex operator algebras and its physical relevance are introduced and discussed systematically in chapter 2, the first chapter tries to sketch a bigger picture, relating much of the material in the rest of this thesis to its historical and geometrical origins and counterparts. In this way it serves as an introduction, but not one without some technical details. After a very crude positioning of two-dimensional conformal field theory within theoretical physics, the mathematical phenomenon of monstrous moonshine is touched upon, which gave birth to the study of vertex operator algebras in the second half of the last century. But vertex operator algebras are also intimately connected to the geometry of interacting strings, which motivated the abstract definition of conformal field theory by Segal and Vafa. The basic categorial and geometric ideas in these approaches are explained, and the precise way Huang's theory of geometric vertex operator algebras draws a bow from geometry to algebra in conformal field theory is described.
The second chapter provides the necessary background in vertex operator algebra theory which is needed to present and understand the main results in chapter 3, and it tries to do so in a self-contained way. In section 2.1 basic notions such as vertex operator algebras, modules, intertwining operators and fusion rules are introduced, and some of their relations and properties are studied. An exhaustive presentation of the theory of vertex operator algebras cannot be fit into one small chapter, and the focus is set on the particularly important associativity and commutativity properties which are discussed in much detail. Then in section 2.2, the notions of rationality, regularity, $$C_n$$-cofiniteness and the Zhu algebra are introduced, and the relations among these finiteness properties for vertex operator algebras are pointed out. A brief introduction to some aspects of $${\cal W}$$-algebras is also given in this context. Finally in section 2.3, a skeleton of $$P(z)$$-tensor product theory is presented. This theory allows to give conditions on the existence and associativity of the nonmeromorphic operator product expansion such that if these conditions are satisfied, the nonmeromorphic operator product expansion can be derived from first principles.
The third chapter combines results from sections 2.2 and 2.3 in a study of all triplet algebras $\{{\cal W}(2,(2p - 1)^{\otimes 3})\}_{p\in \mathbb{Z}_{\geq 2}}$. Most of the material in this chapter is an extended description of the work published in [CF]. The triplet algebras are introduced and it is shown how the list of properties sufficient for the existence and associativity of the nonmeromorphic operator product expansion can almost be reduced to the $$C_2$$-cofiniteness of the vertex operator algebras $${\cal W}(2,(2p - 1)^{\otimes 3})$$. A careful study of singular vectors and characters for the triplet algebras is then applied to prove that they are indeed all $$C_2$$-cofinite. This together with the consequences for the nonmeromorphic operator product expansion is the main result in my thesis. While the explicitly known singular vectors and commutation relations for the triplet algebra $${\cal W}(2,(2p - 1)^{\otimes 3})$$ make the proof rather easy in this case, obtaining the result for all other triplet algebras requires some subtle arguments. Chapter 3 ends with an application of the study of singular vectors to establish an upper bound on the dimension of the Zhu algebras for the triplet algebras.
The triplet algebras are all examples of logarithmic conformal field theories as follows from their explicitly known characters. This fact is not stressed in chapter 3 since the arguments used to establish its main results do not depend crucially on indecomposable structures — even though the triplet algebras are the first family of logarithmic conformal field theories for which $$C_2$$-cofiniteness and nonmeromorphic operator product expansion are proven. Chapter 4 on the other hand is devoted to the specific properties of logarithmic conformal field theories. More precisely, the possibility of a generalized vertex operator algebraic approach to logarithmic conformal field theory that places the defining features of such theories at the most fundamental level is investigated. This is in contrast to known attempts to describe logarithmic conformal field theory in terms of vertex operator algebra theory, which introduces the indecomposable structure and logarithms only at the level of modules and intertwining operators. Part of the reason a more fundamental approach has not yet been presented in the literature are certain expansion issues and difficulties to properly introduce logarithms into the core structure of vertex operator algebras. I am not able to master all these problems, but some results on logarithmic mode algebras are obtained which may help to describe logarithmic conformal field theory in the indicated way in the future.
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Journal: Diploma Thesis (December 2005)

• Towards the Construction of Local Logarithmic Conformal Field Theories
Authors: Anne-Lý Do
Sources: Abstract In this thesis first steps are taken towards the construction of local logarithmic conformal field theories (LCFTs). After resuming some of the central concepts and definitions of conformal field theory we explore local non-chiral LCFTs whose chiral halves possess arbitrary Jordan rank. Our approach rests upon the analysis of the conformal symmetry and aims for comprehensive generality. Beside some rather general assumptions on the structure of a chiral LCFT the deduced statements about its local non-chiral equivalent can be called generic.
In the first part of this thesis we investigate the field content of local LCFTs and present two methods how to construct local representations of the symmetry algebra as subrepresentations of the tensor product of chiral and anti-chiral Jordan cells. The gained Hilbert spaces are not identical but isomorphic. The symplectic fermion modell suggests one of the solutions to be more intuitive.
In the second part we study the possibility to assemble generic chiral correlation functions to local correlation functions of the corresponding non-chiral theory. Beside the constraint of locality, invariance under the global conformal group, duality and monodromy invariance have to be implemented. We propose a constructive method, discuss the generality of our solution and check its consistency with previous findings.
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Journal: Diploma Thesis (October 2007)

• Logarithmic Conformal Field Theory with Supersymmetry
Authors: Dmitriy Drichel
Sources: Abstract In this thesis, the Neveu-Schwarz sectors of $$N=1,2,3$$ extended super Virasoro algebras with indecomposable representations are studied. We answer a series of questions related to the structure of LSCFTs. Thus, our work consists of a treatment of $$N=1,2,3$$ extended super Virasoro theories in general and logarithmic theories in particular. We find that the logarithmic $$N=1$$ theories are probably just supersymmetric extensions of already known $$N=0$$ models. Contrary to previous conjectures made in the literature we conclude that this fact holds for theories with $$N>1$$. This implies that there are no logarithmic fields with respect to the supersymmetric current. A treatment of indecomposable $$\mathfrak{su}(2)$$ representations is presented, and we find that, again, this structure does not appear in $$N=3$$ theories. We find the two- and three-point functions of the $$N=3$$ theories by solving the superconformal Ward identities and comment on irreducible representations. Furthermore, we find no “hidden connection” between logarithmic theories and supersymmetric theories previously conjectured in the literature on the basis of apparent similarity between supersymmetric fields and logarithmic fields in the nilpotent variable formalism.   ::   pdf
Journal: Diploma Thesis (February 2010)

• On explicit results at the intersection of the $$\boldsymbol{\mathbb{Z}_2}$$ and $$\boldsymbol{\mathbb{Z}_4}$$ orbifold subvarieties in $$\boldsymbol{K3}$$ moduli space
Authors: Holger Eberle
Sources: Abstract We examine the recently found point of intersection between the $$\mathbb{Z}_2$$ and $$\mathbb{Z}_4$$ orbifold subvarieties in the $$K3$$ moduli space more closely. First we give an explicit identification of the coordinates of the respective $$\mathbb{Z}_2$$ and $$\mathbb{Z}_4$$ orbifold theories at this point. Secondly we construct the explicit identification of conformal field theories at this point and show the orthogonality of the two subvarieties.   ::   pdf   ::   arXiv:hep-th/0407170
Journal: JHEP 08 (2004) 015

• Conformal Field Theory Between Supersymmetry and Indecomposable Structures
Authors: Holger Eberle
Sources: Abstract This thesis considers conformal field theory in its supersymmetric extension as well as in its relaxation to logarithmic conformal field theory.
Compactification of superstring theory on four-dimensional complex manifolds obeying the Calabi-Yau conditions yields the moduli space of $$N = (4,4)$$ superconformal field theories with central charge $c = 6$ which consists of two continuously connected subspaces. This thesis is concerned with the subspace of $$K3$$ compactifications which is not well known yet. In particular, we inspect the intersection point of the $$\mathbb{Z}_2$$ and $$\mathbb{Z}_4$$ orbifold subvarieties within the $$K3$$ moduli space, explicitly identify the two corresponding points on the subvarieties geometrically, and give an explicit isomorphism of the three conformal field theory models located at that point, a specific $$\mathbb{Z}_2$$ and a $$\mathbb{Z}_4$$ orbifold model as well as the Gepner model $$(2)^4$$. We also prove the orthogonality of the two subvarieties at the intersection point. This is the starting point for the programme to investigate generic points in $$K3$$ moduli space. We use the coordinate identification at the intersection point in order to relate the coordinates of both subvarieties and to explicitly calculate a geometric geodesic between the two subvarieties as well as its generator. A generic point in $$K3$$ moduli space can be reached by such a geodesic originating at a known model. We also present advances on the conformal field theoretic side of deformations along such a geodesic using conformal deformation theory. Since a consistent regularisation of the appearing deformation integrals has not been achieved yet, the completion of this programme is still an open problem.
Moreover, we regard a relaxation of conformal field theory to logarithmic conformal field theory. The latter allows the indecomposable action of the $$L_0$$ Virasoro mode within a representation of the conformal symmetry. In particular, we study general augmented $$c_{p,q}$$ minimal models which generalise the well-known (augmented) $$c_{o,1}$$ model series. We calculate logarithmic nullvectors in both types of models. But most importantly, we investigate the low lying Virasoro representation content and fusion algebra of two general augmented $$c_{p,q}$$ models, the augmented $$c_{2,3} = 0$$ model as well as the augmented Yang-Lee model at $$c_{2,5} = -22/5$$. These exhibit a much richer structure as the $$c_{p,1}$$ models with indecomposable representations up to rank $$3$$. We elaborate several of these new rank $$3$$ representations in great detail and uncover astonishing features. Furthermore, we argue that irreducible representations corresponding to the Kac table domain of the proper minimal models cannot be included into the theory. In particular, the true vacuum representation is rather given by a rank $$1$$ indecomposable but not irreducible subrepresentation of a rank $$2$$ representation. We generalise these generic examples to give the representation content and the fusion algebra of general augmented $$c_{p,q}$$ models as a conjecture. Finally, we open a new connection between logarithmic conformal field theory and quantum spin chains by relating some representations of the augmented $$c_{2,3} = 0$$ model to the representation content of a $$c = 0$$ model which describes an $$XXZ$$ quantum spin chain.
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Journal: PhD Thesis (May 2006)

• Fermionic Sum Representations of Characters in Logarithmic Conformal Field Theory
Authors: Carsten Grabow
Sources: Abstract Based on our article [FGK07], published in Nuclear Physics B, fermionic quasi-particle sum representations consisting of only one single fundamental fermionic form are presented for all characters of the logarithmic conformal field theory models with central charge $$c_{p,1}$$, $$p \geq 2$$.
These new representations are embedded in the surrounding field of Nahm's conjecture and modular forms in general. In this context, it is also shown that it is possible to correctly extract dilogarithm identities, which supports the derived fermionic character expressions even more.
In addition, other building blocks of the fermionic characters, with regard to the $$\mathrm{SU}(2)$$ Wess-Zumino-Witten conformal field theory and Kac-Peterson characters of the affine Lie algebras are presented, which might be of importance for the future work on yet missing fermionic expressions.
To conclude, a conjecture for a physical quasi-particle interpretation for the new fermionic character expressions of the $$c_{p,1}$$ models is made, involving symplectic fermions.
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Journal: Diploma Thesis (February 2007)

• Fusion Algebras and Verlinde-Formula in Logarithmic Conformal Field Theories
Authors: Holger Knuth
Sources: Abstract The extended minimal models of the $$c_{(p,1)}$$-series are examples for conformal field theories with correlation functions exhibiting logarithmic divergences. As their — and other logarithmic CFTs' — basic feature one finds a number of indecomposable, but reducible, representations of their chiral symmetry algebra, the triplet $${\cal W}$$-algebra, which is the maximal extended symmetry algebra of the Virasoro algebra at central charge $$c = c_{(p,1)}$$. These representations together with the irreducible ones form a set, which shows strong evidence of closing under fusion.
An algorithm of Gaberdiel and Kausch to calculate the fusion products has only been completely carried out by them for the cases $$c_{(2,1)} = -2$$ and $$c_{(3,1)} = -7$$, because of the strongly increasing complexity towards higher $$p$$, also proving the closure under fusion for these two cases.
Virtually at the same time Flohr proposed a simpler method of calculation using the Verlinde formula and a specific choice of the $$S$$-matrix, which also took advantage of his work on modular invariant partition sums at that time. Three years ago Fuchs et al., who studied non-semisimple fusion algebras in general, presented another Verlinde-like proposal for the fusion of only the irreducible representations in $$c_{(p,1)}$$ models.
After an general introduction to conformal field theories and a one specifically to $$c_{(p,1)}$$ models I present the last two approaches. Their findings are in correspondence to the known results of the algorithm. I point out the direct connection between both methods, which has been rather befogged until now, as I give for the first time Flohr's $$S$$-matrix in closed form and clarify the meaning of some objects in Fuchs' ansatz with respect to the conformal field theories. In the process I examine diverse aspects of both methods in detail. This thesis will describe at length the path to an extension of Fuchs' approach, which we have found, including the indecomposable representations. It is shown, that the results of this extension are the same as the ones found with Flohr's method. The advantages and disadvantages of the different Verlinde-like formulas for non-semisimple fusion algebras are discussed, in particular also in comparison to semisimple fusion algebras.
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Journal: Diploma Thesis (December 2006)

• Dilogarithm Identities and Characters of Exceptional Rational Conformal Field Theories
Authors: Michael Köhn
Sources: Abstract Based on our recently published article Fermionic Expressions for the Characters of $$c_{p,1}$$ Logarithmic Conformal Field Theories in the journal Nuclear Physics B, fermionic sum representations for the characters of the $${\cal W}(2,2p - 1, 2p - 1, 2p - 1)$$ series of triplet algebras are obtained, providing further evidence that these logarithmic theories constitute well-defined rational conformal field theories. Furthermore, after an investigation of Nahm's conjecture, fermionic character expressions for other conformal field theories such as the minimal Virasoro models and $$\mathrm{SU}(2)$$ Wess-Zumino-Witten (WZW) models are given, some of the latter also being new. In combination with their known bosonic counterparts, fermionic character expressions give rise to so-called bosonic-fermionic $$q$$-series identities, closely related to the famous Rogers-Ramanujan and Andrews-Gordon identities. Additionally, it is displayed how fermionic character expressions imply dilogarithm identities for the effective central charge of the conformal field theory in question. In the case of the triplet algebras, this results in an infinite series of dilogarithm identities. Since a proof for this series of identities already exists, this strongly supports the corresponding fermionic character expressions. In general, fermionic sum representations for characters give rise to an interpretation of the corresponding theory in terms of quasi-particles which obey generalized exclusion statistics. This quasi-particle content is discussed for conformal field theories which admit fermionic character expressions.   ::   pdf
Journal: Diploma Thesis (February 2007)

• Aspects of Logarithmic Conformal Field Theory
Authors: Marco Krohn
Sources: Abstract In this thesis we consider various aspects of logarithmic conformal field theories (LCFTs). After recalling some important definitions and relations of (logarithmic) conformal field theories we study possible extensions of conformal ghost systems to LCFTs and the generic structure of the four-point function.
In the first part of this thesis we analyze the $$b$$-$$c$$ ghost system of central charge $$c = -2$$ and compare it with the $$\theta$$-$$\bar\theta$$ system, which is known to be a LCFT. For that purpose we study two different ways of constructing logarithmic conformal field theories: via symplectic fermions and by deforming the energy-momentum tensor. While the two approaches lead to the same results for $$c = -2$$, they are not equivalent for $$c = -26$$. In the latter case the construction by symplectic fermions leads to unsurmountable problems, but a variation of the deformation method yields operators satisfying the commutation relations of the Virasoro algebra. However, the operators do not act in a consistent way on the Hilbert space of states which forces us to study the theory on nontrivial Riemann surfaces. This leads to a consistent LCFT for $$c = -26$$ which in particular differs from known LCFTs by the structure of the Virasoro modes. The mode $$L_0$$ turns out to be diagonal and it is even possible to choose the deformation parameters appearing in the result in such a way that the global conformal group is non-logarithmic.
In the second part we investigate the generic structure of $$4$$-point correlation functions of fields residing in indecomposable representations of arbritrary rank. To fix the generic structure we need to solve a generalization of the global conformal Ward identities (GCWI). The solutions are recursively determined by using an algorithm which we present in this part of this thesis. We also compute all results for a Jordan-rank $$r = 2$$ and $$r = 3$$ theory, making use of permutation symmetry and introduce a graphical representation. By doing so we obtain much shorter results and get a better understanding of the structure of the solutions. In case all four fields are logarithmic partner fields additional constants in the solution may appear. We discuss the origin of these additional degrees of freedom and investigate the influence of the discrete symmetry on these terms. Finally we explicitly determine the four-point correlator for arbitrary Jordan-rank for the case where up two fields are of logarithmic type and suggest the form for the case of three logarithmic fields.

Keywords: logarithmic conformal field theory, ghost system, correlation function.
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Journal: PhD Thesis (June 2005)

• Symplectic Fermions — Symmetries of a Vertex Operator Algebra
Authors: Johannes Meisig
Sources: Abstract The model of $$d$$ symplectic fermions constructed by Abe [1] gives an example of a $$C_2$$-cofinite vertex operator algebra admitting logarithmic modules. While the case $$d = 1$$ is a rigorous formulation of the well known triplet algebra, the case $$d > 1$$ has not yet been analyzed from the perspective of $${\cal W}$$-algebras. It is shown that in the latter case the ${\cal W}(2, 2^{\otimes(2d^2-d-1)}, 3^{\otimes(2d^2+d)})$-algebra is realized. With respect to its zero mode algebra, it is proven that the zero modes corresponding to the vectors of weight $$3$$ form the $$2d$$-dimensional symplectic Lie algebra. Furthermore, the zero modes corresponding to the vectors of weight $$2$$ form an irreducible representation of this Lie algebra. The use of the zero mode algebra for the classification of irreducible representations of the $${\cal W}$$-algebra is compared to the approach using Zhu's algebra. The isomorphism between Zhu's algebra and the zero mode algebra as Lie algebras, which is expected from general considerations, is confirmed by explicit calculations.   ::   pdf
Journal: Diploma Thesis (March 2007)

• Conformal Field Theory on the Plane and the Torus: On a Relation of Four-Point Correlation Functions and Torus Amplitudes
Authors: Sergej Mikheev
Sources: Abstract In the presented work the relation of CFT four-point correlation functions on the plane to torus zero- and one-point amplitudes in the minimal models with central charge $$c = 1/2$$, $$c = −22/5$$, $$c = 7/10$$ and $$c = 4/5$$ was studied.
We focus here solely on four-point functions of conformal fields which are degenerate at level two. As a consequence, the conformal blocks of these correlators are given as solutions to the hypergeometric differential equation. It shows that the modular parameter used to compute the four-point correlators in terms of minimal characters yields only in a few cases the desired result. Hence, a generalized modular equation is formulated. We show how the generalized modular parameter can be chosen as a characteristic of each minimal model mentioned above. In the concluding chapter some numerical calculations with the new modular quantities are performed.
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Journal: Diploma Thesis (August 2009)

• Stochastic Löwner Evolution as an approach to Conformal Field Theory
Authors: Annekathrin Müller-Lohmann
Sources: Abstract The main focus on this work lies on the relationship between two-dimensional boundary Conformal Field Theories (bCFTs) and Schramm-Löwner Evolutions (SLEs) as motivated by their connection to the scaling limit of Statistical Physics models at criticality. The bCFT approach used for the past 25 years is based on the algebraic formulation of local objects such as fields and their correlations in these models. Introduced in 1999, SLE describes the physical properties from a probabilistic point of view, studying measures on growing curves, i.e. global objects such as cluster surfaces.
After a short motivation of the topic, followed by a more detailed introduction to two-dimensional boundary Conformal Field Theory and Schramm-Löwner Evolution, we present the results of our original work. We extend the method of obtaining SLE variants from a change of measure of the single SLE to derive the most general bCFT model that can be related to SLE. Morevoer, we interpret the change of the measure in the context of physics and Probability Theory. In addition, we discuss the meaning of bulk fields in bCFT as bulk force-points for the SLE variant SLE$$(\kappa,\vec{\rho})$$.
Furthermore, we investigate the short-distance expansion of the boundary condition changing fields, creating cluster interfaces that can be described by SLE, with other boundary or bulk fields. Thereby we derive new SLE martingales related to the existence of boundary fields with vanishing descendant on level three. We motivate that the short-distance scaling law of these martingales as adjustment of the measure can be interpreted as the SLE probability of curves coming close to the location of the second field.
Finally, we extend the algebraic κ-relation for the allowed variances in multiple SLE, arising due to the commutation requirement of the infinitesimal growth operators, to the joint growth of two SLE traces. The analysis straightforwardly suggests the form of the infinitesimal Löwner mapping of joint processes, presumably related to the growth of curves due to the existence of boundary condition changing fields with vanishing descendant on level three.
We conclude with a summary of our results, ranging our work into the context of preceding papers and giving an outlook to promising new methods and open questions.

Keywords: Schramm-Löwner Evolution, boundary Conformal Field Theory, Critical Phenomena
PACS: 11.25.Hf, 02.50.Ey
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Journal: PhD Thesis (December 2008)

• CFT Interpretation of Merging Multiple SLE Traces
Authors: Annekathrin Müller-Lohmann
Sources: Abstract In this paper we give a physical interpretation of the probability of a Stochastic Löwner Evolution (SLE) trace approaching a marked point in the upper half plane, e.g. on another trace. Our approach is based on the concept of fusion of boundary with bulk or other boundary fields in Conformal Field Theory (CFT). In this context, we also state a porposal how the results can be extended to situations that happen not sufficiently far away from the boundary.   ::   pdf   ::   arXiv:0711.3165 [math-ph]
Journal: submitted to J.Phys A: Math.Gen.   ::   ITP-UH-24/07, November 2007, 30pp.

• On a possible interpretation of fusion in stochastic Löwner evolution
Authors: Annekathrin Müller-Lohmann
Sources: Abstract We suggest how to interpret Stochastic Löwner Evolution traces approaching a marked point in the upper half plane $$\mathbb{H}$$. We show that this may be related to the fusion of boundary with bulk fields in Conformal Field Theory by taking a look at the probability of such an event.   ::   pdf   ::   arXiv:0707.0443 [hep-th]
Journal: submitted to JSTAT   ::   ITP-UH-13/07, July 2007, 8pp.

• Conformal Field Theory and Percolation
Authors: Annekathrin Müller-Lohmann
Sources: Abstract In this thesis, important features of two dimensional bond percolation on an infinite square lattice at its critical point within a conformal field theory (CFT) approach are presented. This includes a level three null vector interpretation for Watts' differential equation [78] describing the horizontal vertical crossing probability within this setup. A unique solution among the minimal models, $$c_{(6,1)} = -24$$, seems to be a good candidate, satisfying the level two differential equation for the horizontal crossing probability derived by Cardy [7] as well.
Commonly assumed to be a truly scale invariant problem, percolation nevertheless is usually investigated as a $$c = 0$$ CFT. Moreover this class of CFTs is important for the study of percolation or quenched disorder models in general. Since $$c_{(3,2)} = 0$$ as a minimal model only consists of the identity field, following Cardy [9] different approaches to get a non trivial CFT whose partition functions differ from one as suggested by the work of Pearce and Rittenberg [69] are presented. Concentrating on a similar ansatz for logarithmic behavior as for the triplet series $$(c_{(p,1)})$$, we examine the properties of such a CFT based on the extended Kac-table for $$c_{(9,6)} = 0$$ using a general ansatz for the stress energy tensor residing in a Jordan cell of rank two. We will derive the interesting OPEs in this setup (i.e. of the stress energy tensor and its logarithmic partner) and illustrate it by a bosonic field realization. We will give a motivation why the augmented minimal model seems to be more promising than the previous approaches and present an example of a tensor construction as a fourth ansatz to solve the $$c \rightarrow 0$$ problem as well.
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Journal: Diploma Thesis (November 2005)

• A Novel Conformal Field Theory Approach to Bulk Wave Functions in the Fractional Quantum Hall Effect
Authors: Klaus Osterloh
Sources: Abstract In this thesis, a novel approach is proposed to represent bulk wave functions of fractional quantum Hall states in terms of conformal field theory correlators. It starts from the Laughlin states and their generalization following Jain's picture of composite fermions. These effective particles are naturally identified within the $$b/c$$-spin conformal field theories. The enigmatic phenomenon of fractional statistics is described by twist fields which inherently appear in the spin systems. A geometrical interpretation is obtained in which bulk wave functions are understood as holomorphic functions over a ramified covering of the complex plane. To extend Jain's main series, the concept of composite fermions that pair to spin singlets is introduced. This is naturally adopted by the particular $$b/c$$-spin system with central charge $$c = -2$$ as known for the Haldane-Rezayi state with filling fraction $$\nu = 5/2$$. In this way, the new conformal field theory proposal covers the set of experimentally confirmed fractional quantum Hall states in the lowest Landau level. Concerning their stability with respect to energy gaps of the ground states, a natural ordering is deduced where unobserved filling fractions are precisely avoided. The scheme is compatible with classifications in terms of effective Chern-Simons theories. It leads to severe restrictions of the coupling $$K$$-matrices and, in addition, the $$b/c$$-spin approach can be extended to describe non-Abelian fractional quantum Hall states imposing physical constraints on them.   ::   pdf
Journal: Diploma Thesis (October 2002)

• Investigation of the Wigner Crystal/Fermi Liquid Transition in a Quantum Dot
Authors: Stephan Rafler
Sources: Abstract N/A   ::   pdf   [in German]
Journal: Diploma Thesis (September 2002)

• Logarithmic Extensions and New Indecomposable Structures of Two-Dimensional Conformal Ghost Systems with Additional $$\boldsymbol{\mathbb{Z}_n}$$ Symmetry
Authors: Julia Voelskow
Sources: Abstract In this thesis, I examined the behaviour of ghost systems as two dimensional conformal field theories under additive modifications of the energy-momentum tensor. The corresponding modification fields are required to have modes in the universal enveloping algebra of the algebra of the ghost modes. This condition can be satisfied, if the ghost systems in consideration are defined on a Riemann surface, represented as a ramified covering of the Riemann sphere. The unmodified energy-momentum tensor acts inconsistently on the space of states: Its action on an (enlarged) space of states contradicts the associativity of the underlying operator product algebra. Contrary to this, the modified energy-momentum tensor exhibits the desired rank-2 Jordan cells on the space of states of a $$\mathbb{Z}_n$$ twisted ghost system with central charge $$c = -2$$.
Similar modifications of the energy-momentum tensor are possible for higher spin $$\lambda$$: I have found a more general ansatz containing the above-mentioned as a special case. Consistency conditions on the mode content of this ansatz and on the functional dependence on the other parameters are given explicitly. It could be shown for all modifications which are required to be decomposed by $$b$$-modes only that Jordan cells are not possible for fermionic ghost systems except the one at $$c = -2$$. Nevertheless, indecomposable structures with respect to other Virasoro generators do exist. I have been able to construct quasi-primary states, which have to satisfy inhomogeneous Ward identities with respect to the translation operator. To my knowledge, such indecomposable structres have not been described in the framework of logarithmic conformal field theory.
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Journal: Diploma Thesis (January 2005)

• Conformal Ghosts on the Sphere
Authors: Kirsten Vogeler
Sources: Abstract This thesis is about the relation of geometry and logarithmic conformal field theories. I consider two different geometric settings: in part I the topological $$A$$-model with exmbedding $x : \mathbb{R}^1 \times S^1 \rightarrow \mathbb{CP}^1$, and in part II conformal, fermionic ghosts on the torus.
The $$A$$-model can be transformed such that the path integral yields a $$\delta$$ distribution on the moduli space of instantons. Integration out the dependency on S1, one obtains Morse theory on the universal cover $$\widetilde{(\mathrm{L}\mathbb{CP}^1)}$$ of loop space. Its state space can be derived perturbatively in the charts of this manifold, and can be modeled by the representations of the chiral de Rham complex. Assuming that the representation theory of the A-models and the chiral de Rham complex are identical, I consider the chiral de Rham complex in the following. The state spaces are local, induced representations of the symmetry generated by the gradient vector field of the Mores function. I prove that the Hamiltonian gains additional terms when these local representations are generalized as distributions on $$\widetilde{(\mathrm{L}\mathbb{CP}^1)}$$, and discuss their geometric significance as cohomology operators in a complex of globally extended local representation spaces. Eventually, I show that a logarithmic extension of the chiral de Rham complex corresponds the additional terms in the Hamiltonian.
The conformal, fermionic ghosts of part II transform in irreducible representations of the monodromy group $$\mathbb{Z}_2$$. I show that the conformal field theory of these fields has to be logarithmically extended as soon as the representations of the monodromy group are allowed to move freely on the parameter space $$\mathbb{CP}^1 - \{0,1,\infty\}$$ of the torus. The triplet model constitutes a minimal logarithmic extension of this theory and is fundamental for my last chapter. Therein, I obtain the spectral curve of $$\mathrm{SU}(2)$$ Seiberg-Witten theory in terms of characters of the triplet model. Further, I trace back the prepotential to that model by expressing it as a function of the torus modulus of the spectral curve.
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Journal: PhD Thesis (June 2010)

## Other Publications

• On the Compacatness of Moduli
Authors: Ch.K.I. Flohr, B.U. Flohr, M.A.I. Flohr
Sources: Abstract Let $$b''$$ be a non-bounded path. V.U. Beltrami's derivation of injective topological spaces was a milestone in symbolic set theory. We show that $$\textrm{m}\left(\frac12,\ldots,-i\right) = \int\int\int \lim_{\tilde{t}\rightarrow\emptyset} \tilde{\cal E}({{\cal K}'}^4,\bar d\!\infty)\ \ dA_W\,.$$ Now in this setting, the ability to extend completely Pascal polytopes is essential. The goal ofthe present paper is to classify tangential manifolds.   ::   pdf   ::   arXiv:1213.1415 [non-sense]
Journal:` submitted to J.Abstract.Nonsense   ::   Oct 2012, 8pp.