A Note on Four-Point Functions in Logarithmic Conformal Field Theory

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.Comment: Contribution to the Proceedings of the 37th International Symposium Ahrenshoop on the Theory of Elementary Particles, 7p

Notes on Generalised Nullvectors in logarithmic CFT

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.Comment: 27 pages, 3 figures; v2: Corrected two typos, added one reference to the conclusio

Logarithmic Primary Fields in Conformal and Superconformal Field Theory

In this note, some aspects of the generalization of a primary field to the logarithmic scenario are discussed. This involves understanding how to build Jordan blocks into the geometric definition of a primary field of a conformal field theory. The construction is extended to N=1,2 superconformal theories. For the N=0,2 theories, the two-point functions are calculated.Comment: 17 pages LaTeX 2e, references added, journal details adde

Fusion & Tensoring of Conformal Field Theory and Composite Fermion Picture of Fractional Quantum Hall Effect

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 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.Comment: 13 pages, LaTeX (or better LaTeX2e), no figures, also available at http://www.sns.ias.edu/~flohr

Indecomposable Representations in Z_n Symmetric b,c Ghost Systems via Deformations of the Virasoro Field

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.Comment: 28 pages, 1 figure, uses empheq.sty, mhtools.sty, mhsetup.st

Logarithmic Conformal Field Theory - or - How to Compute a Torus Amplitude on the Sphere

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 be 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.Comment: LaTeX, 38 pp. 3 figures (provided as eps and as pdf). Contribution to the Ian Kogan Memorial Volume "From Fields to Strings: Circumnavigating Theoretical Physics

More Curiosities at Effective c = 1

The moduli space of all rational conformal quantum field theories with effective central charge c_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.Comment: 13 pp. LaTeX with 2 PostScript figures (finally in compressed bitmap format to save disk space