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

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

Operator Product Expansion and Zero Mode Structure in Logarithmic CFT

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.Comment: Contribution to the Proceedings of the 36th International Symposium Ahrenshoop on the Theory of Elementary Particles, 7p