Magnetic fields and fractional statistics in boundary conformal field theory

We study conformal field theories describing two massless one-dimensional fields interacting at a single spatial point. The interactions we include are periodic functions of the bosonized fields separately plus a “magnetic” interaction that mixes the two fields. Such models arise in open-string theory and dissipative quantum mechanics and perhaps in edge state tunneling in the fractional quantized Hall effect. The partition function for such theories is a Coulomb gas with interchange phases arising from the magnetic field. These “fractional statistics” have a profound effect on the phase structure of the Coulomb gas. In this paper we present new exact and approximate results for this type of generalized Coulomb gas.     

A plasma analogy and Berry matrices for non-abelian quantum Hall states

We present an approach to the computation of the non-abelian statistics of quasiholes in quantum Hall states, such as the Pfaffian state, whose wavefunctions are related to the conformal blocks of minimal model conformal field theories. We use the Coulomb gas construction of these conformal field theories to formulate a plasma analogy for the quantum Hall states. A number of properties of the Pfaffian state follow immediately, including the Berry phases, which demonstrate the quasiholes' fractional charge, the abelian statistics of the two-quasihole state, and equal-time ground state correlation functions. The non-abelian statistics of multi-quasihole states follows from an additional assumption.     

Structure constants for the osp(1|2) current algebra

We study the free field realization of the two-dimensional osp(1|2) current algebra. We consider the case in which the level of the affine osp(1|2) symmetry is a positive integer. Using the Coulomb gas technique we obtain integral representations for the conformal blocks of the model. In particular, from the behaviour of the four-point function, we extract the structure constants for the product of two arbitrary primary operators of the theory. From this result we derive the fusion rules of the osp(1|2) conformal field theory and we explore the connections between the osp(1|2) affine symmetry and the N = 1 superconformal field theories.     

Discrete field variables,the Coulomb gas,and low temperature behaviour

Field theories with discrete field variables are reformulated as continuous field theories coupled to a Coulomb gas. TheU(1) gauge theory in three dimensions and thexy-model possess dual versions that are treated this way. Low lying states of the Coulomb gas determine their behaviour at low temperature. The Coulomb vacuum gives rise to a gauge invariant low temperature expansion in powers of 1/β. After clarifying the meaning of exponential terms, we try to determine exponential contributions to Wilson's loop correlation function. These investigations are based on a simplified model of a generalized Villain type.     

Puzzle of bulk conformal field theories at central charge c = 0

Nontrivial critical models in 2D with a central charge c=0 are described by logarithmic conformal field theories (LCFTs), and exhibit, in particular, mixing of the stress-energy tensor with a "logarithmic" partner under a conformal transformation. This mixing is quantified by a parameter (usually denoted b), introduced in Gurarie [Nucl. Phys. B546, 765 (1999)]. The value of b has been determined over the last few years for the boundary versions of these models: b(perco)=-5/8 for percolation and b(poly)=5/6 for dilute polymers. Meanwhile, the existence and value of b for the bulk theory has remained an open problem. Using lattice regularization techniques we provide here an "experimental study" of this question. We show that, while the chiral stress tensor has indeed a single logarithmic partner in the chiral sector of the theory, the value of b is not the expected one; instead, b=-5 for both theories. We suggest a theoretical explanation of this result using operator product expansions and Coulomb gas arguments, and discuss the physical consequences on correlation functions. Our results imply that the relation between bulk LCFTs of physical interest and their boundary counterparts is considerably more involved than in the nonlogarithmic case.     

Harmonic measure and winding of random conformal paths: A Coulomb gas perspective

We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration.     

Monte Carlo simulations of conformal theory predictions for the three-state Potts model

The critical properties of the three-state Potts model are investigated using Monte Carlo simulations. Special interest is given to the measurement of three-point correlation functions and associated universal objects, i.e., structure constants. The results agree well with predictions coming from conformal field theory, confirming, for this example, the correctness of the Coulomb gas formalism and the bootstrap method.     

Asymptotic Completeness for Infraparticles in Two-Dimensional Conformal Field Theory

We formulate a new concept of asymptotic completeness for two-dimensional massless quantum field theories in the spirit of the theory of particle weights. We show that this concept is more general than the standard particle interpretation based on Buchholz’ scattering theory of waves. In particular, it holds in any chiral conformal field theory in an irreducible product representation and in any completely rational conformal field theory. This class contains theories of infraparticles to which the scattering theory of waves does not apply.     

Holomorphic Chern-Simons-Witten theory: From 2D to 4D conformal field theories

It is well known that rational 2D conformal field theories are connected with Chern-Simons theories defined on 3D real manifolds. We consider holomorphic analogues of Chern-Simons theories defined on 3D complex manifolds (six real dimensions) and describe 4D conformal field theories connected with them. All these models are integrable. We describe analogues of the Virasoro and affine Lie algebras, the local action of which on fields of holomorphic analogues of Chern-Simons theories becomes non-local after pushing down to the action on fields of integrable 4D conformal field theories. Quantization of integrable 4D conformal field theories and relations to string theories are briefly discussed.     

Generalized Cauchy determinant formula and its applications to Coulomb gas problems

Using only elementary methods, we generalize the classical Cauchy determinant formula and indicate applications to two-dimensional systems including the neutral asymmetric Coulomb gas and conformal quantum field theory.     

Universal exchange algebra for Bloch waves and Liouville theory

We derive a universal formula for the exchange algebra in the Bloch wave basis. The main tool we use is a lattice version of the Coulomb gas picture of conformal field theory, making its quantum group structure explicit from the very beginning. Calulations are then reduced to a factorization problem inU _q (sl ₂).     

On representations of conformal field theories and the construction of orbifolds

We consider representations of meromorphic bosonic chiral conformal field theories and demonstrate that such a representation is completely specified by a state within the theory. The necessary and sufficient conditions upon this state are derived and, because of their form, we show that we may extend the representation to a representation of a suitable larger conformal field theory. In particular, we apply this procedure to the (untwisted) lattice conformal field theories (i.e. corresponding to the propagation of a bosonic string on a torus), and deduce that Dong's proof of the uniqueness of the twisted representation for the reflection-twisted projection of the Leech lattice conformal field theory generalises to an arbitrary even (self-dual) lattice. As a consequence, we see that the reflection-twisted lattice theories of Dolan, Goddard and Montague are truly self-dual, extending the analogies with the theories of lattices and codes which were being pursued. Some comments are also made on the general concept of the definition of an orbifold of a conformal field theory in relation to this point of view.     

Solving 1D plasmas and 2D boundary problems using Jack polynomials and functional relations

The general one-dimensional log-sine gas is defined by restricting the positive and negative charges of a two-dimensional Coulomb gas to live on a circle. Depending on charge constrannts, this problem is equivalent to different boundary field theories.We study the electrically neutral case, which is equivalent to a two-dimensional free boson with an impurity cosine potential. We use two different methods: a perturbative one based on Jack symmetric functions, and a nonperturbative one based on the thermodynamic Bethe ansatz and functional relations. The first method allows us to find an explicit series expression for all coefficients in the virial expansion of the free energy and the experimentally measurable conductance. Some results for correlation functions are also presented. The second method gives an expression for the full free energy, which yields a surprising fluctuation-dissipation relation between the conductance and the free energy.     

Quantum group symmetries and non-local currents in 2D QFT

We construct and study the implications of some new non-local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions, including the sine-Gordon theory and its generalization to affine Toda theory. These non-local currents provide a non-perturbative formulation of the theories. The symmetry algebras correspond to the quantum affine Kac-Moody algebras. TheS-matrices are completely characterized by these symmetries. FormalS-matrices for the imaginary-coupling affine Toda theories are thereby derived. The application of theseS-matrices to perturbed coset conformal field theory is studied. Non-local charges generating the finite dimensional Quantum Group in the Liouville theory are briefly presented. The formalism based on non-local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d.     

Vacuum structure of the Coulomb gas in two dimensions

We study the plasma phase of the two-dimensional Coulomb gas in the small density limit. The analysis is done using the correspondence of the Coulomb gas with the 1 + 1 sine-Gordon model, which has been exactly solved by the quantum inverse method. We construct the correct vacuum of the field theory, improving the former results. We obtain exact results for the Coulomb gas, which confirm the previous perturbative calculations.     

Current algebra and conformal discrete series

We describe a large class of two-dimensional conformal field theories based on a current algebra construction of Virasoro representations due to Goddard, Kent, and Olive. The basic tool is a generalization of the Feigin-Fuchs representation. All the theories are organized by chiral algebras, the simplest examples being the Virasoro and super-Virasoro algebras.     

Numerical renormalization group for continuum one-dimensional systems

We present a renormalization group (RG) procedure which works naturally on a wide class of interacting one-dimension models based on perturbed (possibly strongly) continuum conformal and integrable models. This procedure integrates Wilson's numerical renormalization group with Zamolodchikov's truncated conformal spectrum approach. The key to the method is that such theories provide a set of completely understood eigenstates for which matrix elements can be exactly computed. In this procedure the RG flow of physical observables can be studied both numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the presence of a magnetic field.     

The O(n) Model on the Annulus

We use Coulomb gas methods to derive an explicit form for the scaling limit of the partition function of the critical O(n) model on an annulus, with free boundary conditions, as a function of its modulus. This correctly takes into account the magnetic charge asymmetry and the decoupling of the null states. It agrees with an earlier conjecture based on Bethe ansatz and quantum group symmetry, and with all known results for special values of n. It gives new formulae for percolation (the probability that a cluster connects the two opposite boundaries) and for self-avoiding loops (the partition function for a single loop wrapping non-trivially around the annulus.) The limit n→0 also gives explicit examples of partition functions in logarithmic conformal field theory.