Generally covariant vs. gauge structure for conformal field theories

We introduce the natural lift of spacetime diffeomorphisms for conformal gravity and discuss the physical equivalence between the natural and gauge natural structure of the theory. Accordingly, we argue that conformal transformations must be introduced as gauge transformations (affecting fields but not spacetime point) and then discuss special structures implied by the splitting of the conformal group.     

Some aspects of conformal field theories on the plane and higher genus Riemann surfaces

We review some aspects of conformal field theories on the plane as well as on higher genus Riemann surfaces.     

Semi-infinite cohomology in conformal field theory and 2D gravity

We discuss various techniques for computing the semi-infinite cohomology of highest weight modules which arise in the BRST quantization of two dimensional field theories. In particular, we concentrate on two such theories - the G/H coset models and 2D gravity coupled to c ≤ 1 conformal matter.     

Conformally invariant field theories in two dimensions and corresponding statistical mechanical models

Belavin, Zamlodochikov and Polyakov have recently proposed a class of conformally invariant field theories in two dimension with exactly determined rational critical indices. We establish a tentative identification of a subset of these theories in terms of the O(n) model and theq-state Potts model in 2-dimensions for appropriaten andq. The results of this work were reported in the conference on “Structural Similarities in Exactly Solved Models” at I.T.P. Santa Barbara, August 1984.     

Arithmetic of Calabi–Yau varieties and rational conformal field theory

It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi–Yau manifolds and the underlying conformal field theory. Specifically, it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse–Weil L-function determined by Artin’s congruent zeta function of the algebraic variety. In this context, a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.     

Boundary conformal field theory and tunneling of edge quasiparticles in non-Abelian topological states

We explain how (perturbed) boundary conformal field theory allows us to understand the tunneling of edge quasiparticles in non-Abelian topological states. The coupling between a bulk non-Abelian quasiparticle and the edge is due to resonant tunneling to a zero mode on the quasiparticle, which causes the zero mode to hybridize with the edge. This can be reformulated as the flow from one conformally invariant boundary condition to another in an associated critical statistical mechanical model. Tunneling from one edge to another at a point contact can split the system in two, either partially or completely. This can be reformulated in the critical statistical mechanical model as the flow from one type of defect line to another. We illustrate these two phenomena in detail in the context of the ν=5/2 quantum Hall state and the critical Ising model. We briefly discuss the case of Fibonacci anyons and conclude by explaining the general formulation and its physical interpretation.     

An exact solution of the renormalization-group equations for the mean field theory of stable and metastable states

An exact solution of the renormalization-group equations corresponding to the mean field theory of stable and metastable states is given which yields the correct free energies for these states. An unusual feature of this solution is that the renormalized Hamiltonian in the two-phase region becomes a multivalued function of the order parameter for all values of the length rescaling parameter beyond a certain critical value. This is closely related to the multivaluedness of the free energy as a function of magnetic field which characterizes the classical theory of metastable and unstable states. As a consequence of this multivaluedness, the trajectory flow in the space of coupling constants exhibits unusual bifurcation. This leads to difficulties in evaluating the metastable and unstable free energies by a trajectory integral of the spin-independent term, which can be resolved by an extension of the standard formalism.This work was supported by NSF grant #550-346-01 (JDG) and a U.S.-Japan Cooperative Science grant (KK and JDG).     

Parametric smoothness and self-scaling of the statistical properties of a minimal climate model: What beyond the mean field theories?

A quasi-geostrophic intermediate complexity model of the mid-latitude atmospheric circulation is considered, featuring simplified baroclinic conversion and barotropic convergence processes. The model undergoes baroclinic forcing towards a given latitudinal temperature profile controlled by the forced equator-to-pole temperature difference T_E. As T_E increases, a transition takes place from a stationary regime-Hadley equilibrium-to a periodic regime, and eventually to a chaotic regime where evolution takes place on a strange attractor. The attractor dimension, metric entropy, and bounding box volume in phase space have a smooth dependence on T_E, which results in power-law scaling properties. Power-law scalings with respect to T_E are detected also for the statistical properties of global physical observables — the total energy of the system and the averaged zonal wind. The scaling laws, which constitute the main novel result of the present work, can be thought to result from the presence of a statistical process of baroclinic adjustment, which tends to decrease the equator-to-pole temperature difference and determines the properties of the attractor of the system. The self-similarity could be of great help in setting up a theory for the overall statistical properties of the general circulation of the atmosphere and in guiding-on a heuristic basis-both data analysis and realistic simulations, going beyond the unsatisfactory mean field theories and brute force approaches. A leading example for this would be the possibility of estimating the sensitivity of the output of the system with respect to changes in the parameters.     

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502–4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré–Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré–Birkhoff normal forms for these systems up to and including terms of , where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of .     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

The rank-size scaling law and entropy-maximizing principle

The rank-size regularity known as Zipf’s law is one of the scaling laws and is frequently observed in the natural living world and social institutions. Many scientists have tried to derive the rank-size scaling relation through entropy-maximizing methods, but they have not been entirely successful. By introducing a pivotal constraint condition, I present here a set of new derivations based on the self-similar hierarchy of cities. First, I derive a pair of exponent laws by postulating local entropy maximizing. From the two exponential laws follows a general hierarchical scaling law, which implies the general form of Zipf’s law. Second, I derive a special hierarchical scaling law with the exponent equal to 1 by postulating global entropy maximizing, and this implies the pure form of Zipf’s law. The rank-size scaling law has proven to be one of the special cases of the hierarchical scaling law, and the derivation suggests a certain scaling range with the first or the last data point as an outlier. The entropy maximization of social systems differs from the notion of entropy increase in thermodynamics. For urban systems, entropy maximizing suggests the greatest equilibrium between equity for parts/individuals and efficiency of the whole.     

Isotropisation of flat homogeneous Bianchi type <Emphasis Type="Italic">I</Emphasis> model with a non minimally coupled and massive scalar field

In previous works, we studied the isotropisation of some Bianchi class A models with a minimally coupled scalar field. In this paper we extend these results, in the special case of a Bianchi type I model, to a non minimally coupled scalar field. The Universe isotropisation for the Brans-Dicke and low energy string theories are studied.