Constraints on a generally covariant four dimensional sigma model

A four dimensional sigma model coupled to the tensor metric field is considered in terms of harmonic maps. An upper bound on the dimensionless coupling appearing in the model is found. For Einstein spaces this coupling is expressed by geometrical quantities.     

Weak charge quantization as an instanton of the interacting sigma model

Coulomb blockade in a quantum dot attached to a diffusive conductor is considered in the framework of the nonlinear sigma model. It is shown that the weak charge quantization on the dot is associated with instanton configurations of the Q field in the conductor. The instantons have a finite action and are replica nonsymmetric. It is argued that such instantons may play a role in the transition regime to the interacting insulator.     

Infinite-dimensional geometry and generally covariant quantum closed string-field theory

A version, based on infinite-dimensional geometry, of the generally covariant closed string-field theory is proposed.     

Distribution of instanton sizes in a simplified instanton gas model

We investigate the distribution of instanton sizes in the framework of a simplified model for ensembles of instantons. This model takes into account the non-diluteness of instantons. The infrared problem for the integration over instanton sizes is dealt with in a self-consistent manner by approximating instanton interactions by a repulsive hard core potential. This leads to a dynamical suppression of large instantons. The characteristic features of the instanton size distribution are studied by means of analytic and Monte Carlo methods. In one dimension exact results can be derived. In any dimension we find a power law behaviour for small sizes, consistent with the semi-classical results. At large instanton sizes the distribution decays exponentially. The results are compared with those from lattice simulations. Received: 15 May 2000 / Revised version: 17 July 2000 / Published online: 25 September 2000     

Exact solutions in gravity with a sigma model source

We consider a D-dimensional model of gravity with non-linear “scalar fields” as a matter source. The model is defined on the product manifold M, which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n − 1 factor spaces are Ricci-flat, e.g. when one space M ₁ of dimension d ₁ > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically symmetric solutions is studied and a restricted version of “no-hair theorem” for black holes is proved. For the case d ₁ = 2 a subclass of latent soliton solutions is singled out.     

Multiple meron solutions of the classical Yang-Mills equation

We give a construction for multiple meron-antimeron solutions to the classical SU(2) Yang-Mills field equations. The corresponding charge density is $\text{Q(x) =}\text{1}\text{2}\text{∑ ± γ(x?x}{}_{\mathrm{i}}\text{)}$, with the x_i distinct points on a line. The problem is reduced to the solution of a scalar φ⁴ equation. We obtain a model with merons connected in pairs.     

On generally covariant quantum field theory and generalized causal and dynamical structures

We give an example of a generally covariant quasilocal algebra associated with the massive free field. Maximal, two-sided ideals of this algebra are algebraic representatives of external metric fields. In some sense, this algebra may be regarded as a concrete realization of Ekstein's ideas of presymmetry in quantum field theory. Using ideas from our example and from usual algebraic quantum field theory, we discuss a generalized scheme, in which maximal ideals are viewed as algebraic representatives of dynamical equations or Lagrangians. The considered frame is no quantum gravity, but may lead to further insight into the relation between quantum theory and space-time geometry.     

Construction of instanton and monopole solutions and reciprocity

An elementary argument demonstrating the completeness of the Atiyah-Drinfeld-Hitchin-Manin construction of self-dual instanton solutions to Euclidean gauge theories is presented. The adaptation of this discussion to Nahm's construction for SU(2) monopoles is outlined. These constructions are shown to establish a reciprocity or duality between self-dual theories in zero and four dimensions and in one and three dimensions, respectively.     

A generally covariant formulation of classical electrodynamics for charges of finite extension

A generally covariant formulation of classical electrodynamics for charges of finite extension has been developed. The charges are required to maintain a prescribed “rigid” shape throughout the course of their motion. An action principle is formulated for the coupled system consisting of the charges plus electromagnetic and gravitational fields. The action principle yields a complicated set of coupled integro-differential equations for the motion and fields. A perturbation expansion in powers of the size of the charge distribution is obtained. In the limit that the size of the charge tends to zero, only a few kinematical features survive in the equations of motion. The resulting equations of motion have the DeWitt-Brehme [Ann. Phys.9 (1969), 220] form, but with additional curvature-coupling terms which were omitted by them owing to an algebraic error.     

Meron solutions of the sigma model and singular harmonic maps

We study the meron solutions of theO(3) sigma model in two-dimensions using the singular harmonic maps. We find new solutions of the topological charge (the Brouwer degree) zero having one dimensional singularities of the fold and collapse types. In general any singularity of a solution of the sigma model is locally equivalent with one of the four types of singular points of a harmonic map.     

Algebraic classification of ℂP n instanton solutions

The finite condition for two-dimensional ℂP ⁿ models is discussed noting that one can impose boundary conditions such that the domain of the field is a compact Riemann surface S _g. Holomorphic maps φ: S _g →ℂP ⁿ give finite energy solutions of the classical field equations, which are classified according to standard methods of algebraic geometry. The moduli problem is discussed in detail for S _g =ℂP ¹=S ².     

Nucleon form factors in a covariant diquark-quark model

In a model where constituent quarks and diquarks interact through quark exchange the Bethe-Salpeter equation in ladder approximation for the nucleon is solved. Quark and diquark confinement is effectively parametrized by choosing appropriately modified propagators. The coupling to external currents is implemented via non-trivial vertex functions for quarks and diquarks to ensure gauge invariance at the constituent level. Nucleon matrix elements are evaluated in a generalized impulse approximation, and electromagnetic, pionic and axial form factors are calculated.     

The Skyrme model on a four-dimensional lattice

The SU(2) Skyrme model is formulated on a four-dimensional lattice with the nearest and next-to-nearest neighbor interactions in the action. To establish a continuum limit, the phase structure is studied and the order of phase transition is determined by various methods. Coexistence of two phases and hysteresis effect are observed at the critical point, indicating a first-order phase transition in the parameter space.     

The Skyrme model on a four-dimensional lattice

The SU(2) Skyrme model is formulated on a four-dimensional lattice with the nearest and next-to-nearest neighbor interactions in the action. To establish a continuum limit, the phase structure is studied and the order of phase transition is determined by various methods. Coexistence of two phases and hysteresis effect are observed at the critical point, indicating a first-order phase transition in the parameter space.