On the fixed points and invariants of Dykhne transformations and the stability of the solutions of the problem of the effective conductivity of randomly inhomogeneous two-phase media

The fixed points and invariants of Dykhne transformations are determined. It is established that they correspond to the exact solutions and duality relations for the effective characteristics of an inhomogeneous medium. The stability of the exact solutions for the effective conductivity, which are fixed points of the Dykhne transformations, is studied by bifurcation-theory methods, and a classification of these fixed points by stability type is given. It is shown that the effective conductivity tensor of a two-phase medium in magnetic and ac electric fields for certain parameters of the medium can be an unstable point of the “saddle” type.     

Characterizing symmetry breaking patterns in a lattice by dual degrees of freedom

A duality transformation in quantum field theory is usually established first through partition functions. It is always important to explore the dual relations between various correlation functions in the transformation. Here, we explore such a dual relation to study quantum phases and phase transitions in an extended boson Hubbard model at 1/3 (2/3) filling on a triangular lattice. We develop systematically a simple and effective way to use the vortex degrees of freedom on dual lattices to characterize both the density wave and valence bond symmetry breaking patterns of the boson insulating states in the direct lattices. In addition to a checkerboard charge density wave (X-CDW) and a stripe CDW, we find a novel CDW-VBS phase which has both local CDW and local valence bond solid (VBS) orders. Implications for Quantum Monte Carlo simulations are addressed. The possible experimental realizations of cold atoms loaded on optical lattices are discussed.     

Quantum E(2) group and its Pontryagin dual

The quantum deformation of the group of motions of the plane and its Pontryagin dual are described in detail. It is shown that the Pontryagin dual is a quantum deformation of the group of transformations of the plane generated by translations and dilations. An explicit expression for the unitary bicharacter describing the Pontryagin duality is found. The Heisenberg commutation relations are written down.Supported in equal parts by the grant of the Ministry of Education of Poland and by Schweizerischer Nationalfonds.     

The bond-algebraic approach to dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ?₂ Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.     

Broken supersymmetry in the matrix model on a circle

We consider the discretization of aD=2 surface using polygons. We map the surface onto superspace and integrate over surfaces of arbitrary genus, obtaining a discretized version of the Green-Schwarz string inD=1. Taking an unusual critical limit of the supersymmetric matrix model involved, we construct exact solutions, to all perturbative orders, for the discretized superstring in one dimension, both when the target space is a real line and when the theory is represented in terms of matrix variables on a circle of finite radius. We comment on the behavior of the compactfied perturbative expansion under duality transformations.BITNET: BELLUCCI at IRMLNF     

An indirect approach for quantum-mechanical eigenproblems: duality transforms

We suggest an indirect approach for solving eigenproblems in quantum mechanics. Unlike the usual method, this method is not a technique for solving differential equations. There exists a duality among potentials in quantum mechanics. The first example is the Newton–Hooke duality revealed by Newton in Principia. Potentials that are dual to each other form a duality family consisting of infinite numbers of family members. If one potential in a duality family is solved, the solutions of all other potentials in the family can be obtained by duality transforms. Instead of directly solving the eigenequation of a given potential, we turn to solve one of its dual potentials which is easier to solve. The solution of the given potential can then be obtained from the solution of this dual potential by a duality transform. The approach is as follows: first to construct the duality family of the given potential, then to find a dual potential which is easier to solve in the family and solve it, and finally to obtain the solution of the given potential by the duality transform. In this paper, as examples, we solve exact solutions for general polynomial potentials.     

Su (2) Poisson–Lie T Duality

Poisson–Lie target space duality is a framework where duality transformations are properly defined. In this Letter, we investigate the dual pair of -models defined by the double SO(3,1) in the Iwasawa decomposition.     

Gerbes and Duality

We describe a global approach to the study of duality transformations between antisymmetric fields with transitions and argue that the natural geometrical setting for the approach is that of gerbes; these objects are mathematical constructions generalizing U(1) bundles and are similarly classified by quantized charges. We address the duality maps in terms of the potentials rather than on their field strengths and show the quantum equivalence between dual theories which in turn allows a rigorous proof of a generalized Dirac quantization condition on the couplings. Our approach needs the introduction of an auxiliary form satisfying a global constraint which in the case of 1-form potentials coincides with the quantization of the magnetic flux. We apply our global approach to refine the proof of the duality equivalence between the d=11 supermembrane and d=10 IIA Dirichlet supermembrane.     

Duality Relations for Non-Ohmic Composites, with Applications to Behavior near Percolation

Keller, Dykhne, and others have exploited duality to derive exact results for the effective behavior of two-dimensional Ohmic composites. This paper addresses similar issues in the non-Ohmic context. We focus primarily on three different types of nonlinearity: (a) the weakly nonlinear regime; (b) power-law behavior; and (c) dielectric breakdown. We first make the consequences of duality explicit in each setting. Then we draw conclusions concerning the critical exponents and scaling functions of dual pairs of random non-Ohmic composites near a percolation threshold. These results generalize, unify, and simplify relations previously derived for nonlinear resistor networks. We also discuss some self-dual nonlinear composites. Our treatment is elementary and self-contained; however, we also link it with the more abstract mathematical discussions of duality by Jikov and Kozlov.     

Spectral ansatz in quantum electrodynamics: Spectral functions and gauge transformations

Using the ansatz of Delbourgo and Salam for the vertex function in quantum electrodynamics, we find approximations to the spectral functions of the electron propagator for covariant gauges. The consistency with the integral relations for the change of the exact spectral functions under covariant gauge transformations is investigated. The approximated spectral functions appear not to be gauge covariant in general.     

Dual properties of inclusive spectra

The dual properties of the inclusive reaction π^?p → pX are studied over a wide energy range by exploiting the scaling behaviour. Semi-local duality is found to be well satisfied. An energy-dependent triple-Regge analysis reveals a strong triple-pomeron coupling. There is some evidence of an abnormal component in which diffractively produced resonances are dual to pomeron exchange. Combining duality with factorization leads to relations between production cross sections of meson and baryon resonances in πp and pp collisions, which are compatible with existing experimental data.     

Colored extension of GL q(2) and its dual algebra

We address the problem of duality between the colored extension of the quantized algebra of functions on a group and that of its quantized universal enveloping algebra, i.e., its dual. In particular, we derive explicitly the algebra dual to the colored extension of GL _q(2) using the colored RLL relations and exhibit its Hopf structure. This leads to a colored generalization of the R-matrix procedure to construct a bicovariant differential calculus on the colored version of GL _q(2). In addition, we also propose a colored generalization of the geometric approach to quantum group duality given by Sudbery and Dobrev.     

O(d,d)-invariance in Inhomogeneous String Cosmologies with Perfect Fluid

In the first part of the present paper, we showthat O(d,d)-invariance usually known in a homogeneouscosmological background written in terms of proper timecan be extended to backgrounds depending on one or several coordinates [which may be anyspace-like or time-like coordinate(s)]. In all cases,the presence of a perfect fluid is taken into accountand the equivalent duality transformation in Einstein frame is explicitly given. In the second part,we present several concrete applications to somefour-dimensional metrics, including inhomogeneous ones,which illustrate the different duality transformations discussed in the first part. Note that most ofthe dual solutions given here do not seem to be known inthe literature.     

On a relation between massive Yang-Mills theories and dual string models

The relations between mass terms in Yang-Mills theories, projective representations of the group of gauge transformations, boundary conditions on vector potentials and Schwinger terms in local charge algebra commutation relations are discussed. The commutation relations (with Schwinger terms) are similar to the current algebra commutation relations of the SU(N) extended dual string model.     

Perturbative prepotential and monodromies in N = 2 heterotic superstring

We discuss the prepotential describing the effective field theory of N = 2 heterotic superstring models. At the one loop-level the prepotential develops logarithmic singularities due to the appearance of charged massless states at particular surfaces in the moduli space of vector multiplets. These singularities modify the classical duality symmetry group which now becomes a representation of the fundamental group of the moduli space minus the singular surfaces. For the simplest two-moduli case, this fundamental group turns out to be a certain braid group and we determine the resulting full duality transformations of the prepotential, which are exact in perturbation theory.     

Evidence for nonperturbative string symmetries

String theory appears to admit a group of discrete field transformations — calledS dualities — as exact nonperturbative quantum symmetries. Mathematically, they are rather analogous to the better-knownT duality symmetries, which hold perturbatively. In this Letter the evidence forS duality is reviewed and some speculations are presented.     

Path-integral derivation of quantum duality in nonlinear sigma-models

We show that a previously derived shift in the dilaton field, which necessarily augments the classical effects of duality transformation on the geometry of a nonlinear sigma-model if conformal invariance is to be preserved at the one-loop level, can be extended without change to the case of sigma-models with Wess-Zumino-Witten term (torsion) before and after duality. We also construct a path-integral implementation of the duality transformation, and discover the origin of the dilaton shift in a functional determinant resulting from the elimination of the first-order field. The path-integral formulation in principle allows a derivation of “quantum” duality transformations which preserve conformal invariance to all orders in α', the string tension parameter.     

Renormalization and symmetry conditions in supersymmetric QED

For supersymmetric gauge theories a consistent regularization scheme that preserves supersymmetry and gauge invariance is not known. In this article we tackle this problem for supersymmetric QED within the framework of algebraic renormalization. For practical calculations, a non-invariant regularization scheme may be used together with counterterms from all power-counting renormalizable interactions. From the Slavnov–Taylor identity, expressing gauge invariance, supersymmetry and translational invariance, simple symmetry conditions are derived that are important in a twofold respect: they establish exact relations between physical quantities that are valid to all orders, and they provide a powerful tool for the practical determination of the counterterms. We perform concrete one-loop calculations in dimensional regularization, where supersymmetry is spoiled at the regularized level, and show how the counterterms necessary to restore supersymmetry can be read off easily. In addition, a specific example is given how the supersymmetry transformations in one-loop order are modified by non-local terms. Received: 23 July 1999 / Published online: 14 October 1999     

Conductivity of a “colored” plane

The conductivity of a “colored” plane, i.e., a plane divided into domains differing in conductivity, is calculated. The exact relation between the effective conductivities of the cited and dual (with inverse conductivities) systems is derived for the isotropic case (i.e., the effective conductivity tensor is proportional to the unit matrix). The conductivity of two-colored systems such as a “chessboard” or triangular lattice is exactly calculated to give σ=(σ₁σ₂)^1/2. The particular case of a “hexagon, ” as well as the duality relations for anisotropic systems and for a system in a magnetic field are discussed.