Duality and Self-Duality (Energy Reflection Symmetry) of Quasi-Exactly Solvable Periodic Potentials

A class of spectral problems with a hidden Lie-algebraic structure is considered. We define a duality transformation which maps the spectrum of one quasi-exactly solvable (QES) periodic potential to that of another QES periodic potential. The self-dual point of this transformation corresponds to the energy-reflection symmetry found previously for certain QES systems. The duality transformation interchanges bands at the bottom (top) of the spectrum of one potential with gaps at the top (bottom) of the spectrum of the other, dual, potential. Thus, the duality transformation provides an exact mapping between the weak coupling (perturbative) and semiclassical (nonperturbative) sectors.     

Spin-chirality duality in a spin ladder with four-spin cyclic exchange

Effect of four-spin cyclic exchange on magnetism is studied in the two-leg S=1/2 ladder. We develop an exact spin-chirality duality transformation, under which the system is self-dual when the four-spin exchange J4 is half of the two-spin exchange. Using the density-matrix renormalization-group method and the duality relation, we find that the four-spin exchange makes the vector-chirality correlation dominant. A "chirality short-range resonating-valence-bond" phase is identified for the first time at large J4.     

Another Proof of the Equivalence Between the Duality Symmetric and Maxwell's Action

We show that the duality symmetric action is equivalent to the Maxwell's action at the quantum level by means of the Dirac's method, and find out that the set of Dirac's brackets on the full phase space is invariant under the dual transformation.     

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.     

Four-dimensional singular oscillator and generalized MIC-Kepler system

It is shown that the generalized MIC-Kepler system and four-dimensional singular oscillator are dual to each other and the duality transformation is the generalized version of the Kustaanheimo-Stiefel transformation. The text was submitted by the authors in English.     

On the uniqueness of the Invariant Equilibrium State and surface tension

Symmetric Equilibrium States and their properties under duality transformation are investigated. Necessary and sufficient conditions are derived for equilibrium states to be transformed into equilibrium states by duality. It is shown that ferromagnetic systems satisfying those conditions have correlation functions bounded by those corresponding to the (+) and free boundary conditions. It is then proved than any Invariant Equilibrium State of a ferromagnetic system is transformed into an equilibrium state by duality and is thus unique if the states defined by the (+), and free boundary conditions coincide on the symmetric algebra. The existence of surface tension between two pure phases is established.     

Remarks on duality of 1 dimensional quantum spin models

We present some results on duality maps and ground states of 1 dimensional quantum spin models. We also give some applications to Kramers Wannier duality and the nonlocal transformation that Kennedy and Tasaki discovered in their study of Haldane phase of quantum antiferromagnetic spin models.     

Mean field methods for strings and monopoles in abelian lattice gauge theories

Selfconsistent approximations, which join the Bethe-Peierls method and duality transformation, are applied to disorder parameters related to strings (domain boundaries) and monopoles in Z(N) and U(1) lattice gauge theories. The two-phase and the three-phase diagrams are reproduced in three and four dimensions. Nice results are obtained for the internal energy and the monopole charge density. A formulation for gauge theories of the selfconsistent Monte Carlo method is introduced in order to improve these approximations.     

Finite-elements method: Conduction and liberation of joule heat in regular structures

A method is proposed and used to calculate the effective conductivity of certain regular matrix structures. The distributions of current and isopotential lines are constructed and the duality transformation is examined. A method is devised for calculating the heat evolved when current flows in the given structures. Topograms are constructed to describe relative specific dissipated power and its spectral functions, which might serve as a criterion of the uniformity of the heat-hource distribution. Ural Polytechnic Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 17–23, May, 1996     

Electromagnetic Duality in General Relativity

By resolving the Riemann curvature relative to a unit timelike vector into electric and magnetic parts, we consider duality relations analogous to those in electromagnetic theory. It turns out that the duality transformation implies the Einstein vacuum equation without the cosmological term. The vacuum equation is invariant under interchange of active and passive electric parts, giving rise to the same vacuum solutions but with the opposite sign for the gravitational constant. Further, by modifying the equation it is possible to construct interesting dual solutions to vacuum as well as to flat spacetimes.     

Critical line of the triangular potts model with two- and three-site interactions: An alternate derivation

A double star-triangle relation which is a generalization of the usual single star-triangle relation is introduced. Together with a duality transformation it is possible to rederive the critical line of the Potts model with two- and three-site interactions on alternate triangles.     

Duality transformations in electrodynamics

Electrodynamics admitting a duality transformation group is considered. For such an electrodynamics an extension of the classical Rainich-Misner-Wheeler theory is presented.On leave of absence from the University of Warsaw, Warsaw, Poland.     

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.     

A Groupoid Approach to Noncommutative T-Duality

Topological T-duality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dual-torus bundle. We give a new geometric construction of T-dualization, which allows the duality to be extended in the following two directions. First, bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized. Second, bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated, though in this case the base space of the bundles is best viewed as a topological stack. Some methods developed for the construction may be of independent interest. These are a Pontryagin type duality that interchanges commutative principal bundles with gerbes, a nonabelian Takai type duality for groupoids, and the computation of certain equivariant Brauer groups. The research reported here was supported in part by National Science Foundation grants DMS-0703718 and DMS-0611653.     

Twisted Duality Symmetry and Integrability of Hybrid Superstring on AdS2×S2

In this paper, we show that there exists a twisted duality symmetry between the Maurer-Cartan equations and the equations of motion in the hybrid formalism for the type IIB superstring in an AdS₂×S² background with Ramond-Ramond flux. As a result, from the twisted duality transformation, we construct the Lax connection with the spectral parameter, which ensures the integrability of the system.     

Local duality, chiral supersymmetry and fermion-like formulation of non-Abelian pure gauge fields

This work generalizes the fermion-like formulation of the Maxwell theory to the non-Abelian Yang–Mills theory without matter fields. This is a new representation of the Lie algebra valued electric and magnetic fields. The resulting equations of motion are invariant under the chiral transformation. In this formulation, duality is a kind of chirality. We may also define local duality transformations in terms of space-time dependent parameters. There is an N=1 supersymmetry for the Dirac-like operator in this representation. Received: 19 June 2000 / Revised version: 10 October 2000 / Published online: 15 March 2001     

Spectral duality in integrable systems from AGT conjecture

We describe relationships between integrable systems with N degrees of freedom arising from the Alday-Gaiotto-Tachikawa conjecture. Namely, we prove the equivalence (spectral duality) between the N-cite Heisenberg spin chain and a reduced gl _N Gaudin model both at classical and quantum level. The former one appears on the gauge theory side of the Alday-Gaiotto-Tachikawa relation in the Nekrasov-Shatashvili (and further the Seiberg-Witten) limit while the latter one is natural on the CFT side. At the classical level, the duality transformation relates the Seiberg-Witten differentials and spectral curves via a bispectral involution. The quantum duality extends this to the equivalence of the corresponding Baxter-Schrödinger equations (quantum spectral curves). This equivalence generalizes both the spectral self-duality between the 2 × 2 and N × N representations of the Toda chain and the famous Adams-Harnad-Hurtubise duality.     

The variational principle in dual relativistic electrodynamics

Relativistic electrodynamics is supplemented by the principle of equivalence of fields and currents, a generalization of the Love principle which is widely used in radiophysics. This is derived by requiring the invariance of the Lagrangian both to Lorentz transformations and to the duality transformation obtained from the generalized Larmor-Pistol'kors duality principle.     

Kagome晶格上混合自旋模型的自由费密近似解

本文考虑Kagome晶格上的混合自旋模型,运用decimation和对偶变换,证明它等价于一个八顶角模型。由八顶角模型的自由费密近似解给出了该模型的自由能及临界条件的近似表达式。 关键词：