A field-theoretic model for Hodge theory

We demonstrate that the four-dimensional (4D) ((3+1)-dimensional) free Abelian 2-form gauge theory presents a tractable field-theoretical model for the Hodge theory where the well-defined symmetry transformations correspond to the de Rham cohomological operators of differential geometry. The conserved charges, corresponding to the above continuous symmetry transformations, obey an algebra that is reminiscent of the algebra obeyed by the cohomological operators. The discrete symmetry transformation of the theory represents the realization of the Hodge duality operation that exists in the relationship between the exterior and co-exterior derivatives of differential geometry. Thus, we provide the realizations of all the mathematical quantities, associated with the de Rham cohomological operators, in the language of the symmetries of the present 4D free Abelian 2-form gauge theory.     

Basic Brackets of a 2D Model for the Hodge Theory Without its Canonical Conjugate Momenta

We deduce the canonical brackets for a two (1+1)-dimensional (2D) free Abelian 1-form gauge theory by exploiting the beauty and strength of the continuous symmetries of a Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density that respects, in totality, six continuous symmetries. These symmetries entail upon this model to become a field theoretic example of Hodge theory. Taken together, these symmetries enforce the existence of exactly the same canonical brackets amongst the creation and annihilation operators that are found to exist within the standard canonical quantization scheme. These creation and annihilation operators appear in the normal mode expansion of the basic fields of this theory. In other words, we provide an alternative to the canonical method of quantization for our present model of Hodge theory where the continuous internal symmetries play a decisive role. We conjecture that our method of quantization is valid for a class of field theories that are tractable physical examples for the Hodge theory. This statement is true in any arbitrary dimension of spacetime.     

Dualities and the phase diagram of the p-clock model

A new “bond-algebraic” approach to duality transformations provides a very powerful technique to analyze elementary excitations in the classical two-dimensional XY and p-clock models. By combining duality and Peierls arguments, we establish the existence of non-Abelian symmetries, the phase structure, and transitions of these models, unveil the nature of their topological excitations, and explicitly show that a continuous U(1) symmetry emerges when p?5. This latter symmetry is associated with the appearance of discrete vortices and Berezinskii-Kosterlitz-Thouless-type transitions. We derive a correlation inequality to prove that the intermediate phase, appearing for p?5, is critical (massless) with decaying power-law correlations.     

Discrete Riemann Surfaces and the Ising Model

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality. Received: 23 May 2000/ Accepted: 21 November 2000     

Spinor-vector duality in heterotic string vacua

Classification of the N=1 space–time supersymmetric fermionic Z₂×Z₂ heterotic-string vacua with symmetric internal shifts, revealed a novel spinor-vector duality symmetry over the entire space of vacua, where the S_t↔V duality interchanges the spinor plus anti-spinor representations with vector representations. In this paper we demonstrate that the spinor-vector duality exists also in fermionic Z₂ heterotic string models, which preserve N=2 space–time supersymmetry. In this case the interchange is between spinorial and vectorial representations of the unbroken SO(12) GUT symmetry. We provide a general algebraic proof for the existence of the S_t↔V duality map. We present a novel basis to generate the free fermionic models in which the ten-dimensional gauge degrees of freedom are grouped into four groups of four, each generating an SO(8) modular block. In the new basis the GUT symmetries are produced by generators arising from the trivial and non-trivial sectors, and due to the triality property of the SO(8) representations. Thus, while in the new basis the appearance of GUT symmetries is more cumbersome, it may be more instrumental in revealing the duality symmetries that underly the string vacua.     

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.     

Symmetry properties and bi-Hamiltonian structure of the Toda lattice

We carry out a systematic analysis of the Toda lattice equations developing a method which extends the symmetry approach formalism to discrete one-dimensional systems. We find a hereditary operator which admits a symplectic-implectic factorization. As a consequence of this property, we derive the Hamiltonian and the bi-Hamiltonian structure, together with the constants of motion and a set of infinitely-many commuting Lie-Bäcklund symmetries of the Toda chain.     

Duality symmetry group of two dimensional heterotic string theory

The equations of motion of the massless sector of the two dimensional string theory, obtained by compactifying the heterotic string theory on an eight dimensional torus, is known to have an affine o(8,24) symmetry algebra generating an O(8,24) loop group. In this paper we study how various known discrete S- and T- duality symmetries of the theory are embedded in this loop group. This allows us to identify the generators of the discrete duality symmetry group of the two dimensional string theory.     

A symmetry principle for topological quantum order

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. These symmetries may be actual symmetries of the Hamiltonian characterizing the system, or emergent symmetries. To this end, we introduce the concept of low-dimensional Gauge-like symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU(N) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev’s Toric code model and Wen’s plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We exactly recast some known non-local string correlators in terms of local correlation functions. We discuss relations to problems in graph theory.     

Duality and Hidden Symmetries in Interacting Particle Systems

In the context of Markov processes, both in discrete and continuous setting, we show a general relation between duality functions and symmetries of the generator. If the generator can be written in the form of a Hamiltonian of a quantum spin system, then the “hidden” symmetries are easily derived. We illustrate our approach in processes of symmetric exclusion type, in which the symmetry is of SU(2) type, as well as for the Kipnis-Marchioro-Presutti (KMP) model for which we unveil its SU(1,1) symmetry. The KMP model is in turn an instantaneous thermalization limit of the energy process associated to a large family of models of interacting diffusions, which we call Brownian energy process (BEP) and which all possess the SU(1,1) symmetry. We treat in details the case where the system is in contact with reservoirs and the dual process becomes absorbing.     

Operator Systems from Discrete Groups

We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group ${\mathbb{F}_n}$ on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group ${\mathbb{Z}_2}$ . We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.     

A Remark on Schwarz's Topological Field Theory

The standard evaluation of the partition function Z of Schwarz's topological field theory results in the Ray–Singer analytic torsion. Here we present an alternative evaluation which results in Z=1. Mathematically, this amounts to a novel perspective on analytic torsion: it can be formally written as a ratio of volumes of spaces of differential forms which is formally equal to 1 by Hodge duality. An analogous result for Reidemeister combinatorial torsion is also obtained.     

CMS: Jet/b,Jet/<Emphasis Type="Italic">τ</Emphasis>, Missing E<Subscript>T</Subscript>

It is often claimed [1] that the (Hodge type) duality operation is defined only in even dimensional spacetimes and that self-duality is further restricted to twice-odd dimensional spacetime theories. The purpose of this paper is to extend the notion of both duality symmetry as well as self-duality.By considering tensorial doublets, we introduce a novel well-defined notion of self-duality based on a duality Hodge-type operation in arbitrary dimension and for any rank of these tensors. Thus, a generalized Self-Dual Action is defined such that equations of motion are the claimed generalized self-duality relations. We observe in addition, that taking the proper limit on the parameters of this action, it always provides us with a master-action, which interpolates models well-studied in physics; by considering a particular limit, we find an action which describes an interesting type of relation, referred to as semi-self-duality, which results to be the parent action between Maxwell-type actions.Finally, we apply these ideas to construct manifest Hodge-type self-dual solutions in a (2+1)-dimensional version of the Maxwell’s theory.     

An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

An integrable discrete system obtained by the algebraization of the difference operator is studied. The system is named discrete generalized nonlinear Schrödinger (GNLS) equation, which can be reduced to classical discrete nonlinear Schrödinger (NLS) equation. Furthermore, all of the linear reductions for the discrete GNLS equation are given through the theory of circulant matrices and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones.     

Chan–Paton soliton gauge states of the compactified open string

We study the mechanism of the enhanced gauge symmetry of the bosonic open string compactified on a torus by analyzing the zero-norm soliton (non-zero winding of the Wilson line) gauge states in the spectrum. Unlike the closed string case, we find that the soliton gauge state exists only at massive levels. These soliton gauge states correspond to the existence of enhanced massive gauge symmetries with transformation parameters containing both Einstein and Yang–Mills indices. In the T-dual picture, these symmetries exist only at some discrete values of compactified radii when N D-branes are coincident. Received: 14 May 1999 / Published online: 17 March 2000     

Symmetries and exact solutions of discrete nonconservative systems

Based on the property of the discrete model entirely inheriting the symmetry of the continuous system,we present a method to construct exact solutions with continuous groups of transformations in discrete nonconservative systems.The Noether's identity of the discrete nonconservative system is obtained.The symmetric discrete Lagrangian and symmetric discrete nonconservative forces are defined for the system.Generalized quasi-extremal equations of discrete nonconservative systems are presented.Discrete conserved quantities are derived with symmetries associated with the continuous system.We have also found that the existence of the one-parameter symmetry group provides a reduction to a conserved quantity;but the existence of a two-parameter symmetry group makes it possible to obtain an exact solution for a discrete nonconservative system.Several examples are discussed to illustrate these results.     

The Three-Component Defocusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions

We present a rigorous theory of the inverse scattering transform (IST) for the three-component defocusing nonlinear Schrödinger (NLS) equation with initial conditions approaching constant values with the same amplitude as \({x\to\pm\infty}\). The theory combines and extends to a problem with non-zero boundary conditions three fundamental ideas: (i) the tensor approach used by Beals, Deift and Tomei for the n-th order scattering problem, (ii) the triangular decompositions of the scattering matrix used by Novikov, Manakov, Pitaevski and Zakharov for the N-wave interaction equations, and (iii) a generalization of the cross product via the Hodge star duality, which, to the best of our knowledge, is used in the context of the IST for the first time in this work. The combination of the first two ideas allows us to rigorously obtain a fundamental set of analytic eigenfunctions. The third idea allows us to establish the symmetries of the eigenfunctions and scattering data. The results are used to characterize the discrete spectrum and to obtain exact soliton solutions, which describe generalizations of the so-called dark-bright solitons of the two-component NLS equation.     

An Interacting Gauge Field Theoretic Model for Hodge Theory: Basic Canonical Brackets

We derive the basic canonical brackets amongst the creation and annihilation operators for a two （1 ＋ 1）- dimensional （2D） gauge field theoretic model of an interacting Hodge theory where a U（1） gauge field （Aμ） is coupled with the fermionic Dirac fields （ψ and ψ）. In this derivation, we exploit the spin-statistics theorem, normal ordering and the strength of the underlying six infinitesimal continuous symmetries （and the concept of their generators） that are present in the theory. We do not use the definition of the canonical conjugate momenta （corresponding to the basic fields of the theory） anywhere in our whole discussion. Thus, we conjecture that our present approach provides an alternative to the canonical method of quantization for a class of gauge field theories that are physical examples of Hodge theory where the continuous symmetries （and corresponding generators） provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level.     

两个力学量算符具有共同本征态系的条件

量子力学是人们了解微观世界的一门重要的课程.在量子力学中,由于任何实物体都具有波粒二象性,必须用算符来表示力学量,因此算符理论显得尤为重要.在研究多个算符时,有一个非常重要的定理:两个力学量算符能够具有共同本征函数系的充分必要条件是这两个个力学量算符能够互相对易.本文分析了在使用该定理时可能存在的一些问题,并对这些问题进行了澄清.