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At least 55 records · Page 3Linked to original sources
arXiv1996

Complex structure and the construction of the $:ϕ^4_4:$ quantum field theory in four-dimensional space-time

We announce results about the nonperturbative mathematically rigorous construction of the $:\!ϕ^4_4\!:$ quantum field theory in four-dimensional space-time. The complex structure of solutions of the classical nonlinear (real-valued) wave equation and quantization are closely connected among themselves and allow to construct non-perturbatively the quantum field theory with interaction $:\!ϕ^4_4\!:$ in four-dimensional space-time. We consider vacuum averages, in particular, we construct Wightman functions and matrix elements of the scattering operator as generalized functions for finite energies. The constructed theory is obviously nontrivial.

hep-th↗
arXiv1997

Poisson Structures for Dispersionless Integrable Systems and Associated W-Algebras

In analogy to the KP theory, the second Poisson structure for the dispersionless KP hierarchy can be defined on the space of commutative pseudodifferential operators $L=p^n+\sum_{j=-\infty}^{n-1}u_j p^j$. The reduction of the Poisson structure to the symplectic submanifold $u_{n -1}=0$ gives rise to the w-algebras. In this paper, we discuss properties of this Poisson structure, its Miura transformation and reductions. We are particularly interested in the following two cases: a) L is pure polynomial in p with multiple roots and b) L has multiple poles at finite distance. The w-algebra corresponding to the case a) is defined as $w_ {[m_1,m_2, ... ,m_r]}$, where m_i means the multiplicity of roots and to the case b) is defined by $w(n,[m_1,m_2, ... ,m_r])$ where m_i is the multiplicity of poles. We prove that w(n,[m_1, m_2, ... , m_r])$-algebra is isomorphic via a transformation to $w_{[m_1,m_2, ... ,m_r]} \bigoplus w_{n+m} \bigoplus U(1) with $m=\sum m_i$. We also give the explicit free fields representations for these w-algebras.

hep-th↗
arXiv1997

Three-body Generalizations of the Sutherland Problem

The three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is shown to be exactly solvable. When written in appropriate variables, its eigenfunctions can be expressed in terms of Jack symmetric polynomials. The exact solvability of the problem is explained by a hidden $sl(3,R)$ symmetry. A generalized Sutherland three-particle problem including both two- and three-body trigonometric potentials and internal degrees of freedom is then considered. It is analyzed in terms of three first-order noncommuting differential-difference operators, which are constructed by combining SUSYQM supercharges with the elements of the dihedral group~$D_6$. Three alternative commuting operators are also introduced.

hep-th↗
arXiv1997

String-Orthogonal Polynomials, String equations, and 2-Toda symmetries

1. The 2-Toda lattice and its generic symmetries 2. A Larger class of symmetries for special initial conditions 3. Borel decomposition of Moment matrices, tau-functions and string-orthogonal polynomials 4. From string-orthogonal Polynomials to the 2-Toda lattice and the string equation 5. Virasoro constraints on two-matrix integrals

hep-th↗
arXiv1997

The Dynamics of Classical Chiral $QCD_{2}$ Currents

In this paper the dynamics of the classical chiral $QCD_{2}$ currents is studied. We describe how the dynamics of the theory can be summarized in an equation of the Lax form, thereby demonstrating the existence of an infinite set of conserved quantities. Next, the $r$ matrix of a fundamental Poisson relation is obtained and used to demonstrate that the conserved charges Poisson commute. An underlying diffeomorphism symmetry of the equations of motion which is not a symmetry of the action is used to provide a geometric interpretation for the case of gauge group SU(2). This enables us to show that the solutions to the classical equations of motion can be identified with a large class of curves, to demonstrate an auto-Bäcklund transformation and to demonstrate a non linear superposition principle. A link between the spectral problem for $QCD_{2}$ and the solution to the closed curve problem is also demonstrated. We then go on to provide a systematic inverse scattering treatment. This formalism is used to obtain the reflectionless single boundstate eigenvalue soliton solution.

hep-th↗
arXiv1997

Quantum Integrable Systems: Basic Concepts and Brief Overview

An overview of the quantum integrable systems (QIS) is presented. Basic concepts of the theory are highlighted stressing on the unifying algebraic properties, which not only helps to generate systematically the representative Lax operators of different models, but also solves the related eigenvalue problem in an almost model independent way. Difference between the approaches in the integrable ultralocal and nonultralocal quantum models are explained and the interrelation between the QIS and other subjects are focussed on.

hep-th↗
arXiv1997

Gauge Symmetry and Integrable Models

We establish the isomorphism between a nonlinear $σ$-model and the abelian gauge theory on an arbitrary curved background, which allows us to derive integrable models and the corresponding Lax representations from gauge theoretical point of view. In our approach the spectral parameter is related to the global degree of freedom associated with the conformal or Galileo transformations of the spacetime. The B$\ddot{\rm a}$cklund transformations are derived from Chern-Simons theory where the spectral parameter is defined in terms of the extract compactified space dimension coordinate.

hep-th↗
arXiv1997

Sine-Gordon Solitons and Black Holes

In the talk, the relationship between black holes in Jackiw-Teitelboim(JT) dilaton gravity and solitons in sine-Gordon field theory is described. The well-known connection between solutions of the sine-Gordon equation and constant curvature metrics is reviewed and expanded. In particular, it is shown that solutions to the dilaton field equations for a given metric in JT theory also solve the sine-Gordon equation linearized about the corresponding soliton. Next, it is shown that from the B${\ddot a}$cklund transformations relating different soliton solutions, it is possible to construct a flat SL(2,R) connection which forms the basis for the gauge theory formulation of JT dilaton gravity. Finally, the supersymmetric generalization of sine-Gordon theory is reviewed, leading to a new approach to the problem of the origin of low dimensional black hole entropy.

hep-th↗
arXiv1998

Classical limit of the Knizhnik-Zamolodchikov-Bernard equations as hierarchy of isomonodromic deformations. Free fields approach

We investigate the classical limit of the Knizhnik-Zamolodchikov-Bernard equations, considered as a system of non-stationar Schröodinger equations on singular curves, where times are the moduli of curves. It has a form of reduced non-autonomous hamiltonian systems which include as particular examples the Schlesinger equations, Painlevé VI equation and their generalizations. In general case, they are defined as hierarchies of isomonodromic deformations (HID) with respect to changing the moduli of underling curves. HID are accompanying with the Whitham hierarchies. The phase space of HID is the space of flat connections of $G$ bundles with some additional data in the marked points. HID can be derived from some free field theory by the hamiltonian reduction under the action of the gauge symmetries and subsequent factorization with respect to diffeomorphisms of curve. This approach allows to define the Lax equations associated with HID and the linear system whose isomonodromic deformations are provided by HID. In addition, it leads to description of solutions of HID by the projection method. In some special limit HID convert into the Hitchin systems. In particular, for $\SL$ bundles over elliptic curves with a marked point we obtain in this limit the elliptic Calogero $N$-body system

hep-th↗
arXiv1997

Multi-spin correlation functions for the Z-Invariant Ising model

Continuing our work hep-th/9609135 where a explicit formula for the two-point functions of the two dimensional Z-invariant Ising model were found. I obtain here different results for the higher correlation functions and several consistency checks are done.

hep-th↗
arXiv1998

Calogero model and sL(2,R) algebra

The Calogero model with external harmonic oscillator potential is discussed from sL(2,R) algebra point of view. Explicit formulae for functions with exponential time behaviour are given; in particular, the integrals of motion are constructed and their involutiveness demonstrated. The superintegrability of the model appears to be a simple consequence of the formalism.

hep-th↗
arXiv1999

Quantum Field Theory of Topological Defects as Inhomogeneous Condensates

In the framework of the Closed-Time-Path formalism, we show how topological defects may arise in Quantum Field Theory as result of a localized (inhomogeneous) condensation of particles. We demonstrate our approach on two examples; kinks in the $2D λψ^{4}$ theory (both at zero and finite temperature) and vortices in the complex $4D λψ^{4} $ theory.

hep-th↗
arXiv1998

Bispectrality of KP Solitons

It is by now well known that the wave functions of rational solutions to the KP hierarchy (those which can be achieved as limits of the pure n-soliton solutions) satisfy an additional eigenvalue equation for ordinary differential operators in the spectral parameter. This property is known as ``bispectrality'' and has proved to be both interesting and useful. In this note, it is shown that certain (non-rational) soliton solutions of the KP hierarchy satisfy an eigenvalue equation for a non-local operator constructed by composing ordinary differential operators in the spectral parameter with translation operators in the spectral parameter, and therefore have a form of bispectrality as well. Considering the results relating ordinary bispectrality to the self-duality of the rational Calogero-Moser particle system, it seems likely that this new form of bispectrality should be related to the duality of the Ruijsenaars system.

math-ph↗
arXiv1998

Darboux Transformations of Bispectral Quantum Integrable Systems

We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct non-trivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero-Moser.

math-ph↗
arXiv1998

On solutions of the Schlesinger Equations in Terms of $Θ$-Functions

In this paper we construct explicit solutions and calculate the corresponding $τ$-function to the system of Schlesinger equations describing isomonodromy deformations of $2\times 2$ matrix linear ordinary differential equation whose coefficients are rational functions with poles of the first order; in particular, in the case when the coefficients have four poles of the first order and the corresponding Schlesinger system reduces to the sixth Painlevé equation with the parameters $1/8, -1/8, 1/8, 3/8$, our construction leads to a new representation of the general solution to this Painlevé equation obtained earlier by K. Okamoto and N. Hitchin, in terms of elliptic theta-functions.

math-ph↗
arXiv1998

More On Generalized Symmetries of Partial Differential Equations and Quasiexact Solvability

For the class of systems of PDEs, for which infinitesimal translations (with respect to some (in)dependent variables) possess specific finite-dimensional invariant subspaces of the space of generalized symmetries of the system considered. We establish when there exist generalized symmetries from these subspaces, which depend from the above-mentioned variables.

math.DG↗
arXiv1998

On Time-Dependant Symmetries of Schroedinger Equation

We show that the number of symmetry operators of order not higher that q of the nonstationary n-dimensional (n=1,2,3,4) Schroedinger equation (SE) with nonvanishing potentials is finite and does not exceed that of SE with zero potentials for arbitrary q=0,1,2,... . This result is applied for the determination of the general form of time dependance of the symmetry operators of SE with time-independant potentials.

math.AP↗