Affine Toda Field Theories Research Articles

Polynomial Recursion Equations in Form Factors of ADE-Toda Field Theories

It is shown that the problem of calculating form factors in ADE affine Toda field theories can be reduced to the nonperturbative recursive calculation of polynomials symmetric in each sort of variables. We determine these recursion equations explicitly for the ADE series and characterize the polynomial solutions by an interplay between the weight space of the underlying Lie algebra and representations of the symmetric group.

Q-deformed Coxeter element in non-simply laced affine Toda field theories

The Lie algebraic structures of the S-matrices for the affine Toda field theories based on the dual pairs ( X N (1), Y M ( l) ) are discussed. For the non-simply laced horizontal subalgebra X N and the simply laced horizontal subalgebra Y M , we introduce a “ q-deformation” of a Coxeter element and a “ p-deformation” of a twisted Coxeter element respectively. Using these deformed objects, expressions for the generating function of the multiplicities of the building block of the S-matrices are obtained. The relation between the deformed version of Dorey's rule and the generalized Dorey's rule due to Chari and Pressley are discussed.

Quantum affine symmetry and scattering amplitudes of the imaginary coupled d4(3) affine Toda field theory

An exact S-matrix is conjectured for the imaginary coupled d 4 (3) affine Toda field theory, using the U q ( g 2 (1)) symmetry. It is shown that this S-matrix is consistent with the results for the case of real coupling using the breather-particle correspondence. For q a root of unity it is argued that the theory can be restricted to yield Φ(11∥14) perturbations of WA 2 minimal models and the restriction is performed for the (3, p′) minimal models

The R-matrix of the Uq( d4(3)) algebra and g2(1) affine Toda field theory

The R-matrix of the Uq( d 4 (3)) algebra is constructed in the 8-dimensional fundamental representation. Using this result, an exact S-matrix is conjectured for the imaginary coupled g 2 (1) affine Toda field theory, the structure of which is found to be very similar to the previously investigated S-matrix of d 4 (3) Toda theory. It is shown that this S-matrix is consistent with the results for the case of real coupling using the breather-particle correspondence. For q a root of unity it is argued that the theory can be restricted to yield Φ(11‖12) perturbations of WA 2 minimal models.

S-matrices of non-simply laced affine Toda theories by folding

The exact factorisable quantum S-matrices are known for simply laced as well as non-simply laced affine Toda field theories. Non-simply laced theories are obtained from the affine Toda theories based on simply laced algebras by folding the corresponding Dynkin diagrams. The same process, called classical ‘reduction’, provides solutions of a non-simply laced theory from the classical solutions with special symmetries of the parent simply laced theory. In the present note we shall elevate the idea of folding and classical reduction to the quantum level. To support our views we have made some interesting observations for S-matrices of non-simply laced theories and give a prescription for obtaining them through the folding of simply laced ones.

Form factors in Dn(1) affine Toda field theories

We derive closed recursion equations for the symmetric polynomials occurring in the form factors of D n (1) affine Toda field theories. These equations follow from kinematical and bound state residue equations for the full form factor. We also discuss the equations arising from second- and third-order forward channel poles of the S-matrix. The highly symmetric case of D 4 (1) form factors is treated in detail. We calculate explicitly cases with a few particles involved.

On soliton quantum S-matrices in simply laced affine Toda field theories

Exact solutions to the quantum S-matrices for solitons in simply laced affine Toda field theories are obtained, except for certain factors of simple type which remain undetermined in some cases. These are found by postulating solutions which are consistent with the semi-classical limit, ℏ → 0, and the known time delays for a classical two-soliton interaction. This is done by a ‘ q-deformation’ procedure, to move from the classical time delay to the exact S-matrix, by inserting a special function called the ‘regularised’ quantum dilogarithm, which only holds when | q| = 1. It is then hecked that the solutions satisfy the crossing, unitarity and bootstrap constraints of S-matrix theory. These properties essentially follow from analogous properties satisfied by the classical time delay. Furthermore, the lowest mass breather S-matrices are computed by the bootstrap, and it is shown, modulo the undetermined factors, that these agree with the particle S-matrices known already in the affine Toda field theories, in all simply laced cases.

Form factors of exponential operators and exact wave function renormalization constant in the Bullough-Dodd model

We compute the form factors of exponential operators e kgϕ( x) in the two-dimensional integrable Bullough-Dodd model ( a 2 (2) affine Toda field theory). These form factors are selected among the solutions of general non-derivative scalar operators by their asymptotic cluster property. Through analytical continuation to complex values of the coupling constant these solutions permit to compute the form factors of scaling relevant primary fields in the lightest-breather sector of integrable / gf 1,2 and / gf 1,5 deformations of conformal minimal models. We also obtain the exact wave-function renormalization constant Z( g) of the model and the properly normalized form factors of the operators ϕ( x) and : ϕ 2( x) :.

Inverse scattering and solitons in An−1 affine Toda field theories

We implement the inverse scattering method in the case of the A n affine Toda field theories, by studying the space-time evolution of simple poles in the underlying loop group. We find the known single-soliton solutions, as well as additional solutions with non-linear modes of oscillation around the standard solution, by studying the particularly simple case where the residue at the pole is a rank-one projection. We show that these solutions with extra modes have the same mass and topological charges as the standard solutions, so we do not shed any light on the missing topological charge problem in these models. From the monodromy matrix it is shown that these solutions have the same higher conserved charges as the standard solutions. We also show that the integrated energy-momentum density can be calculated from the central extension of the loop group.

Boundary Reflection Matrices for Non-Simply-Laced Affine Toda Field Theories (Abstract)

Boundary Reflection Matrices for Non-Simply-Laced Affine Toda Field Theories (Abstract)

Remarks on excited states of affine Toda solitons

The identification in affine Toda field theory of the quantum particle with the lowest breather allows us to re-interpret discrete modes of excitation of solitons breathers bound to solitons, and thus to investigate them through the proposed solition-breather S-matrices. There are implications for the physical spectrum and for the semiclassical soliton mass corrections.

Square root singularity in boundary reflection matrix

Two-particle scattering amplitudes for integrable relativistic quantum field theory in 1+1 dimensions can normally have at most singularities of poles and zeros along the imaginary axis in the complex rapidity plane. It has been supposed that single-particle amplitudes of the exact boundary reflection matrix exhibit the same structure. In this paper, single-particle amplitudes of the exact boundary reflection matrix corresponding to the Neumann boundary condition for affine Toda field theory associated with twisted affine algebras a 2 n (2) are conjectured, based on one-loop result, as having a new kind of square root singularity.

NONLINEAR SIGMA MODELS ON A HALF PLANE

In the context of integrable field theory with boundary, the integrable nonlinear sigma models in two dimensions, for example the O(N), the principal chiral, the CPN−1 and the complex Grassmannian sigma models, are discussed on a half plane. In contrast to the well-known cases of sine-Gordon, nonlinear Schrödinger and affine Toda field theories, these nonlinear sigma models in two dimensions are not classically integrable if restricted on a half plane. It is shown that the infinite set of nonlocal charges characterizing the integrability on the whole plane is not conserved for the free (Neumann) boundary condition. If we require that these nonlocal charges be conserved, then the solutions become trivial.

Affine Toda solitons and systems of Calogero-Moser type

The solitons of affine Toda field theory are related to the spin-generalised Ruijsenaars-Schneider (or relativistic Calogero-Moser) models. This provides the sought after extension of the correspondence between the sine-Gordon solitons and the Ruijsenaars-Schneider model.

Boundary reflection matrix foradeaffine Toda field theory

We present a complete set of conjectures for the exact boundary reflection matrix for $ade$ affine Toda field theory defined on a half line with the Neumann boundary condition.

Functional equations of form factors for diagonal scattering theories

The form factor bootstrap approach is applied for diagonal scattering theories. We consider the ADE theories and determine the functional equations satisfied by the minimal two-particle form factors. We also determine the parameterization of the singularities in two-particle form factors. For A 2 (1) affine Toda field theory, which is the simplest non-self-conjugate theory, form factors are derived up to four-body ones and identification of the operator is done. Generalizing this identification to the AN N (1) affine Toda cases, we find the two-particle form factors. We also determine the additional pole structure of form factors which comes from the double pole of the S-matrices of the A N (1) theory. For A N theories, the existence of the conserved Z N+1 charge leads to the division of the set of form factors into N + 1 decoupled sectors.

Boundary reflection matrices for nonsimply laced affine Toda field theories.

The boundary reflection matrices for nonsimply laced affine Toda field theories defined on a half line with the Neumann boundary condition are investigated. The boundary reflection matrices for some pairs of the models are evaluated up to one loop order by perturbation theory. Then the exact boundary reflection matrices which are consistent with the one loop result are found under the assumption of ``duality'' and tested against algebraic consistency such as the boundary bootstrap equation and boundary crossing-unitarity relation. \textcopyright{} 1996 The American Physical Society.

Background field boundary conditions for affine Toda field theories

Classical integrability is investigated for affine Toda field theories in the presence of a constant background tensor field. This leads to a further set of discrete possibilities containing no free parameters other than the bulk coupling constant.

BRAID RELATIONS IN AFFINE TODA FIELD THEORY

We provide explicit realizations for the operators which when exchanged give rise to the scattering matrix. For affine Toda field theory we present two alternative constructions, one related to a free bosonic theory and the other formally to the quantum affine Heisenberg algebra of [Formula: see text].

Non-Canonical Folding of Dynkin Diagrams and Reduction of Affine Toda Theories

The equation of motion of affine Toda field theory is a coupled equation for $r$ fields, $r$ is the rank of the underlying Lie algebra. Most of the theories admit reduction, in which the equation is satisfied by fewer than $r$ fields. The reductions in the existing literature are achieved by identifying (folding) the points in the Dynkin diagrams which are connected by symmetry (automorphism). In this paper we present many new reductions. In other words the symmetry of affine Dynkin diagrams could be extended and it leads to non-canonical foldings. We investigate these reductions in detail and formulate general rules for possible reductions. We will show that eventually most of the theories end up in $a_{2n}^{(2)}$ that is the theory cannot have a further dimension $m$ reduction where $m<n$.