Affine Toda Field Theories Research Articles

Common structures of quantum field theories and lattice systems through boundary symmetry

The sine-Gordon model and affine Toda field theories on the half-line, on the one hand, the XXZ spin chain with nondiagoual boundary terms, and interacting many-body lattice systems with a flow, on the other, have a common characteristic. They possess nonlocal conserved boundary charges, generating the Askey-Wilson algebra, a coideal subalgebra of the bulk quantized affine symmetry. We argue that the boundary Askey-Wilson symmetry is the deep algebraic property allowing for integrability of the physical system in consideration.

Integrable defects in affine Toda field theory and infinite-dimensional representations of quantum groups

Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang–Baxter equations, and second by solving a linear intertwining relation between a finite-dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite-dimensional representation expressed in terms of sets of generalised ‘quantum’ annihilation and creation operators. The principal examples analysed are based on the a 2 ( 2 ) and a n ( 1 ) affine Toda models but examples of similar infinite-dimensional representations for quantum Borel algebras for all other affine Toda theories are also provided.

$\mathcal{PT}$ Invariant Complex E 8 Root Spaces

We provide a construction procedure for complex root spaces invariant under antilinear transformations, which may be applied to any Coxeter group. The procedure is based on the factorisation of a chosen element of the Coxeter group into two factors. Each of the factors constitutes an involution and may therefore be deformed in an antilinear fashion. Having the importance of the E(8)-Coxeter group in mind, such as underlying a particular perturbation of the Ising model and the fact that for it no solution could be found previously, we exemplify the procedure for this particular case. As a concrete application of this construction we propose new generalisations of Calogero-Moser Sutherland models and affine Toda field theories based on the invariant complex root spaces and deformed complex simple roots, respectively.

Generalized q-Onsager Algebras and Boundary Affine Toda Field Theories

Generalizations of the q-Onsager algebra are introduced and studied. In one of the simplest case and q = 1, the algebra reduces to the one proposed by Uglov–Ivanov. In the general case and q ≠ 1, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) is exhibited. Boundary (soliton non-preserving) integrable quantum Toda field theories are then considered in light of these results. For the first time, all defining relations for the underlying non-Abelian symmetry algebra are explicitly obtained. As a consequence, based on purely algebraic arguments all integrable (fixed or dynamical) boundary conditions are classified.

Comments on defects in the ar Toda field theories

A simple, basic argument is given, based solely on energy–momentum considerations, to recover conditions under which ar affine or conformal Toda field theories can support defects of integrable type. Associated triangle relations are solved to provide expressions for transmission matrices that generalize previously known examples calculated for the sine-Gordon model and the a2 affine Toda model.

Boundary Lax pairs for the [formula omitted] Toda field theories

Based on the recent formulation of a general scheme to construct boundary Lax pairs, we develop this systematic construction for the A n ( 1 ) affine Toda field theories (ATFT). We work out explicitly the first two models of the hierarchy, i.e. the sine-Gordon ( A 1 ( 1 ) ) and the A 2 ( 1 ) models. The A 2 ( 1 ) Toda theory is the first non-trivial example of the hierarchy that exhibits two distinct types of boundary conditions. We provide here novel expressions of boundary Lax pairs associated to both types of boundary conditions.

PT Symmetry and QCD: Finite Temperature and Density

The relevance of PT symmetry to quantum chromodynamics (QCD), the gauge theory of the strong interactions, is explored in the context of finite temperature and density. Two significant problems in QCD are studied: the sign problem of finite-density QCD, and the problem of confinement. It is proven that the effective action for heavy quarks at finite density is PT -symmetric. For the case of 1+1 dimensions, the PT -symmetric Hamiltonian, although not Hermitian, has real eigenvalues for a range of values of the chemical potential µ, solving the sign problem for this model. The effective action for heavy quarks is part of a potentially large class of generalized sine-Gordon models which are non-Hermitian but are PT -symmetric. Generalized sine-Gordon models also occur naturally in gauge theories in which magnetic monopoles lead to confinement. We explore gauge theories where monopoles cause confinement at arbitrarily high temperatures. Several different classes of monopole gases exist, with each class leading to different string tension scaling laws. For one class of monopole gas models, the PT -symmetric affine Toda field theory emerges naturally as the effective theory. This in turn leads to sine-law scaling for string tensions, a behavior consistent with lattice simulations.

Hitchin's equations and integrability of BPSZNstrings in Yang-Mills theories

We show that ZN string's BPS equations are equivalent to the Hitchin's equations (or self-duality equation) and also to the zero curvature condition. We construct the general form for BPS ZN string solutions for arbitrary simple gauge groups with non-trivial center. Depending on the vacuum solutions considered, the ZN string's BPS equations reduce to different two dimensional integrable field equations. For a particular vacuum we obtain the equation of affine Toda field theory.

An(1)affine Toda field theories with integrable boundary conditions revisited

Generic classically integrable boundary conditions for the An(1) affine Toda field theories (ATFT) are investigated. The present analysis rests primarily on the underlying algebra, defined by the classical version of the reflection equation. We use as a prototype example the first non-trivial model of the hierarchy i.e. the A2(1) ATFT, however our results may be generalized for any An(1) (n > 1). We assume here two distinct types of boundary conditions called some times soliton preserving (SP), and soliton non-preserving (SNP) associated to two distinct algebras, i.e. the reflection algebra and the (q) twisted Yangian respectively. The boundary local integrals of motion are then systematically extracted from the asymptotic expansion of the associated transfer matrix. In the case of SNP boundary conditions we recover previously known results. The other type of boundary conditions (SP), associated to the reflection algebra, are novel in this context and lead to a different set of conserved quantities that depend on free boundary parameters. It also turns out that the number of local integrals of motion for SP boundary conditions is `double' compared to those of the SNP case.

BPSZNstring tensions, sine law and Casimir scaling, and integrable field theories

We consider a Yang-Mills-Higgs theory with spontaneous symmetry breaking of the gauge group $G\ensuremath{\rightarrow}U(1{)}^{r}\ensuremath{\rightarrow}{C}_{G}$, with ${C}_{G}$ being the center of $G$. We study two vacua solutions of the theory which produce this symmetry breaking. We show that for one of these vacua, the theory in the Coulomb phase has the mass spectrum of particles and monopoles which is exactly the same as the mass spectrum of particles and solitons of two-dimensional affine Toda field theory, for suitable coupling constants. That result holds also for $\mathcal{N}=4$ super Yang-Mills theories. On the other hand, in the Higgs phase, we show that for each of the two vacua the ratio of the tensions of the BPS ${Z}_{N}$ strings satisfy either the Casimir scaling or the sine law scaling for $G=SU(N)$. These results are extended to other gauge groups: for the Casimir scaling, the ratios of the tensions are equal to the ratios of the quadratic Casimir constant of specific representations; for the sine law scaling, the tensions are proportional to the components of the left Perron-Frobenius eigenvector of Cartan matrix ${K}_{ij}$ and the ratios of tensions are equal to the ratios of the soliton masses of affine Toda field theories.

On purely transmitting defects in affine Toda field theory

Affine Toda field theories with a purely transmitting integrable defect are considered and the model based on a_2 is analysed in detail. After providing a complete characterization of the problem in a classical framework, a suitable quantum transmission matrix, able to describe the interaction between an integrable defect and solitons, is found. Two independent paths are taken to reach the result. One is an investigation of the triangle equations using the S-matrix for the imaginary coupling bulk affine Toda field theories proposed by Hollowood, and the other uses a functional integral approach together with a bootstrap procedure. Evidence to support the results is collected in various ways: for instance, through the calculation of the transmission factors for the lightest breathers. While previous discoveries within the sine-Gordon model motivated this study, there are several new phenomena displayed in the a_2 model including intriguing disparities between the classical and the quantum pictures. For example, in the quantum framework, for a specific range of the coupling constant that excludes a neighbourhood of the classical limit, there is an unstable bound state.

About the self-dual Chern–Simons system and Toda field theories on the noncommutative plane

The relation of the noncommutative self-dual Chern–Simons (NCSDCS) system to the noncommutative generalizations of Toda and of affine Toda field theories is investigated more deeply. This paper continues the programme initiated in (Cabrera-Carnero I 2005 J. High Energy Phys. JHEP10(2005)071), where it was presented how it is possible to define Toda field theories through second order differential equation systems starting from the NCSDCS system. Here we show that using the connection of the NCSDCS to the noncommutative chiral model, exact solutions of the Toda field theories can also be constructed by means of the noncommutative extension of the uniton method proposed in (Lee K-M 2004 J. High Energy Phys. JHEP08(2004)054) by Lee. Particularly some specific solutions of the nc Liouville model are explicitly constructed.

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars- Schneider equations which correspond to the $N$-soliton solutions of $A^{(1)}_n$ affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.

Abelian Toda field theories on the noncommutative plane

Generalizations of GL(n) abelian Toda and $\widetilde{GL}(n)$ abelian affine Toda field theories to the noncommutative plane are constructed. Our proposal relies on the noncommutative extension of a zero-curvature condition satisfied by algebra-valued gauge potentials dependent on the fields. This condition can be expressed as noncommutative Leznov-Saveliev equations which make possible to define the noncommutative generalizations as systems of second order differential equations, with an infinite chain of conserved currents. The actions corresponding to these field theories are also provided. The special cases of GL(2) Liouville and $\widetilde{GL}(2)$ sinh/sine-Gordon are explicitly studied. It is also shown that from the noncommutative (anti-)self-dual Yang-Mills equations in four dimensions it is possible to obtain by dimensional reduction the equations of motion of the two-dimensional models constructed. This fact supports the validity of the noncommutative version of the Ward conjecture. The relation of our proposal to previous versions of some specific Toda field theories reported in the literature is presented as well.

Affine Toda field theories related to Coxeter groups of noncrystallographic type

We propose affine Toda field theories related to the noncrystallographic Coxeter groups H 2 , H 3 and H 4 . The classical mass spectrum, the classical three-point couplings and the one-loop corrections to the mass renormalisation are determined. The construction is carried out by means of a reduction procedure from crystallographic to noncrystallographic Coxeter groups. The embedding structure explains for various affine Toda field theories that their particles can be organised in pairs, such that their relative masses differ by the golden ratio.

On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

A family of real Hamiltonian forms (RHF) for the special class of affine 1 + 1-dimensional Toda field theories is constructed. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF is related to a special ℤ2-symmetry of the system of roots for the relevant Kac-Moody algebra. A number of explicit nontrivial examples of RHF of ATFT are presented.

APPLICATIONS OF QUANTUM INTEGRABLE SYSTEMS

We present two applications of quantum integrable systems. First, we predict that it is possible to generate high harmonics from solid state devices by demonstrating that the emission spectrum of a minimally coupled laser field of frequency ω to an impurity system of a quantum wire, contains multiples of the incoming frequency. Second, by evaluating expressions for the conductance in the high temperature regime we show that multiples of the characteristic filling fractions of the Jain sequence, which occur in the fractional quantum Hall effect, can be obtained from quantum wires which are described by minimal affine Toda field theories.

Affine Toda field theory from tree unitarity

Elasticity property (i.e. no-particle creation) is used in the tree level scattering of scalar particles in 1+1 dimensions to construct the affine Toda field theory(ATFT) associated with root systems of groups $a_2^{(2)}$ and $c_2^{(1)}$. A general prescription is given for constructing ATFT (associated with rank two root systems) with two self conjugate scalar fields. It is conjectured that the same method could be used to obtain the other ATFT associated with higher rank root systems.

Universal boundary reflection amplitudes

For all affine Toda field theories we propose a new type of generic boundary bootstrap equations, which can be viewed as a very specific combination of elementary boundary bootstrap equations. These equations allow to construct general solutions for the boundary reflection amplitudes, which are valid for theories related to all simple Lie algebras, that is simply laced and non-simply laced. We provide a detailed study of these solutions for concrete Lie algebras in various representations. The boundary bootstrap equations relating different types of exited boundary states are not automatically solved by our expressions.

Affine Toda field theories with defects

A Lagrangian approach is proposed and developed to study defects within affine Toda field theories. In particular, a suitable Lax pair is constructed together with examples of conserved charges. It is found that only those models based on $a_r^{(1)}$ data appear to allow defects preserving integrability. Surprisingly, despite the explicit breaking of Lorentz and translation invariance, modified forms of both energy and momentum are conserved. Some, but apparently not all, of the higher spin conserved charges are also preserved after the addition of contributions from the defect. This fact is illustrated by noting how defects may preserve a modified form of just one of the spin 2 or spin -2 charges but not both of them.