Affine Toda field theory in the presence of reflecting boundaries (original) (raw)

USP-IFQSC/TH/93-12 Affine Toda Field Theory in the Presence of Reflecting Boundaries

1993

We show that the “boundary crossing-unitarity equation ” recently proposed by Ghoshal and Zamolodchikov is a consequence of the boundary bootstrap equation for the S-matrix and the wall-bootstrap equation. We solve this set of equations for all affine Toda theories related to simply laced Lie algebras, obtaining explicit formulas for the W-matrix which encodes the scattering of a particle with the boundary in the ground state. For each theory there are two solutions to these equations, related by CDD-ambiguities, each giving rise to different kind of physics.

On the universal representation of the scattering matrix of affine Toda field theory

By exploiting the properties of q-deformed Coxeter elements, the scattering matrices of affine Toda field theories with real coupling constant related to any dual pair of simple Lie algebras may be expressed in a completely generic way. We discuss the governing equations for the existence of bound states, i.e. the fusing rules, in terms of q-deformed Coxeter elements, twisted q-deformed Coxeter elements and undeformed Coxeter elements. We establish the precise relation between these different formulations and study their solutions. The generalized S-matrix bootstrap equations are shown to be equivalent to the fusing rules. The relation between different versions of fusing rules and quantum conserved quantities, which result as nullvectors of a doubly q-deformed Cartan like matrix, is presented. The properties of this matrix together with the so-called combined bootstrap equations are utilised in order to derive generic integral representations for the scattering matrix in terms of quantities of either of the two dual algebras. We present extensive case-by-case data, in particular on the orbits generated by the various Coxeter elements.

Boundary bound states in affine Toda field theory

We demonstrate that the generalization of the Coleman-Thun mechanism may be applied to the situation, when considering scattering processes in 1+1-dimensions in the presence of reflecting boundaries. For affine Toda field theories we find that the binding energies of the bound states are always half the sum over a set of masses having the same colour with respect to the bicolouration of the Dynkin diagram. For the case of E 6 -affine Toda field theory we compute explicitly the spectrum of all higher boundary bound states. The complete set of states constitutes a closed bootstrap.

The fusing rule and the scattering matrix of affine Toda theory

Affine Toda theory is an integrable theory with many interesting features. Classically, the presence of trilinear couplings is given by Dorey's "fusing rule", whatever the simple Lie algebra concerned. This paper discusses the structure of this rule, alternative solutions and formulations, and the relationship to the quantum conservation laws. This insight is applied to the conjectured scattering matrix of the quantum theory. The crossing and bootstrap properties are verified in a general way, valid for any Lie algebra, but the analyticity properties require the extra assumption that the algebra be simply laced. Various identities satisfied by a Coxeter element play a crucial role.

Boundary One-Point Functions, Scattering, and Background Vacuum Solutions in Toda Theories

International Journal of Modern Physics A, 2003

The parametric families of integrable boundary affine Toda theories are considered. We calculate boundary one-point functions and propose boundary S-matrices in these theories. We use boundary one-point functions and S-matrix amplitudes to derive boundary ground state energies and exact solutions describing classical vacuum configurations.

Universal boundary reflection amplitudes

Nuclear Physics B - NUCL PHYS B, 2004

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.

The S-matrix coupling dependence for a, d and e affine Toda field theory

Physics Letters B, 1991

Toda field theories are solvable 1 + 1 dimensional quantum field theories closely related to integrable deformations of conformal field theory. The S-matrix elements for an affine Toda field theory are believed to depend on the coupling constant fl through one universal function B(fl) which cannot be determined by unitarity, crossing and the bootstrap. From the requirement of nonexistence of extra poles in the physical region its form is conjectured to be B(fl)= (2~t)-lfl:/(1 + f12/4/t). We show that the above conjecture is correct up to one-loop order (i.e., f14) of perturbation for simply laced, i.e., a, d and e affine Toda field theories using a general argument which exhibits much of the richness of these theories.

Classically integrable boundary conditions for affine Toda field theories

Nuclear Physics B, 1995

Boundary conditions compatible with classical integrability are studied both directly, using an approach based on the explicit construction of conserved quantities, and indirectly by first developing a generalisation of the Lax pair idea. The latter approach is closer to the spirit of earlier work by Sklyanin and yields a complete set of conjectures for permissible boundary conditions for any affine Toda field theory.