We derive the non-local conserved charges in the sine-Gordon model and affine Toda field theories on the half-line.They generate new kinds of symmetry algebras that are coideals of the usual quantum groups.We show how intertwiners of tensor product representations of these algebras lead to solutions of the reflection equation.We describe how this method for finding solutions to the reflection equation parallels the previously know method of using intertwiners of quantum groups to find solutions to the Yang-Baxter equation. * Speaker.1 Earlier calculations for the sine-Gordon model at the free fermion point were done in [16]
We derive the non-local conserved charges in the sine-Gordon model and affine Toda field theories on the half-line.They generate new kinds of symmetry algebras that are coideals of the usual quantum groups.We show how intertwiners of tensor product representations of these algebras lead to solutions of the reflection equation.We describe how this method for finding solutions to the reflection equation parallels the previously know method of using intertwiners of quantum groups to find solutions to the Yang-Baxter equation. * Speaker.1 Earlier calculations for the sine-Gordon model at the free fermion point were done in [16]
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.
The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non–local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary, one of the non–local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian. LAPTH-1029/04 Quantum groups have attracted much interest recently because of their intrinsic mathematical interest [1]–[3], but also because they provide a powerful scheme for deriving solutions of the Yang-Baxter equation [4]–[6] R12(λ1 − λ2) R13(λ1) R23(λ2) = R23(λ2) R13(λ1) R12(λ1 − λ2). (1.1) R acts on V ⊗2, whereas the Yang–Baxter equation on V ⊗3, and as customary R12 = R ⊗ I, R23 = I ⊗ R. The
It is known that integrable models associated to rational R matrices give rise to certain non-abelian symmetries known as Yangians. Analogously ‘boundary ’ symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton non-preserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary non-local charges directly at the quantum level. The non-local charges are simply coproduct realizations of particular boundary quantum algebras called ‘boundary ’ or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the symmetry of the open spin chain, namely we show that a number of the boundary non-local charges are in fact conserved quantities.
A procedure to construct K -matrices from the generalized q -Onsager algebra is proposed. This procedure extends the intertwiner techniques used to obtain scalar ( c -number) solutions of the reflection equation to dynamical (non- c -number) solutions. It shows the relation between soliton non-preserving reflection equations or twisted reflection equations and the generalized q -Onsager algebras. These dynamical K -matrices are important to quantum integrable models with extra degrees of freedom located at the boundaries; for instance, in the quantum affine Toda field theories on the half-line, they yield the boundary amplitudes. As examples, the cases of and are treated in detail.
This is a review on infinite non-abelian symmetries in two-dimensional field theories. We show how any integrable QFT enjoys the existence of infinitely many {\bf conserved} charges. These charges {\bf do not commute} between them and satisfy a Yang--Baxter algebra. They are generated by quantum monodromy operators and provide a representation of q−deformed affine Lie algebras $U_q({\hat\G})$. We review the work by de Vega, Eichenherr and Maillet on the bootstrap construction of the quantum monodromy operators in classically scale invariant theories where the classical monodromy matrix is conserved. Then, the recent generalization to the sine--Gordon (or massive Thirring) model, where such operators do not possess a classical analogue is given (This provides a representation of SU^(2)q). It is then reported on the recent work by Destri and de Vega, where both commuting and non-commuting integrals of motion are systematically obtained by Bethe Ansatz in the light-cone approach. The eigenvalues of the six--vertex alternating transfer matrix τ(ł) are explicitly computed on a generic physical state through algebraic Bethe ansatz. In the thermodynamic limit τ(ł) turns out to be a two--valued periodic function. One determination generates the local abelian charges, including energy and momentum, while the other yields the abelian subalgebra of the (non--local) YB algebra. In particular, the bootstrap results coincide with the ratio between the two determinations of the lattice transfer matrix.[Based on a Lecture at the Clausthal Symposium].
It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.