Quantum group symmetryof integrable models

on the half line

Talk presented at the Workshop on Integrable Theories, Solitons and Duality

at IFT, Sao Paulo, 4 July 2002

Gustav W DeliusAlan GeorgeNiall MacKay

hep-th/0112023and unpublished

ContentsI. Boundary quantum groups

Describing an algebraic structure that leads to the solutions of the reflection equation

II. Non-local charges in affine Toda field theory on the half-lineUsing boundary perturbation theory to find non-local charges unbroken by the boundary conditions

Boundary quantum groups Drinfeld and Jimbo explained how to

obtain solutions of the Yang-Baxter equation as intertwiners of representations of quantum groups

We will introduce boundary quantum groups to similarly obtain solutions of the reflection equation as intertwiners

Yang-Baxter equation

=

where =

Reflection equation

=

where

Drinfeld-Jimbo quantum groups

Boundary quantum groups

Properties required of boundary quantum group

Finding boundary quantum groupsTo find boundary quantum groups and their

representations we determine the symmetry algebras of physical models on the half line:

Rational: Principal chiral models This leads to twisted Yangians Y(g,h) as explained by Niall MacKay in his talk at this conference.

Trigonometric: Affine Toda field theorieswe will derive boundary quantum groups

Sine-Gordon model

Non-local charges

[Bernard and LeClair, Comm.Math.Phys. 142 (1991) 99]

Quantum affine sl2

Neumann boundary condition

Free field two-point functions

Perturbing operator

Boundary quantum group

General boundary condition

Boundary perturbation theory

Boundary quantum group

Affine Toda theory action

Non-local charges

[Bernard and LeClair, Comm.Math.Phys. 142 (1991) 99]

Quantum affine algebra

[Felder and LeClair, Int.J.Mod.Phys. A7 (1992) 239]

Boundary conditions

Non-local charges

Conventions

Boundary quantum groups

How do we establish non-local charges non-perturbatively? We calculated quantum group charges

using first order perturbation theory only.

We then used these to calculate the reflection matrices in the vector representation.

But how can we rule out that there are higher order corrections which will be needed to treat higher representations?

Reconstructing symmetry generators from reflection matrix

Reconstructing symmetry generators from reflection matrix

The an(1) case

Reflection equation algebras

Summary We introduced boundary quantum groups as certain coideal

subalgebras of quantum affine algebras Reflection matrices arise as intertwiners of representations of

boundary quantum groups We used boundary perturbation theory to find the non-local

charges in affine Toda field on the half-line. They give us boundary quantum groups for trigonometric reflection matrices .

We determined the reflection matrices for the vector representation of an

(1), cn(1), dn

(1), and a2n-1(2).

We showed that our expressions for the boundary non-local charges are exact.

Things to do next Establish the boundary quantum groups for bn

(1), a2n(2), dn+1

(2) and the exceptional cases (in progress with Alan George)

Find the boundary quantum groups corresponding to Dirichlet type boundary conditions

Apply the tensor product graph method to the construction of higher reflection matrices

Study the representation theory of boundary quantum groups (multiplets of boundary states)

Study the mathematical structure of boundary quantum groups (Universal K matrix? Classification?)