Quantum Group Symmetryon the half line

A study in integrable quantum field theory with a boundary

Talk given on 08/05/02 to theEdinburgh Mathematical Physics Group

Gustav W DeliusDepartment of MathematicsUniversity of York

Different ways to summarize it

Reflection of solitons off boundaries; Coideal subalgebras of quantum

groups; Multiplet structure of boundary states; Solutions of the reflection equation.

We are studying

Organization of talk Sine-Gordon soliton scattering and

reflection as a warm-up S-matrices, bound states, and

quantum groups Reflection matrices, boundary

bound states and boundary quantum groups

Sine-Gordon Solitons

Lagrangian:

Field equation:

Soliton solution:

Cosine potential

Soliton

Classical Soliton scattering

For example in the sine-Gordon model

Time advance during scattering

The solitons experiencea time advance while scattering through each other.

Classical Soliton reflection

For example in the sine-Gordon model

Method of images

For example in the sine-Gordon model

Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.

Time advance during reflection

For an attractiveboundary conditionThe soliton experiencesa time advanceduring reflection.

Time delay during reflection

For a repulsiveboundary conditionThe soliton experiencesa time advanceduring reflection.

Quantum amplitudes

Scattering amplitude Reflection amplitude

Solitontype

rapidity

Factorization

= Yang-Baxterequation

= Reflectionequation

Cherednik, Theor.Math.Phys. 61 (1984) 977Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.

One way to obtain amplitudes is to solve:

Bound states

breatherBoundary breather

Poles in theamplitudesorrespondingto bound states

Classical breather solution

Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.

Scattering matrix

The solitons with rapidity span representation spaces

Highest weightof representation

rapidity

Schur’s Lemma

Quantum Group Symmetry

Theory: Symmetry:

Tensor product decomposition

Example: fundamental reps of sl(n)

Several irreducible reprs of sl(n) are tied togetherinto a single irreducible representation of

where

At special values of the S-matrix projects onto subrepresentations.

Tensor product graph for Cn

Introducing a boundary The boundary condition will break

the quantum group symmetry to a subalgebra

Depends onboundary parameters

Reflection matrix

Sometimes particle comesback in conjugate representation

Boundary states form multipletsof resdual symmetry algebra

Coideal subalgebra

Boundary quantum groups

Trigonometric:

Realized in affine Toda field theory with boundary condition

Derived using boundary conformal perturbation theory

Delius, MacKay, hep-th/0112023

Boundary quantum group

Rational:

Obtained from principal chiral model on G withthe field at the boundary constrained to lie in H.

Boundary bound states

where

Delius, MacKay, Short, Phys.Lett. B 522(2001)335-344.

Three things to remember Boundary breaks quantum group

symmetry to a coideal subalgebra. Solutions of reflection equation can

now be obtained from symmetry. Spectrum of boundary states is

determined to branching rules.

Affine Toda theory

Generalize the sine-Gordon potential

For example sl(3):Simple roots of affine Lie algebra

Affine Toda solitons

In this case there are six fundamental solitons interpolating along the green and the blue arrows.

Example sl(3):

In general it is believed that the solitons fill out the fundamental representationsof the Lie algebra.

Affine Toda theory action

Nonlocal charges

Quantum affine algebra