The work presented in this talk has connections to several very different areas of physics and mathematics. The results originated from a desire to determine the reflection amplitudes of solitons in affine Toda theories off boundaries. In the process of this work we discovered that the symmetry that governs affine Toda theories on the half line are certain coideal subalgebras of quantum affine algebras. The study of these algebras and their representations is a wide new field for mathematical study. The mathematical results so achieved feed back into the physics because they lead to an understanding of the multiplet structure of boundary bound states. This deeper understanding of boundary bound states in an integrable quantum field theory are likely to have wider relevance. The other outcome of the discovery of the quantum group symmetry of the boundary quantum field theories is a practical method for finding solutions to the reflection equation. Solutions to the reflection equation are needed not only to describe reflection amplitudes but also for defining integrable lattice models and quantum spin chains with boundaries. A very similar story as for the solitons in affine Toda theory also arises for the particles in prinipal chiral models. We have discovered that the symmetry algebra preserved by certain boundary conditions are twisted Yangians. However this would be material for a separate talk.
Quantum Group Symmetry on the half line A study in integrable quantum field theory with a boundary Talk given on 08/05/02 to the Edinburgh Mathematical Physics Group Gustav W Delius Department of Mathematics University of York
Different ways to summarize it <ul><li>Reflection of solitons off boundaries; </li></ul><ul><li>Coideal subalgebras of quantum groups; </li></ul><ul><li>Multiplet structure of boundary states; </li></ul><ul><li>Solutions of the reflection equation. </li></ul>We are studying
Organization of talk <ul><li>Sine-Gordon soliton scattering and reflection as a warm-up </li></ul><ul><li>S-matrices, bound states, and quantum groups </li></ul><ul><li>Reflection matrices, boundary bound states and boundary quantum groups </li></ul>
Tensor product decomposition Example: fundamental reps of sl(n) where At special values of the S-matrix projects onto subrepresentations . Several irreducible reprs of sl(n) are tied together into a single irreducible representation of
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 with the 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 <ul><li>Boundary breaks quantum group symmetry to a coideal subalgebra. </li></ul><ul><li>Solutions of reflection equation can now be obtained from symmetry. </li></ul><ul><li>Spectrum of boundary states is determined to branching rules. </li></ul>
Affine Toda theory <ul><li>Generalize the sine-Gordon potential </li></ul>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 representations of the Lie algebra.