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- 1. 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
- 5. Classical Soliton scattering For example in the sine-Gordon model
- 6. Time advance during scattering The solitons experience a time advance while scattering through each other.
- 7. Classical Soliton reflection For example in the sine-Gordon model
- 8. Method of images For example in the sine-Gordon model Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.
- 9. Time advance during reflection For an attractive boundary condition The soliton experiences a time advance during reflection.
- 10. Time delay during reflection For a repulsive boundary condition The soliton experiences a time advance during reflection.
- 11. Quantum amplitudes Scattering amplitude Reflection amplitude Soliton type rapidity
- 12. Factorization = Yang-Baxter equation = Reflection equation Cherednik, Theor.Math.Phys. 61 (1984) 977 Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841. One way to obtain amplitudes is to solve:
- 13. Bound states breather Boundary breather Poles in the amplitudes orresponding to bound states
- 14. Classical breather solution Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.
- 15. Scattering matrix The solitons with rapidity span representation spaces Highest weight of representation rapidity
- 16. Schur’s Lemma
- 17. Quantum Group Symmetry Theory: Symmetry:
- 18. 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
- 19. Tensor product graph for C n
- 21. Reflection matrix Sometimes particle comes back in conjugate representation Boundary states form multiplets of resdual symmetry algebra
- 23. Boundary quantum groups Trigonometric: Realized in affine Toda field theory with boundary condition Derived using boundary conformal perturbation theory Delius, MacKay, hep- th /0112023
- 24. Boundary quantum group Rational: Obtained from principal chiral model on G with the field at the boundary constrained to lie in H.
- 25. Boundary bound states where Delius, MacKay, Short, Phys. Lett . B 522(2001)335-344 .
- 28. 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.
- 29. Affine Toda theory action
- 30. Nonlocal charges
Notas do Editor
- 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.