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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
- 2. 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
- 3. 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>
- 4. Sine-Gordon Solitons <ul><li>Lagrangian: </li></ul><ul><li>Field equation: </li></ul><ul><li>Soliton solution: </li></ul>Cosine potential Soliton
- 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
- 20. Introducing a boundary <ul><li>The boundary condition will break the quantum group symmetry to a subalgebra </li></ul>Depends on boundary parameters
- 21. Reflection matrix Sometimes particle comes back in conjugate representation Boundary states form multiplets of resdual symmetry algebra
- 22. Coideal subalgebra
- 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 .
- 26. 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>
- 27. Affine Toda theory <ul><li>Generalize the sine-Gordon potential </li></ul>For example sl(3): Simple roots of affine Lie algebra
- 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
- 31. Quantum affine algebra

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