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Quantum Group Symmetry on the half line A study in integrable quantum field theory with a boundary Talk given on 08/05/02 ...
Different ways to summarize it <ul><li>Reflection of solitons off boundaries; </li></ul><ul><li>Coideal subalgebras of qua...
Organization of talk <ul><li>Sine-Gordon soliton scattering and reflection as a warm-up </li></ul><ul><li>S-matrices, boun...
Sine-Gordon Solitons <ul><li>Lagrangian: </li></ul><ul><li>Field equation: </li></ul><ul><li>Soliton solution: </li></ul>C...
Classical Soliton scattering For example in the sine-Gordon model
Time advance during scattering The solitons experience a 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  attractive boundary condition The soliton experiences a time advance during reflect...
Time delay during reflection For a  repulsive boundary condition The soliton experiences a time advance during reflection.
Quantum amplitudes Scattering amplitude Reflection amplitude Soliton type rapidity
Factorization = Yang-Baxter equation = Reflection equation Cherednik, Theor.Math.Phys. 61 (1984) 977 Ghoshal & Zamolodchik...
Bound states breather Boundary  breather Poles in the amplitudes orresponding to bound states
Classical breather solution Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.
Scattering matrix The solitons with rapidity   span representation spaces  Highest weight of representation rapidity
Schur’s Lemma
Quantum Group Symmetry Theory: Symmetry:
Tensor product decomposition Example: fundamental reps of sl(n) where At special values of    the S-matrix projects onto ...
Tensor product graph for C n
Introducing a boundary <ul><li>The boundary condition will break the quantum group symmetry  to a subalgebra  </li></ul>De...
Reflection matrix Sometimes particle comes back in conjugate representation Boundary states  form multiplets of resdual sy...
Coideal subalgebra
Boundary quantum groups Trigonometric: Realized in affine Toda field theory with boundary condition Derived using boundary...
Boundary quantum group Rational: Obtained from principal chiral model on G with the field at the boundary constrained to l...
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>Solutio...
Affine Toda theory <ul><li>Generalize the sine-Gordon potential </li></ul>For example sl(3): Simple roots of  affine Lie a...
Affine Toda solitons In this case there are  six fundamental solitons  interpolating along the  green and the blue arrows....
Affine Toda theory action
Nonlocal charges
Quantum affine algebra
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  1. 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. 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. 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. 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. 5. Classical Soliton scattering For example in the sine-Gordon model
  6. 6. Time advance during scattering The solitons experience a time advance while scattering through each other.
  7. 7. Classical Soliton reflection For example in the sine-Gordon model
  8. 8. Method of images For example in the sine-Gordon model Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.
  9. 9. Time advance during reflection For an attractive boundary condition The soliton experiences a time advance during reflection.
  10. 10. Time delay during reflection For a repulsive boundary condition The soliton experiences a time advance during reflection.
  11. 11. Quantum amplitudes Scattering amplitude Reflection amplitude Soliton type rapidity
  12. 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. 13. Bound states breather Boundary breather Poles in the amplitudes orresponding to bound states
  14. 14. Classical breather solution Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.
  15. 15. Scattering matrix The solitons with rapidity  span representation spaces Highest weight of representation rapidity
  16. 16. Schur’s Lemma
  17. 17. Quantum Group Symmetry Theory: Symmetry:
  18. 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. 19. Tensor product graph for C n
  20. 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. 21. Reflection matrix Sometimes particle comes back in conjugate representation Boundary states form multiplets of resdual symmetry algebra
  22. 22. Coideal subalgebra
  23. 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. 24. Boundary quantum group Rational: Obtained from principal chiral model on G with the field at the boundary constrained to lie in H.
  25. 25. Boundary bound states where Delius, MacKay, Short, Phys. Lett . B 522(2001)335-344 .
  26. 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. 27. Affine Toda theory <ul><li>Generalize the sine-Gordon potential </li></ul>For example sl(3): Simple roots of affine Lie algebra
  28. 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. 29. Affine Toda theory action
  30. 30. Nonlocal charges
  31. 31. Quantum affine algebra

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