Sao paulo


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Sao paulo

  1. 1. Quantum group symmetry of integrable models on the half line Talk presented at the Workshop on Integrable Theories, Solitons and Duality at IFT, Sao Paulo, 4 July 2002 Gustav W Delius Alan George Niall MacKay hep-th/0112023 and unpublished
  2. 2. Contents <ul><li>Boundary quantum groups Describing an algebraic structure that leads to the solutions of the reflection equation </li></ul><ul><li>Non-local charges in affine Toda field theory on the half-line Using boundary perturbation theory to find non-local charges unbroken by the boundary conditions </li></ul>
  3. 3. Boundary quantum groups <ul><li>Drinfeld and Jimbo explained how to obtain solutions of the Yang-Baxter equation as intertwiners of representations of quantum groups </li></ul><ul><li>We will introduce boundary quantum groups to similarly obtain solutions of the reflection equation as intertwiners </li></ul>
  4. 4. Yang-Baxter equation = where =
  5. 5. Reflection equation where =
  6. 6. Drinfeld-Jimbo quantum groups
  7. 7. Boundary quantum groups
  8. 8. Properties required of boundary quantum group
  9. 9. Finding boundary quantum groups <ul><li>To find boundary quantum groups and their representations we determine the symmetry algebras of physical models on the half line: </li></ul><ul><li>Rational: Principal chiral models This leads to twisted Yangians Y(g,h) as explained by Niall MacKay in his talk at this conference. </li></ul><ul><li>Trigonometric: Affine Toda field theories we will derive boundary quantum groups </li></ul>
  10. 10. Sine-Gordon model
  11. 11. Non-local charges [Bernard and LeClair, Comm.Math.Phys. 142 (1991) 99]
  12. 12. Quantum affine sl 2
  13. 13. Neumann boundary condition
  14. 14. Free field two-point functions
  15. 15. Perturbing operator
  16. 16. Boundary quantum group
  17. 17. General boundary condition
  18. 18. Boundary perturbation theory
  19. 19. Boundary quantum group
  20. 20. Affine Toda theory action
  21. 21. Non-local charges [Bernard and LeClair, Comm.Math.Phys. 142 (1991) 99]
  22. 22. Quantum affine algebra [Felder and LeClair, Int.J.Mod.Phys. A7 (1992) 239]
  23. 23. Boundary conditions
  24. 24. Non-local charges
  25. 25. Conventions
  26. 26. Boundary quantum groups
  27. 27. How do we establish non-local charges non-perturbatively? <ul><li>We calculated quantum group charges using first order perturbation theory only. </li></ul><ul><li>We then used these to calculate the reflection matrices in the vector representation. </li></ul><ul><li>But how can we rule out that there are higher order corrections which will be needed to treat higher representations? </li></ul>
  28. 28. Reconstructing symmetry generators from reflection matrix
  29. 29. Reconstructing symmetry generators from reflection matrix
  30. 30. The a n (1) case
  31. 31. Reflection equation algebras
  32. 32. Summary <ul><li>We introduced boundary quantum groups as certain coideal subalgebras of quantum affine algebras </li></ul><ul><li>Reflection matrices arise as intertwiners of representations of boundary quantum groups </li></ul><ul><li>We used boundary perturbation theory to find the non-local charges in affine Toda field on the half-line. They give us boundary quantum groups for trigonometric reflection matrices . </li></ul><ul><li>We determined the reflection matrices for the vector representation of a n (1) , c n (1) , d n (1) , and a 2n-1 (2) . </li></ul><ul><li>We showed that our expressions for the boundary non-local charges are exact. </li></ul>
  33. 33. Things to do next <ul><li>Establish the boundary quantum groups for b n (1) , a 2n (2) , d n+1 (2) and the exceptional cases (in progress with Alan George) </li></ul><ul><li>Find the boundary quantum groups corresponding to Dirichlet type boundary conditions </li></ul><ul><li>Apply the tensor product graph method to the construction of higher reflection matrices </li></ul><ul><li>Study the representation theory of boundary quantum groups (multiplets of boundary states) </li></ul><ul><li>Study the mathematical structure of boundary quantum groups (Universal K matrix? Classification?) </li></ul>