Quantum group symmetry on the half-line
     Talk at CPT, Durham, 11 January 2001
                     Gustav W Delius
   ...
Outline
  •   General remarks on quantum group symmetry on
      the whole line and on the half line
  •   Application to ...
Quantum Group Symmetry
Let A be the quantum group symmetry algebra
(Yangian or quantum affine algebra) of some QFT.
  • Par...
Action on Particles
     µ
LetVθ  be the space spanned by the particles in
                                    µ
multiplet...
Action on Particles
     µ
LetVθ  be the space spanned by the particles in
                                    µ
multiplet...
S-matrix as intertwiner
The S-matrix has to commute with the action of any
symmetry charge Q ∈ A,
                        ...
Yang-Baxter equation
Schur’s lemma implies that the S-matrix satisfies the
Yang-Baxter equation.
      µ                   ...
On the half-line
Let us now impose an integrable boundary condition.
This will break the symmetry to a subalgebra B ⊂ A.

...
Reflection Matrix as Intertwiner
The reflection matrix has to commute with the action
                        ˆ
of any symme...
Coideal property
The residual symmetry algebra B does not have to be
a Hopf algebra. However it must be a left coideal of
...
The Reflection Equation
The reflection equation is again a consequence of
Schur’s lemma
                          id ⊗K ν (θ...
Mathematical Problem
Given A find its coideal subalgebras B such that for a
set of representations on has that
            ...
Boundary Bound States
Particles can bind to the boundary, creating multiplets
of boundary bound states. These span represe...
Quantum Group Symmetry
Let A be the quantum group symmetry algebra
(Yangian or quantum affine algebra) of some QFT.
  • Par...
Outline
  •   General remarks on quantum group symmetry on
      the whole line and on the half line
  •   Application to ...
Affine Toda theories
  •   Review of non-local charges
  •   Neumann boundary condition
  •   General boundary condition as...
Toda Action
             1                ¯ + λ
         S=                 2
                        d z ∂φ∂φ          d2...
Non-local Charges
[Bernard & LeClair, Commun. Math. Phys. 142 (1991) 99]
                ∞                                ...
Quantum Affine Algebra
Together with the topological charge

                          βˆ         ∞
                     Tj...
Neumann boundary
Any field configuration invariant under x → −x
satisfies the Neumann condition ∂x φ = 0 at x = 0.
Therefore ...
Boundary Perturbation
The more general integrable boundary conditions
         Bowcock, Corrigan, Dorey & Rietdijk, Nucl.P...
Conserved Charges
It can now be checked in first order boundary
perturbation theory that the charges
                      ...
Coideal property
Using the coproduct

            ∆(Qi ) = Qi ⊗ 1 + q Ti ⊗ Qi ,
            ∆(Qi ) = Qi ⊗ 1 + q Ti ⊗ Qi ,
...
Calculating Reflection Matrices
Using the representation matrices
 µ ˆ
πθ (Qi ) = x ei+1 i + x−1 ei i+1 + ˆi ((q −1 − 1) ei...
Solution
If all | i | = 1 then one finds the solution

     i          −1            (n+1)/2                  −(n+1)/2     ...
Outline
  •   General remarks on quantum group symmetry on
      the whole line and on the half line
  •   Application to ...
Principal Chiral Models
                       1
                    L = Tr ∂µ g −1 ∂ µ g
                       2
G × G s...
Boundary
Boundary condition g(0) ∈ H where H ⊂ G such that G/H is a
symmetric space. The Lie algebra splits g = h ⊕ k. Wri...
Reflection Matrices
The reflection matrices have to take the form
             µ[λ]                                 µ[λ]    ...
Outline
  •   General remarks on quantum group symmetry on
      the whole line and on the half line
  •   Application to ...
Reconstruction of symmetry
Let us assume that for one particular representation Vθµ we know
the reflection matrix K µ (θ) :...
Generators for B
Introducing matrix indices:
                        ¯¯
            (Bθ )α β = (Lµ )α γ (K µ (θ))γ δ (Lµ )...
Charges in affine Toda
Applying the above construction to the vector solitons
in affine Toda theory and expanding in powers ...
Points to remember
  •   Boundary breaks quantum group symmetry A to
      a subalgebra B .




                          ...
Points to remember
  •   Boundary breaks quantum group symmetry A to
      a subalgebra B .
  •   B is not a Hopf algebra ...
Points to remember
  •   Boundary breaks quantum group symmetry A to
      a subalgebra B .
  •   B is not a Hopf algebra ...
Points to remember
  •   Boundary breaks quantum group symmetry A to
      a subalgebra B .
  •   B is not a Hopf algebra ...
Points to remember
  •   Boundary breaks quantum group symmetry A to
      a subalgebra B .
  •   B is not a Hopf algebra ...
Points to remember
  •   Boundary breaks quantum group symmetry A to
      a subalgebra B .
  •   B is not a Hopf algebra ...
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  1. 1. Quantum group symmetry on the half-line Talk at CPT, Durham, 11 January 2001 Gustav W Delius gwd2@york.ac.uk Department of Mathematics, University of York Quantum group symmetry on the half-line – p.1/33
  2. 2. Outline • General remarks on quantum group symmetry on the whole line and on the half line • Application to affine Toda theories • Application to principal chiral models • Reconstruction of residual symmetry from reflection matrices Quantum group symmetry on the half-line – p.2/33
  3. 3. Quantum Group Symmetry Let A be the quantum group symmetry algebra (Yangian or quantum affine algebra) of some QFT. • Particle multiplets span representations of A • Multiparticle states transform in tensor product representations given by coproduct of A • S-matrices are intertwiners of tensor product representations • Boundary breaks symmetry to subalgebra B • Residual symmetry algebra B is coideal • Reflection matrices are determined by their intertwining property • Boundary bound states span representations of B Quantum group symmetry on the half-line – p.3/33
  4. 4. Action on Particles µ LetVθ be the space spanned by the particles in µ multiplet µ with rapidity θ. Each Vθ carries a representation πθ : A → End(Vθµ ). µ Quantum group symmetry on the half-line – p.4/33
  5. 5. Action on Particles µ LetVθ be the space spanned by the particles in µ multiplet µ with rapidity θ. Each Vθ carries a representation πθ : A → End(Vθµ ). µ Asymptotic two-particle states span tensor product µ spaces Vθ ⊗ Vθν . The symmetry acts on these through the coproduct ∆ : A → A ⊗ A. Quantum group symmetry on the half-line – p.4/33
  6. 6. S-matrix as intertwiner The S-matrix has to commute with the action of any symmetry charge Q ∈ A, µ ν µ (πθ ⊗πθ )(∆(Q)) µ Vθ ⊗ Vθν −− − − − − − − −→ Vθ ⊗ Vθν   S µν (θ−θ ) S µν (θ−θ ) ν µ (πθ ⊗πθ )(∆(Q)) Vθν ⊗ Vθµ − − − − − Vθν ⊗ Vθµ − − − −→ This determines the S-matrix uniquely up to an overall factor (which is then fixed by unitarity, crossing symmetry and closure of the bootstrap). Quantum group symmetry on the half-line – p.5/33
  7. 7. Yang-Baxter equation Schur’s lemma implies that the S-matrix satisfies the Yang-Baxter equation. µ S µν (θ−θ ) ⊗ id µ Vθ ⊗ Vθν ⊗ Vθλ −− − −→ −−−− Vθν ⊗ Vθ ⊗ Vθλ     id⊗S νλ (θ −θ ) id ⊗ S µλ (θ−θ ) µ µ Vθ ⊗ Vθλ ⊗ Vθν Vθν ⊗ Vθλ ⊗ Vθ     S µλ (θ−θ ) ⊗ id S νλ (θ −θ )⊗id µ id ⊗ S µν (θ−θ ) µ Vθλ ⊗ Vθ ⊗ Vθν −− − −→ −−−− Vθλ ⊗ Vθν ⊗ Vθ Quantum group symmetry on the half-line – p.6/33
  8. 8. On the half-line Let us now impose an integrable boundary condition. This will break the symmetry to a subalgebra B ⊂ A. On the half-line a particle with positive rapidity θ will eventually hit the boundary and be reflected into another particle with opposite rapidity −θ. This is described by the reflection matrices µ µ µ¯ K (θ) : Vθ → V−θ . Quantum group symmetry on the half-line – p.7/33
  9. 9. Reflection Matrix as Intertwiner The reflection matrix has to commute with the action ˆ of any symmetry charge Q ∈ B ⊂ A, µ ˆ πθ (Q) Vθµ − − Vθµ −→   K µ (θ) K µ (θ) ¯ µ ˆ µ¯ π−θ (Q) µ¯ V−θ − − → V−θ −− If the residual symmetry algebra B is "large enough" then this determines the reflection matrices uniquely up to an overall factor. Quantum group symmetry on the half-line – p.8/33
  10. 10. Coideal property The residual symmetry algebra B does not have to be a Hopf algebra. However it must be a left coideal of A in the sense that ˆ ∆(Q) ∈ A ⊗ B ˆ for all Q ∈ B . This allows it to act on multi-soliton states. Quantum group symmetry on the half-line – p.9/33
  11. 11. The Reflection Equation The reflection equation is again a consequence of Schur’s lemma id ⊗K ν (θ ) Vθµ ⊗ Vθν − − − → Vθµ ⊗ V−θ −−− ¯ ν   S µν (θ−θ ) µ¯ ν S (θ+θ )  µ ¯ µ Vθν ⊗ Vθ ν V−θ ⊗ Vθ   id ⊗K µ (θ) id ⊗K µ (θ) µ¯ ¯ µ¯ Vθν ⊗ V−θ ν V−θ ⊗ V−θ   S ν µ (θ+θ ) ¯ ¯¯ νµ S (θ−θ )  µ¯ id ⊗K ν (θ ) µ¯ ¯ V−θ ⊗ Vθν −− −→ −−− V−θ ν ⊗ V−θ Quantum group symmetry on the half-line – p.10/33
  12. 12. Mathematical Problem Given A find its coideal subalgebras B such that for a set of representations on has that µ • tensor products Vθ ⊗ Vθν are generically irreducible, µ µ¯ • intertwiners K (θ) : µ Vθ → V−θ exist. Physical Problem Find the boundary condition corresponding to B. Quantum group symmetry on the half-line – p.11/33
  13. 13. Boundary Bound States Particles can bind to the boundary, creating multiplets of boundary bound states. These span representations V [λ] of the symmetry algebra B . The reflection of particles off these boundary bound states is described by intertwiners ¯ K µ[λ] (θ) : Vθµ ⊗ V [λ] → V−θ ⊗ V [λ] . µ Quantum group symmetry on the half-line – p.12/33
  14. 14. Quantum Group Symmetry Let A be the quantum group symmetry algebra (Yangian or quantum affine algebra) of some QFT. • Particle multiplets span representations of A • Multiparticle states transform in tensor product representations given by coproduct of A • S-matrices are intertwiners of tensor product representations • Boundary breaks symmetry to subalgebra B • Residual symmetry algebra B is coideal • Reflection matrices are determined by their intertwining property • Boundary bound states span representations of B Quantum group symmetry on the half-line – p.13/33
  15. 15. Outline • General remarks on quantum group symmetry on the whole line and on the half line • Application to affine Toda theories • Application to principal chiral models • Reconstruction of residual symmetry from reflection matrices Quantum group symmetry on the half-line – p.14/33
  16. 16. Affine Toda theories • Review of non-local charges • Neumann boundary condition • General boundary condition as perturbation • Derivation of reflection matrices from the quantum group symmetry Quantum group symmetry on the half-line – p.15/33
  17. 17. Toda Action 1 ¯ + λ S= 2 d z ∂φ∂φ d2 z Φpert , 4π 2π where n Φ pert = ˆ 1 αj · φ . exp −iβ j=0 |αj |2 Quantum group symmetry on the half-line – p.16/33
  18. 18. Non-local Charges [Bernard & LeClair, Commun. Math. Phys. 142 (1991) 99] ∞ ∞ 1 ¯j = 1 ¯ ¯ Qj = dx (Jj − Hj ) , Q dx (Jj − Hj ), 4πc −∞ 4πc −∞ where Jj =: exp 2i ¯ 2i ˆα β j · ϕ :, Jj =: exp ˆα β j · ϕ :, ¯ ˆ Hj = β2 λ β 2 −2 : exp i 2 ˆ ˆ − β αj · ϕ − iβαj · ϕ : , ¯ ˆ ˆ β ˆ ¯ j = λ β 2 : exp i H 2 ˆ ˆ − β αj · ϕ − iβαj · ϕ :, ¯ ˆ β 2 −2 ˆ β for j = 0, 1, . . . , n. Quantum group symmetry on the half-line – p.17/33
  19. 19. Quantum Affine Algebra Together with the topological charge βˆ ∞ Tj = dx αj · ∂x φ 2π −∞ they generate the quantum affine algebra Uq (ˆ) with relations g [Ti , Qj ] = αi · αj Qj , ¯ [Ti , Qj ] = −αi · αj Qj ¯ ¯ q 2Ti − 1 Qi Qj − q −αi ·αj Qj Qi = δij 2 , qi − 1 where qi = q αi ·αi /2 , as well as the Serre relations. [Felder & LeClair, Int.J.Mod.Phys. A7 (1992) 239] Quantum group symmetry on the half-line – p.18/33
  20. 20. Neumann boundary Any field configuration invariant under x → −x satisfies the Neumann condition ∂x φ = 0 at x = 0. Therefore the field theory on the half line with Neumann boundary condition can be identified with the parity invariant subsector of the theory on the full line. ¯ Parity acts on the non-local charges as Qi → Qi and thus the combinations ˆ ¯ Qi = Q i + Qi are the conserved charges in the theory on the half line. Quantum group symmetry on the half-line – p.19/33
  21. 21. Boundary Perturbation The more general integrable boundary conditions Bowcock, Corrigan, Dorey & Rietdijk, Nucl.Phys.B445 (1995) 469] n ˆ ˜ ˆ iβ ˜ ∂x φ = −iβλb j αj exp − αj · φ j=0 2 are obtained from the action λb S = SNeumann + dt Φpert boundary (t), 2π where n ˆ iβ ˜ Φpert boundary (t) = j exp − αj · φ(0, t) . j=0 2 Quantum group symmetry on the half-line – p.20/33
  22. 22. Conserved Charges It can now be checked in first order boundary perturbation theory that the charges ¯ Qi = Qi + Qi + ˆi q Ti , where ˆ λb i β 2 ˆi = , ˆ 2πc 1 − β 2 are conserved. They generate the algebra B . Quantum group symmetry on the half-line – p.21/33
  23. 23. Coideal property Using the coproduct ∆(Qi ) = Qi ⊗ 1 + q Ti ⊗ Qi , ∆(Qi ) = Qi ⊗ 1 + q Ti ⊗ Qi , ∆(Ti ) = Ti ⊗ 1 + 1 ⊗ Ti . one calculates ˆ ¯ ˆ ∆(Qi ) = (Qi + Qi ) ⊗ 1 + q Ti ⊗ Qi , which verifies the coideal property ∆(B ) ⊂ A ⊗ B . Quantum group symmetry on the half-line – p.22/33
  24. 24. Calculating Reflection Matrices Using the representation matrices µ ˆ πθ (Qi ) = x ei+1 i + x−1 ei i+1 + ˆi ((q −1 − 1) ei i + (q − 1) ei+1 i+1 + ˆ ˆ the intertwining property Qi K = K Qi gives the following set of linear equations for the entries of the reflection matrix: 0 = ˆi (q −1 − q)K i i + x K i i+1 − x−1 K i+1 i , 0 = K i+1 i+1 − K i i , 0 = ˆi q K i j + x−1 K i+1 j , j = i, i + 1, 0 = ˆi q −1 K j i + x K j i+1 , j = i, i + 1. Quantum group symmetry on the half-line – p.23/33
  25. 25. Solution If all | i | = 1 then one finds the solution i −1 (n+1)/2 −(n+1)/2 k(θ) K i (θ) = q (−q x) − ˆ q (−q x) , q −1 − q K i j (θ) = ˆi · · · ˆj−1 (−q x)i−j+(n+1)/2 k(θ), for j > i, K j i (θ) = ˆi · · · ˆj−1 ˆ (−q x)j−i−(n+1)/2 k(θ), for j > i, which is unique up to an overall numerical factor k(θ). This agrees with Georg Gandenberger’s solution of the reflection equation. If all i = 0 then the solution is diagonal. For other values for the i there are no solutions! Quantum group symmetry on the half-line – p.24/33
  26. 26. Outline • General remarks on quantum group symmetry on the whole line and on the half line • Application to affine Toda theories • Application to principal chiral models • Reconstruction of residual symmetry from reflection matrices Quantum group symmetry on the half-line – p.25/33
  27. 27. Principal Chiral Models 1 L = Tr ∂µ g −1 ∂ µ g 2 G × G symmetry L R jµ = ∂µ g g −1 , jµ = −g −1 ∂µ g, Y (g) × Y (g) symmetry Q(0)a = a j0 dx x 1 a Q(1)a = a j1 dx − f bc b j0 (x) c j0 (y) dy dx 2 Quantum group symmetry on the half-line – p.26/33
  28. 28. Boundary Boundary condition g(0) ∈ H where H ⊂ G such that G/H is a symmetric space. The Lie algebra splits g = h ⊕ k. Writing h-indices as i, j, k, .. and k-indices as p, q, r, ... the conserved charges are (0)i (1)p (1)p 1 h (0)p Q and Q ≡Q + [C2 , Q ], 4 where C2 ≡ γij Q(0)i Q(0)j is the quadratic Casimir operator of g h restricted to h. They generate "twisted Yangian" Y (g, h). Quantum group symmetry on the half-line – p.27/33
  29. 29. Reflection Matrices The reflection matrices have to take the form µ[λ] µ[λ] µ[λ] K (θ) = τ[ν] (θ) P[ν] , V [ν] ⊂V µ ⊗V [λ] where the µ[λ] P[ν] (θ) : V µ ⊗ V [λ] → V [ν] ⊂ V µ ⊗ V [λ] ¯ µ[λ] are Y (g, h) intertwiners. The coefficients can τ[ν] (θ) be determined by the tensor product graph method. [Delius, MacKay and Short, Phys.Lett. B 522(2001)335-344, hep-th/0109115] Quantum group symmetry on the half-line – p.28/33
  30. 30. Outline • General remarks on quantum group symmetry on the whole line and on the half line • Application to affine Toda theories • Application to principal chiral models • Reconstruction of residual symmetry from reflection matrices Quantum group symmetry on the half-line – p.29/33
  31. 31. Reconstruction of symmetry Let us assume that for one particular representation Vθµ we know the reflection matrix K µ (θ) : Vθµ → V−θ . We define the µ ¯ corresponding A-valued L-operators in terms of the universal R-matrix R of A, Lµ = (πθ ⊗ id) (R) ∈ End(Vθµ ) ⊗ A, θ µ Lµ = π−θ ⊗ id (Rop ) ∈ End(V−θ ) ⊗ A. ¯¯ θ µ ¯ µ ¯ From these L-operators we construct the matrices µ ¯¯ Bθ = Lµ (K µ (θ) ⊗ 1) Lµ ∈ End(Vθµ , V−θ ) ⊗ A. µ ¯ θ θ Quantum group symmetry on the half-line – p.30/33
  32. 32. Generators for B Introducing matrix indices: ¯¯ (Bθ )α β = (Lµ )α γ (K µ (θ))γ δ (Lµ )δ β ∈ A. µ θ θ µ We find that for all θ the (Bθ )α β are elements of the coideal subalgebra B which commutes with the reflection matrices. It is easy to check the oideal property: µ ¯¯ ∆ ((Bθ )α β ) = (Lµ )α δ (Lµ )σ β ⊗ (Bθ )δ σ , µ θ θ Also any K ν (θ ) : Vθν → V−θ which satisfies the appropriate ν ¯ reflection equation commutes with the action of the elements µ (Bθ )α β µ µ K ν (θ ) ◦ πθ ((Bθ )α β ) = π−θ ((Bθ )α β ) ◦ K ν (θ ), ν ν ¯ Quantum group symmetry on the half-line – p.31/33
  33. 33. Charges in affine Toda Applying the above construction to the vector solitons in affine Toda theory and expanding in powers of x = eθ gives n µ Bθ = B + x ¯ (q −1 − q) el+1 l ⊗ Ql + Ql + ˆl q Tl + O(x2 ). l=0 This shows that the charges were correct to all orders. Note that the B-matrices satisfy the quadratic relations 1 2 2 ˇ ¯¯ νµ ˇ P R (θ−θ )P µ Bθ Rµ¯ (θ+θ ν ) Bθν = Bθν ˇ ¯ ˇ P Rν µ (θ+θ )P Quantum group symmetry on the half-line – p.32/33
  34. 34. Points to remember • Boundary breaks quantum group symmetry A to a subalgebra B . Quantum group symmetry on the half-line – p.33/33
  35. 35. Points to remember • Boundary breaks quantum group symmetry A to a subalgebra B . • B is not a Hopf algebra but a coideal of A. Quantum group symmetry on the half-line – p.33/33
  36. 36. Points to remember • Boundary breaks quantum group symmetry A to a subalgebra B . • B is not a Hopf algebra but a coideal of A. • Reflection matrices are determined by symmetry, no need to solve the reflection equation. Quantum group symmetry on the half-line – p.33/33
  37. 37. Points to remember • Boundary breaks quantum group symmetry A to a subalgebra B . • B is not a Hopf algebra but a coideal of A. • Reflection matrices are determined by symmetry, no need to solve the reflection equation. • Boundary parameters in affine Toda theory are restricted, otherwise no reflection matrix exists. Quantum group symmetry on the half-line – p.33/33
  38. 38. Points to remember • Boundary breaks quantum group symmetry A to a subalgebra B . • B is not a Hopf algebra but a coideal of A. • Reflection matrices are determined by symmetry, no need to solve the reflection equation. • Boundary parameters in affine Toda theory are restricted, otherwise no reflection matrix exists. • Symmetry algebras B are reflection equation algebras as defined by Sklyanin. Quantum group symmetry on the half-line – p.33/33
  39. 39. Points to remember • Boundary breaks quantum group symmetry A to a subalgebra B . • B is not a Hopf algebra but a coideal of A. • Reflection matrices are determined by symmetry, no need to solve the reflection equation. • Boundary parameters in affine Toda theory are restricted, otherwise no reflection matrix exists. • Symmetry algebras B are reflection equation algebras as defined by Sklyanin. • Twisted Yangians Y (g, h) appear as symmetry algebra in principal chiral models with boundary. Quantum group symmetry on the half-line – p.33/33

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