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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Points to remember
• Boundary breaks quantum group symmetry A to
a subalgebra B .
Quantum group symmetry on the half-line – p.33/33

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. 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. 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. 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. 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