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Quantum group symmetry on the half-lineTalk at CPT, Durham, 11 January 2001

Gustav W [email protected]

Department of Mathematics, University of York

Quantum group symmetry on the half-line – p.1/33

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 SymmetryLet 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 Bε

• Residual symmetry algebra Bε is coideal• Reflection matrices are determined by their

intertwining property• Boundary bound states span representations of Bε


Action on Particles

Let V µθ be the space spanned by the particles in

multiplet µ with rapidity θ. Each V µθ carries a

representation πµθ : A → End(V µ

θ ).

Asymptotic two-particle states span tensor productspaces V µ

θ ⊗ V νθ′ . The symmetry acts on these through

the coproduct ∆ : A → A⊗A.


Action on Particles

Let V µθ be the space spanned by the particles in

multiplet µ with rapidity θ. Each V µθ carries a

representation πµθ : A → End(V µ

θ ).

Asymptotic two-particle states span tensor productspaces V µ

θ ⊗ V νθ′ . The symmetry acts on these through

the coproduct ∆ : A → A⊗A.


S-matrix as intertwinerThe S-matrix has to commute with the action of anysymmetry charge Q ∈ A,

V µθ ⊗ V ν

θ′

(πµθ ⊗πν

θ′)(∆(Q))

−−−−−−−−−→ V µθ ⊗ V ν

θ′ySµν(θ−θ′)

ySµν(θ−θ′)

V νθ′ ⊗ V µ

θ

(πνθ′⊗π

µθ )(∆(Q))

−−−−−−−−−→ V νθ′ ⊗ V µ

θ

This determines the S-matrix uniquely up to an overallfactor (which is then fixed by unitarity, crossing symmetry and

closure of the bootstrap).


Yang-Baxter equationSchur’s lemma implies that the S-matrix satisfies theYang-Baxter equation.

V µθ ⊗ V ν

θ′ ⊗ V λθ′′

Sµν(θ−θ′)⊗ id−−−−−−−−→ V ν

θ′ ⊗ V µθ ⊗ V λ

θ′′yid⊗Sνλ(θ′−θ′′) id⊗Sµλ(θ−θ′′)

yV µ

θ ⊗ V λθ′′ ⊗ V ν

θ′ V νθ′ ⊗ V λ

θ′′ ⊗ V µθySµλ(θ−θ′′)⊗ id Sνλ(θ′−θ′′)⊗id

y

V λθ′′ ⊗ V µ

θ ⊗ V νθ′

id⊗Sµν(θ−θ′)−−−−−−−−→ V λ

θ′′ ⊗ V νθ′ ⊗ V µ

θ


On the half-lineLet us now impose an integrable boundary condition.This will break the symmetry to a subalgebra Bε ⊂ A.

On the half-line a particle with positive rapidity θ willeventually hit the boundary and be reflected intoanother particle with opposite rapidity −θ. This isdescribed by the reflection matrices

Kµ(θ) : V µθ → V µ

−θ.


Reflection Matrix as IntertwinerThe reflection matrix has to commute with the actionof any symmetry charge Q ∈ Bε ⊂ A,

V µθ

πµθ (Q)

−−−→ V µθyKµ(θ)

yKµ(θ)

V µ−θ

πµ−θ(Q)

−−−−→ V µ−θ

If the residual symmetry algebra Bε is "large enough"then this determines the reflection matrices uniquelyup to an overall factor.


Coideal propertyThe residual symmetry algebra Bε does not have to bea Hopf algebra. However it must be a left coideal ofA in the sense that

∆(Q) ∈ A⊗ B for all Q ∈ Bε.

This allows it to act on multi-soliton states.


The Reflection EquationThe reflection equation is again a consequence ofSchur’s lemma

V µθ ⊗ V ν

θ′id⊗Kν(θ′)−−−−−−→ V µ

θ ⊗ V ν−θ′ySµν(θ−θ′) Sµν(θ+θ′)

yV ν

θ′ ⊗ V µθ V ν

−θ′ ⊗ V µθyid⊗Kµ(θ)

yid⊗Kµ(θ)

V νθ′ ⊗ V µ

−θ V ν−θ′ ⊗ V µ

−θySνµ(θ+θ′) Sνµ(θ−θ′)

y

V µ−θ ⊗ V ν

θ′id⊗Kν(θ′)−−−−−−→ V µ

−θ ⊗ V ν−θ′


Mathematical ProblemGiven A find its coideal subalgebras B such that for aset of representations on has that

• tensor products V µθ ⊗ V ν

θ′ are genericallyirreducible,

• intertwiners Kµ(θ) : V µθ → V µ

−θ exist.

Physical ProblemFind the boundary condition corresponding to B.


Boundary Bound StatesParticles can bind to the boundary, creating multipletsof boundary bound states. These span representationsV [λ] of the symmetry algebra Bε. The reflection ofparticles off these boundary bound states is describedby intertwiners

Kµ[λ](θ) : V µθ ⊗ V [λ] → V µ

−θ ⊗ V [λ].


Quantum Group SymmetryLet 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 Bε

• Residual symmetry algebra Bε is coideal• Reflection matrices are determined by their

intertwining property• Boundary bound states span representations of Bε




reflection matrices


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


Toda Action

S =1

4π

∫d2z ∂φ∂φ +

λ

2π

∫d2z Φpert,

where

Φpert =n∑

j=0

exp

(−iβ

1

|αj|2αj · φ

).


Non-local Charges[Bernard & LeClair, Commun. Math. Phys. 142 (1991) 99]

Qj =1

4πc

∫∞

−∞

dx (Jj −Hj) , Qj =1

4πc

∫∞

−∞

dx (Jj − Hj),

where

Jj =: exp(

2i

βαj · ϕ

): , Jj =: exp

(2i

βαj · ϕ

): ,

Hj = λ β2

β2−2

: exp(i(

2

β− β

)αj · ϕ − iβαj · ϕ

): ,

Hj = λ β2

β2−2

: exp(i(

2

β− β

)αj · ϕ − iβαj · ϕ

):,

for j = 0, 1, . . . , n.


Quantum Affine AlgebraTogether with the topological charge

Tj =β

2π

∫∞

−∞

dx αj · ∂xφ

they generate the quantum affine algebra Uq(g) with relations

[Ti, Qj] = αi · αj Qj, [Ti, Qj] = −αi · αj Qj

QiQj − q−αi·αjQjQi = δijq2Ti − 1

q2i − 1

,

where qi = qαi·αi/2, as well as the Serre relations.

[Felder & LeClair, Int.J.Mod.Phys. A7 (1992) 239]


Neumann boundaryAny field configuration invariant under x → −xsatisfies the Neumann condition ∂xφ = 0 at x = 0.Therefore the field theory on the half line withNeumann boundary condition can be identified withthe parity invariant subsector of the theory on the fullline.Parity acts on the non-local charges as Qi 7→ Qi andthus the combinations

Qi = Qi + Qi

are the conserved charges in the theory on the halfline.


Boundary PerturbationThe more general integrable boundary conditions

Bowcock, Corrigan, Dorey & Rietdijk, Nucl.Phys.B445 (1995) 469]

∂xφ = −iβλb

n∑

j=0

εjαj exp

(−

iβ

2αj · φ

)

are obtained from the action

Sε = SNeumann +λb

2π

∫dt Φpert

boundary(t),

where

Φpertboundary(t) =

n∑

j=0

εj exp

(−

iβ

2αj · φ(0, t)

).


Conserved ChargesIt can now be checked in first order boundaryperturbation theory that the charges

Qi = Qi + Qi + εiqTi,

where

εi =λbεi

2πc

β2

1 − β2,

are conserved. They generate the algebra Bε.


Coideal propertyUsing the coproduct

∆(Qi) = Qi ⊗ 1 + qTi ⊗ Qi,

∆(Qi) = Qi ⊗ 1 + qTi ⊗ Qi,

∆(Ti) = Ti ⊗ 1 + 1 ⊗ Ti.

one calculates

∆(Qi) = (Qi + Qi) ⊗ 1 + qTi ⊗ Qi,

which verifies the coideal property

∆(Bε) ⊂ A⊗ Bε.


Calculating Reflection MatricesUsing the representation matrices

πµθ (Qi) = x ei+1

i + x−1 eii+1 + εi ((q

−1 − 1) eii + (q − 1) ei+1

i+1 + 1)

the intertwining property Qi K = K Qi gives the following setof linear equations for the entries of the reflection matrix:

0 = εi(q−1 − q)K i

i + x K ii+1 − x−1 Ki+1

i,

0 = K i+1i+1 − K i

i,

0 = εi q Kij + x−1 Ki+1

j, j 6= i, i + 1,

0 = εi q−1 Kj

i + x Kji+1, j 6= i, i + 1.


SolutionIf all |εi| = 1 then one finds the solution

Kii(θ) =

(q−1 (−q x)(n+1)/2 − ε q (−q x)−(n+1)/2

) k(θ)

q−1 − q,

Kij(θ) = εi · · · εj−1 (−q x)i−j+(n+1)/2 k(θ), for j > i,

Kji(θ) = εi · · · εj−1ε (−q x)j−i−(n+1)/2 k(θ), for j > i,

which is unique up to an overall numerical factor k(θ). Thisagrees with Georg Gandenberger’s solution of the reflectionequation.If all εi = 0 then the solution is diagonal.

For other values for the εi there are no solutions!




reflection matrices


Principal Chiral Models

L =1

2Tr(∂µg

−1∂µg)

G × G symmetry

jLµ = ∂µg g−1, jR

µ = −g−1∂µg,

Y (g) × Y (g) symmetry

Q(0)a =

∫ja0 dx

Q(1)a =

∫ja1dx −

1

2fa

bc

∫jb0(x)

∫ x

jc0(y) dy dx


BoundaryBoundary condition g(0) ∈ H where H ⊂ G such that G/H is asymmetric space. The Lie algebra splits g = h ⊕ k. Writingh-indices as i, j, k, .. and k-indices as p, q, r, ... the conservedcharges are

Q(0)i and Q(1)p ≡ Q(1)p +1

4[Ch

2 , Q(0)p],

where Ch

2 ≡ γijQ(0)iQ(0)j is the quadratic Casimir operator of g

restricted to h. They generate "twisted Yangian" Y (g,h).


Reflection MatricesThe reflection matrices have to take the form

Kµ[λ](θ) =∑

V [ν]⊂V µ⊗V [λ]

τµ[λ][ν] (θ) P

µ[λ][ν] ,

where the

Pµ[λ][ν] (θ) : V µ ⊗ V [λ] → V [ν] ⊂ V µ ⊗ V [λ]

are Y (g, h) intertwiners. The coefficients τµ[λ][ν] (θ) can

be determined by the tensor product graph method.[Delius, MacKay and Short, Phys.Lett. B 522(2001)335-344,

hep-th/0109115]




reflection matrices


Reconstruction of symmetryLet us assume that for one particular representation V µ

θ we knowthe reflection matrix Kµ(θ) : V µ

θ → V µ−θ. We define the

corresponding A-valued L-operators in terms of the universalR-matrix R of A,

Lµθ = (πµ

θ ⊗ id) (R) ∈ End(V µθ ) ⊗A,

Lµθ =

(πµ−θ ⊗ id

)(Rop) ∈ End(V µ

−θ) ⊗A.

From these L-operators we construct the matrices

Bµθ = Lµ

θ (Kµ(θ) ⊗ 1) Lµθ ∈ End(V µ

θ , V µ−θ) ⊗A.


Generators for BIntroducing matrix indices:

(Bµθ )α

β = (Lµθ )α

γ(Kµ(θ))γ

δ(Lµθ )δ

β ∈ A.

We find that for all θ the (Bµθ )α

β are elements of the coidealsubalgebra B which commutes with the reflection matrices.It is easy to check the oideal property:

∆ ((Bµθ )α

β) = (Lµθ )α

δ(Lµθ )σ

β ⊗ (Bµθ )δ

σ,

Also any Kν(θ′) : V νθ′ → V ν

−θ′ which satisfies the appropriatereflection equation commutes with the action of the elements(Bµ

θ )αβ

Kν(θ′) ◦ πνθ′((B

µθ )α

β) = πν−θ′((B

µθ )α

β) ◦ Kν(θ′),


Charges in affine TodaApplying the above construction to the vector solitonsin affine Toda theory and expanding in powers ofx = eθ gives

Bµθ = B + x

n∑

l=0

(q−1 − q) el+1l ⊗(Ql + Ql + εl q

Tl)

+ O(x2).

This shows that the charges were correct to all orders.Note that the B-matrices satisfy the quadraticrelations

PRνµ(θ−θ′)P1

Bµθ Rµν(θ+θ′)

2

Bνθ′ =

2

Bνθ′ PRνµ(θ+θ′)P

1

Bµθ Rµν(θ−θ′),


Points to remember• Boundary breaks quantum group symmetry A to

a subalgebra Bε.

• Bε 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 Bε are reflection equation

algebras as defined by Sklyanin.• Twisted Yangians Y (g,h) appear as symmetry

algebra in principal chiral models with boundary.



a subalgebra Bε.• Bε 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 arerestricted, otherwise no reflection matrix exists.

• Symmetry algebras Bε are reflection equationalgebras as defined by Sklyanin.

• Twisted Yangians Y (g,h) appear as symmetryalgebra in principal chiral models with boundary.



a subalgebra Bε.• Bε 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 arerestricted, otherwise no reflection matrix exists.







restricted, otherwise no reflection matrix exists.








algebras as defined by Sklyanin.







algebras as defined by Sklyanin.• Twisted Yangians Y (g,h) appear as symmetry

algebra in principal chiral models with boundary.


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