iskmaa

Solitons, Boundaries,and Quantum Affine Algebras

Talk at ISKMAA 2002Chennai, 29 January 2001

Gustav W [email protected]

Department of MathematicsUniversity of YorkUnited Kingdom

Solitons, Boundaries,and Quantum Affine Algebras – p.1/27

Aims• To show you an application of quantum affine

algebras Uq(g) and Yangians Y (g),

namely the calculation of soliton scattering amplitudes.

• To present an algebraic method to find solutionsof the reflection equation.

• To motivate you to study certain coidealsubalgebras of Uq(g) and Y (g)

so that I can calculate reflection matrices and learn aboutboundary solitons.


Affine Toda field theoryAssociated to every affine Kac-Moody algebra g thereis a relativistic field equation

∂2xφ − ∂2

t φ =1

2i

n∑

j=0

ηj αj ei(αj ,φ)

where φ = φ(x, t) takes values in the root space of g,α1, . . . , αn are the simple roots of g and α0 = −Ψwhere

Ψ =

{highest root of g (untwisted g)

highest short root of g (twisted g),

( , ) is the Killing form and∑n

j=0 ηjαj = 0.Solitons, Boundaries,and Quantum Affine Algebras – p.3/27

Soliton solutions• There are constant solutions (vacua) with

φ = 2πλ for any fundamental weight λ of g∨.• There are soliton solutions which interpolate

between these vacua. They are stable due to thetopological charge

T [φ] = φ(∞) − φ(−∞) = 2πλ. (1)

• The solitons fall into multiplets, depending ontheir topological charge.

• Multi-soliton solutions describe the scattering ofsolitons.


Quantum solitons• In the quantum theory we associate particle states

with the soliton solutions.• Let V µ

θ be the space spanned by the solitons inmultiplet µ with rapidity θ.

• Asymptotic two-soliton states span tensorproduct spaces V µ

θ ⊗ V νθ′ .

• An incoming two-soliton state in V µθ ⊗ V ν

θ′ withθ > θ′ will evolve during scattering into anoutgoing state in V ν

θ′ ⊗ V µθ with scattering

amplitudes given by the two-soliton S-matrix

Sµν(θ − θ′) : V µθ ⊗ V ν

θ′ → V νθ′ ⊗ V µ

θ .


FactorizationDue to integrability, the multi-soliton S-matrixfactorizes into a product of two-soliton S-matrices.

The compatibility of the two ways to factorizerequires the S-matrix to be a solution of theYang-Baxter equation.


Quantum affine symmetry[Bernard & LeClair, Commun. Math. Phys. 142 (1991) 99]

The quantum affine Toda theory has symmetrycharges which generate the Uq(g

∨) algebra with

q = e2πi 1−~

~ .

We work with the generators Ti, Qi, Qi, i = 0, . . . , n,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.Solitons, Boundaries,and Quantum Affine Algebras – p.7/27

Action on solitons• Each V µ

θ carries a representationπµ

θ : Uq(g) → End(V µθ ).

• The symmetry acts on these through thecoproduct ∆ : Uq(g) → Uq(g) ⊗ Uq(g).

∆(Qi) = Qi ⊗ 1 + qTi ⊗ Qi,

∆(Qi) = Qi ⊗ 1 + qTi ⊗ Qi,

∆(Ti) = Ti ⊗ 1 + 1 ⊗ Ti.


S-matrix as intertwinerThe S-matrix has to commute with the action of anysymmetry charge Q ∈ Uq(g),

V µθ ⊗ V ν

θ′(πµ

θ ⊗πνθ′

)(∆(Q))−−−−−−−−−→ V µ

θ ⊗ V νθ′ySµν(θ−θ′)

ySµν(θ−θ′)

V νθ′ ⊗ V µ

θ

(πνθ′⊗πµ

θ )(∆(Q))−−−−−−−−−→ V ν

θ′ ⊗ V µθ

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 equationBecause the tensor product representations areirreducible for generic rapidities, Schur’s lemmaimplies that the S-matrix satisfies the Yang-Baxterequation

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

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

∂xφ = i

n∑

j=0

εjαj exp

(i

2(αj, φ)

)

where the εj are free parameters.This will break the symmetry to a subalgebraBε ⊂ Uq(g) generated by

[Delius and MacKay, hep-th/0112023]

Qi = Qi + Qi + εiqTi, i = 0, . . . , n.


Coideal propertyThe residual symmetry algebra Bε does not have to be a Hopfalgebra. However it must be a left coideal of Uq(g) in the sensethat

∆(Q) ∈ Uq(g) ⊗ B for all Q ∈ Bε.

This allows it to act on multi-soliton states.

We calculate

∆(Qi) = (Qi + Qi) ⊗ 1 + qTi ⊗ Qi,

which verifies the coideal property.


The reflection matrixOn 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


FactorizationThe multi-soliton reflection matrices factorize.

The compatibility condition between the two ways offactorizing the two-soliton reflection matrix is calledthe reflection equation.


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

V µθ

πµθ (Q)

−−−→ V µθyKµ(θ)

yKµ(θ)

V µ−θ

πµ−θ(Q)

−−−−→ V µ−θ

If Bε is "large enough" so that V µθ and V µ

−θ are ir-reducible, then the reflection matrices are determineduniquely up to an overall factor.


The Reflection EquationIf Bε is "large enough" so that the tensor products areirreducible, then the reflection equation holds

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′


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!


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].


Mathematical ProblemGiven Uq(g) find its coideal subalgebras B such thatfor a set of representations on has that

• tensor products V µθ ⊗ V ν

θ′ are genericallyirreducible,

• intertwiners Kµ(θ) : V µθ → V µ

−θ exist.

This gives solutions to the reflection equation.


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]


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 Uq(g)-valued L-operators in terms of theuniversal R-matrix R of Uq(g),

Lµθ = (πµ

θ ⊗ id) (R) ∈ End(V µθ ) ⊗ Uq(g),

Lµθ =

(πµ−θ ⊗ id

)(Rop) ∈ End(V µ

−θ) ⊗ Uq(g).

From these L-operators we construct the matrices

Bµθ = Lµ

θ (Kµ(θ) ⊗ 1) Lµθ ∈ End(V µ

θ , V µ−θ) ⊗ Uq(g).


Generators for BIntroducing matrix indices:

(Bµθ )α

β = (Lµθ )α

γ(Kµ(θ))γ

δ(Lµθ )δ

β ∈ Uq(g).

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 coideal 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.The B-matrices can be shown to be reflection equationalgebras in the sense of Slyanin.


Points to remember• Uq(g) and Y (g) appear a symmetry algebras of

certain massive quantum field theories.• The soliton S-matrices are solutions of the

Yang-Baxter equation and can be obtained romthe symmetry

• A boundary breaks the symmetry to a left coidealsubalgebra.

• The soliton reflection matrices are solutions ofthe reflection equation and can be obtained fromthe symmetry.

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


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