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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.
Solitons, Boundaries,and Quantum Affine Algebras – p.2/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.4/27
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 .
Solitons, Boundaries,and Quantum Affine Algebras – p.5/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.6/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.8/27
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).
Solitons, Boundaries,and Quantum Affine Algebras – p.9/27
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
Solitons, Boundaries,and Quantum Affine Algebras – p.10/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.11/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.12/27
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
Solitons, Boundaries,and Quantum Affine Algebras – p.13/27
FactorizationThe multi-soliton reflection matrices factorize.
The compatibility condition between the two ways offactorizing the two-soliton reflection matrix is calledthe reflection equation.
Solitons, Boundaries,and Quantum Affine Algebras – p.14/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.15/27
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′
Solitons, Boundaries,and Quantum Affine Algebras – p.16/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.17/27
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!
Solitons, Boundaries,and Quantum Affine Algebras – p.18/27
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].
Solitons, Boundaries,and Quantum Affine Algebras – p.19/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.20/27
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
Solitons, Boundaries,and Quantum Affine Algebras – p.21/27
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).
Solitons, Boundaries,and Quantum Affine Algebras – p.22/27
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]
Solitons, Boundaries,and Quantum Affine Algebras – p.23/27
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).
Solitons, Boundaries,and Quantum Affine Algebras – p.24/27
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′),
Solitons, Boundaries,and Quantum Affine Algebras – p.25/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.26/27
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.
Solitons, Boundaries,and Quantum Affine Algebras – p.27/27