Qghl
-
Upload
gustavdelius -
Category
Technology
-
view
345 -
download
2
description
Transcript of Qghl
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 symmetry on the half-line – p.2/33
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
Quantum group symmetry on the half-line – p.3/33
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.
Quantum group symmetry on the half-line – p.4/33
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.
Quantum group symmetry on the half-line – p.4/33
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).
Quantum group symmetry on the half-line – p.5/33
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
Quantum group symmetry on the half-line – p.6/33
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.
Quantum group symmetry on the half-line – p.7/33
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.
Quantum group symmetry on the half-line – p.8/33
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.
Quantum group symmetry on the half-line – p.9/33
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′
Quantum group symmetry on the half-line – p.10/33
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.
Quantum group symmetry on the half-line – p.11/33
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 symmetry on the half-line – p.12/33
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
Quantum group symmetry on the half-line – p.13/33
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 symmetry on the half-line – p.14/33
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
Toda Action
S =1
4π
∫d2z ∂φ∂φ +
λ
2π
∫d2z Φpert,
where
Φpert =n∑
j=0
exp
(−iβ
1
|αj|2αj · φ
).
Quantum group symmetry on the half-line – p.16/33
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 group symmetry on the half-line – p.17/33
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]
Quantum group symmetry on the half-line – p.18/33
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.
Quantum group symmetry on the half-line – p.19/33
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)
).
Quantum group symmetry on the half-line – p.20/33
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.
Quantum group symmetry on the half-line – p.21/33
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.
Quantum group symmetry on the half-line – p.22/33
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.
Quantum group symmetry on the half-line – p.23/33
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!
Quantum group symmetry on the half-line – p.24/33
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 symmetry on the half-line – p.25/33
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
Quantum group symmetry on the half-line – p.26/33
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).
Quantum group symmetry on the half-line – p.27/33
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]
Quantum group symmetry on the half-line – p.28/33
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 symmetry on the half-line – p.29/33
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.
Quantum group symmetry on the half-line – p.30/33
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′),
Quantum group symmetry on the half-line – p.31/33
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′),
Quantum group symmetry on the half-line – p.32/33
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.
Quantum group symmetry on the half-line – p.33/33
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 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.
Quantum group symmetry on the half-line – p.33/33
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 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.
Quantum group symmetry on the half-line – p.33/33
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 equationalgebras as defined by Sklyanin.
• Twisted Yangians Y (g,h) appear as symmetryalgebra in principal chiral models with boundary.
Quantum group symmetry on the half-line – p.33/33
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 symmetryalgebra in principal chiral models with boundary.
Quantum group symmetry on the half-line – p.33/33
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.
Quantum group symmetry on the half-line – p.33/33