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Quantum Group Symmetryon the half line
A study in integrable quantum field theory with a boundary
Talk given on 08/05/02 to theEdinburgh Mathematical Physics Group
Gustav W DeliusDepartment of MathematicsUniversity of York
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Different ways to summarize it
Reflection of solitons off boundaries; Coideal subalgebras of quantum
groups; Multiplet structure of boundary states; Solutions of the reflection equation.
We are studying
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Organization of talk Sine-Gordon soliton scattering and
reflection as a warm-up S-matrices, bound states, and
quantum groups Reflection matrices, boundary
bound states and boundary quantum groups
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Sine-Gordon Solitons
Lagrangian:
Field equation:
Soliton solution:
Cosine potential
Soliton
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Classical Soliton scattering
For example in the sine-Gordon model
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Time advance during scattering
The solitons experiencea time advance while scattering through each other.
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Classical Soliton reflection
For example in the sine-Gordon model
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Method of images
For example in the sine-Gordon model
Saleur,Skorik,Warner, Nucl.Phys.B441(1995)421.
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Time advance during reflection
For an attractiveboundary conditionThe soliton experiencesa time advanceduring reflection.
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Time delay during reflection
For a repulsiveboundary conditionThe soliton experiencesa time advanceduring reflection.
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Quantum amplitudes
Scattering amplitude Reflection amplitude
Solitontype
rapidity
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Factorization
= Yang-Baxterequation
= Reflectionequation
Cherednik, Theor.Math.Phys. 61 (1984) 977Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.
One way to obtain amplitudes is to solve:
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Bound states
breatherBoundary breather
Poles in theamplitudesorrespondingto bound states
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Classical breather solution
Ghoshal & Zamolodchikov, Int.J.Mod.Phys. A9 (1994) 3841.
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Scattering matrix
The solitons with rapidity span representation spaces
Highest weightof representation
rapidity
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Schur’s Lemma
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Quantum Group Symmetry
Theory: Symmetry:
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Tensor product decomposition
Example: fundamental reps of sl(n)
Several irreducible reprs of sl(n) are tied togetherinto a single irreducible representation of
where
At special values of the S-matrix projects onto subrepresentations.
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Tensor product graph for Cn
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Introducing a boundary The boundary condition will break
the quantum group symmetry to a subalgebra
Depends onboundary parameters
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Reflection matrix
Sometimes particle comesback in conjugate representation
Boundary states form multipletsof resdual symmetry algebra
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Coideal subalgebra
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Boundary quantum groups
Trigonometric:
Realized in affine Toda field theory with boundary condition
Derived using boundary conformal perturbation theory
Delius, MacKay, hep-th/0112023
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Boundary quantum group
Rational:
Obtained from principal chiral model on G withthe field at the boundary constrained to lie in H.
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Boundary bound states
where
Delius, MacKay, Short, Phys.Lett. B 522(2001)335-344.
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Three things to remember Boundary breaks quantum group
symmetry to a coideal subalgebra. Solutions of reflection equation can
now be obtained from symmetry. Spectrum of boundary states is
determined to branching rules.
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Affine Toda theory
Generalize the sine-Gordon potential
For example sl(3):Simple roots of affine Lie algebra
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Affine Toda solitons
In this case there are six fundamental solitons interpolating along the green and the blue arrows.
Example sl(3):
In general it is believed that the solitons fill out the fundamental representationsof the Lie algebra.
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Affine Toda theory action
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Nonlocal charges
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Quantum affine algebra