SOLITONS, BOUNDARIES, AND QUANTUM AFFINE ALGEBRAS

GUSTAV W DELIUSAbstract. This is a condensed write-up of a talk delivered at the Ramanu-jan International Symposium on Kac-Moody Lie algebras and Applications inChennai in January 2002. The talk introduces special coideal subalgebras ofquantum affine algebras which appear in physics when solitons are restrictedto live on a half-line by an integrable boundary condition. We review how thequantum affine symmetry determines the soliton S-matrix in affine Toda fieldtheory and then go on to use the unbroken coideal subalgebra on the half-lineto determine the soliton reflection matrix. This gives a representation theo-retic method for the solution of the reflection equation (boundary Yang-Baxterequation) by reducing it to a linear equation.

[1]  P. Baseilhac,et al.  Sine-Gordon quantum field theory on the half-line with quantum boundary degrees of freedom , 2002, hep-th/0208005.

[2]  P. Sorba,et al.  Coideal subalgebras in quantum affine algebras , 2002, math/0208140.

[3]  G. Delius,et al.  Quantum Affine Reflection Algebras of Type $$d_n^{\left( 1 \right)}$$ and Reflection Matrices , 2002, math/0208043.

[4]  A. Doikou,et al.  Hecke algebraic approach to the reflection equation for spin chains , 2002, hep-th/0206076.

[5]  N. Mackay.  Rational K-matrices and representations of twisted Yangians , 2002, math/0205155.

[6]  Rafael I. Nepomechie,et al.  LETTER TO THE EDITOR: Solutions of the boundary Yang-Baxter equation for arbitrary spin , 2002, hep-th/0204076.

[7]  G. Delius,et al.  Quantum Group Symmetry in sine-Gordon and Affine Toda Field Theories on the Half-Line , 2001, hep-th/0112023.

[8]  G. Delius,et al.  Boundary remnant of Yangian symmetry and the structure of rational reflection matrices , 2001, hep-th/0109115.

[9]  M. Reid,et al.  Solitons : differential equations, symmetries and infinite dimensional algebras , 2000 .

[10]  G. M. Gandenberger.  New non-diagonal solutions to the $a_n^{(1)}$ boundary Yang-Baxter equation , 1999, hep-th/9911178.

[11]  A. Lima-Santos.  Reflection K-matrices for 19-vertex models , 1999, solv-int/9906003.

[12]  Ke Wu,et al.  Classification of solutions to the reflection equation for two-component systems , 1998, hep-th/9808083.

[13]  G. Delius.  Soliton-preserving boundary condition in affine Toda field theories , 1998, hep-th/9809140.

[14]  G. Delius.  Restricting affine Toda theory to the half-line , 1998, hep-th/9807189.

[15]  C. Ahn,et al.  Complete non-diagonal reflection matrices of RSOS/SOS and hard hexagon models , 1997, solv-int/9710024.

[16]  V. Fridkin,et al.  Solutions of the reflection equation for face and vertex models associated with An(1), Bn(1), Cn(1), Dn(1) and An(2) , 1996, hep-th/9601051.

[17]  P. Pearce,et al.  A construction of solutions to reflection equations for interaction-round-a-face models , 1995, hep-th/9512218.

[18]  G. Watts,et al.  Twisted algebra R-matrices and S-matrices forbn(1) affine Toda solitons and their bound states , 1995, hep-th/9509007.

[19]  E. Corrigan,et al.  Background field boundary conditions for affine Toda field theories , 1995, hep-th/9510071.

[20]  N. Mackay,et al.  Exact S-matrices for d (2) n+1 affine Toda solitons and their bound states , 1995, hep-th/9506169.

[21]  E. Corrigan,et al.  Classically integrable boundary conditions for affine Toda field theories , 1995, hep-th/9501098.

[22]  N. Reshetikhin,et al.  Quantum Groups , 1993, hep-th/9311069.

[23]  G. Delius,et al.  On the construction of trigonometric solutions of the Yang-Baxter equation , 1994, hep-th/9405030.

[24]  H. Vega,et al.  EXACT BETHE ANSATZ SOLUTION FOR An−1 CHAINS WITH NON-SUq(n) INVARIANT OPEN BOUNDARY CONDITIONS , 1994, hep-th/9404141.

[25]  A. Zamolodchikov,et al.  Boundary S matrix and boundary state in two-dimensional integrable quantum field theory , 1993, hep-th/9306002.

[26]  N. Turok,et al.  Affine Toda solitons and vertex operators , 1993, hep-th/9305160.

[27]  T. Hollowood.  Quantizing sl(n) solitons and the Hecke algebra , 1992, hep-th/9203076.

[28]  Nicolai Reshetikhin,et al.  Quantum Groups , 1993 .

[29]  D. Bernard,et al.  The quantum double in integrable quantum field theory , 1992, hep-th/9205064.

[30]  T. Hollowood.  Solitons in affine Toda field theories , 1991, hep-th/9110010.

[31]  D. Bernard,et al.  Quantum group symmetries and non-local currents in 2D QFT , 1991 .

[32]  E. Sklyanin.  Boundary conditions for integrable quantum systems , 1988 .

[33]  M. Jimbo.  QuantumR matrix for the generalized Toda system , 1986 .

[34]  神保 道夫.  Quantum R matrix for the generalized Toda system , 1986 .

[35]  N. Turok,et al.  Local conserved densities and zero-curvature conditions for Toda lattice field theories , 1985 .

[36]  I. Cherednik.  Factorizing particles on a half-line and root systems , 1984 .

[37]  R. Rajaraman.  Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory , 1982 .

[38]  Alexander B. Zamolodchikov,et al.  Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models , 1979 .