The Full Set of C N -invariant Factorized S-matrices

We use the method of the tensor product graph to construct rational (Yangian invariant) solutions of the Yang-Baxter equation in fundamental representations of cn and thence the full set of cn-invariant factorized S-matrices. February 1, 2008 Supported by a U.K. Science and Engineering Research Council studentship. Address from 1st August: RIMS, Kyoto Univ., Kyoto 606, Japan. Integrable quantum field theories in 1+1 dimensions are expected to have exact Smatrices in which particle number and the set of asymptotic momenta are conserved, and in which multiparticle interactions factorize into products of two-particle interactions. The condition that this factorization be consistent is then the Yang-Baxter equation (YBE), so that in theories with a global Lie group invariance, such as the principal chiral model, the S-matrices are constructed from group-invariant solutions of the YBE. The spectrum of the theory then consists of multiplets within which the particles have equal mass and which form representations of the group G. One method for constructing these S-matrices is the bootstrap procedure (known as the ‘fusion procedure’ for solutions of the YBE) in which, at appropriate poles, intermediate states of the S-matrix are identified as particle states whose S-matrices can then be constructed. An alternative method is to construct explicitly the action of the underlying charge algebra on particle multiplets, and then use conservation of these charges to deduce the S-matrix. Bernard recently showed that this algebra is precisely Drinfeld’s Yangian Y (A), where A is the Lie algebra of the group G: if we write the action of the charges on states consisting of two asymptotically free particles as ∆, there is a local charge satisfying