QUANTIZING SL(N) SOLITONS AND THE HECKE ALGEBRA

The problem of quantizing a class of two-dimensional integrable quantum field theories is considered. The classical equations of the theory are the complex sl(n) affine Toda equations which admit soliton solutions with real masses. The classical scattering theory of the solitons is developed using Hirota's solution techniques. A form for the soliton S matrix is proposed based on the constraints of S matrix theory, integrability and the requirement that the semiclassical limit is consistent with the semiclassical WKB quantization of the classical scattering theory. The proposed S matrix is an intertwiner of the quantum group associated to sl(n), where the deformation parameter is a function of the coupling constant. It is further shown that the S matrix describes a nonunitary theory, which reflects the fact that the classical Hamiltonian is complex. The spectrum of the theory is found to consist of the basic solitons, excited (or 'breathing') solitons, scalar states (or breathers) and solitons transforming in nonfundamental representations. For some region of coupling constant space only the original solitons are in the spectrum and so the S matrix is complete, in addition arguments are presented which indicate that in a more restricted region the theory is actually unitary. It is also noted that the construction of the S matrix is valid for any representation of the Hecke algebra, allowing the definition of restricted S matrices, which lie in the unitary and complete region.