Quantizing SL(N) Solitons and the Hecke Alegbra

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 semi-classical limit is consistent with the semi-classical 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 non-unitary theory, which reflects the fact that the classical Hamiltonian is complex. The spectrum of the theory is found to consist of the basic solitons, scalar states (or breathers) and excited (or `breathing') solitons. 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, in which case the theory is unitary.