Quantizing SL(N) Solitons and the Hecke Alegbra

TL;DR

This paper develops a quantum scattering theory for complex affine Toda solitons in two dimensions, proposing an $S$-matrix linked to quantum groups and Hecke algebra representations, revealing non-unitary and unitary regimes.

Contribution

It introduces a novel $S$-matrix construction for $sl(n)$ affine Toda solitons based on quantum groups and Hecke algebra representations, extending classical scattering analysis to the quantum domain.

Findings

Proposed an $S$-matrix intertwining quantum group symmetries.

Identified the spectrum including solitons, breathers, and excited states.

Showed the theory's non-unitarity and conditions for unitarity in restricted models.

Abstract

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.