The Analytic Structure of Trigonometric S Matrices

TL;DR

This paper constructs and analyzes the analytic structure of $S$-matrices for quantum group representations of classical Lie algebras, revealing inconsistencies in naive generalizations and proposing a consistent form based on the Coleman-Thun mechanism.

Contribution

It provides explicit constructions of $S$-matrices for vector representations of quantum groups and clarifies their analytic properties across different Lie algebra types.

Findings

Complete $S$-matrices for $a_{m-1}$ and $c_m$ are derived.

Naive generalizations of the Gross-Neveu $S$-matrix are inconsistent for non-simply-laced algebras.

A consistent $S$-matrix form for $c_m$ is proposed using the Coleman-Thun mechanism.

Abstract

$S$-matrices associated to the vector representations of the quantum groups for the classical Lie algebras are constructed. For the $a_{m-1}$ and $c_m$ algebras the complete $S$-matrix is found by an application of the bootstrap equations. It is shown that the simplest form for the $S$-matrix which generalizes that of the Gross-Neveu model is not consistent for the non-simply-laced algebras due to the existence of unexplained singularities on the physical strip. However, a form which generalizes the $S$-matrix of the principal chiral model is shown to be consistent via an argument which uses a novel application of the Coleman-Thun mechanism. The analysis also gives a correct description of the analytic structure of the $S$-matrix of the principle chiral model for $c_m$.