Infinite Quantum Group Symmetry of Fields in Massive 2D Quantum Field Theory

TL;DR

This paper develops a framework to extend quantum group symmetries to fields in 1+1 dimensional integrable quantum field theories, revealing an infinite-dimensional symmetry structure involving Yangians and quantum doubles.

Contribution

It introduces a method to incorporate quantum group symmetries into the space of fields, including the form factors of descendants and the braiding relations via the universal R-matrix.

Findings

Quantum double of the Yangian is a Hopf algebra deformation of a Kac-Moody algebra.

Fields form infinite-dimensional Verma modules, including energy-momentum tensor and currents.

The braiding relations are governed by the universal R-matrix.

Abstract

Starting from a given S-matrix of an integrable quantum field theory in $1+1$ dimensions, and knowledge of its on-shell quantum group symmetries, we describe how to extend the symmetry to the space of fields. This is accomplished by introducing an adjoint action of the symmetry generators on fields, and specifying the form factors of descendents. The braiding relations of quantum field multiplets is shown to be given by the universal $\CR$-matrix. We develop in some detail the case of infinite dimensional Yangian symmetry. We show that the quantum double of the Yangian is a Hopf algebra deformation of a level zero Kac-Moody algebra that preserves its finite dimensional Lie subalgebra. The fields form infinite dimensional Verma-module representations; in particular the energy-momentum tensor and isotopic current are in the same multiplet.