Bi-module Properties of Group-Valued Local Fields and Quantum-Group Difference Equations

TL;DR

This paper constructs explicit quantum-group generators in WZNW theory, demonstrating their bi-module properties and deriving quantum-group difference equations for correlation functions, with solutions provided for two-point functions.

Contribution

It provides an explicit construction of quantum-group generators in WZNW theory and shows their bi-module properties and associated difference equations.

Findings

Correlation functions satisfy quantum-group difference equations.

Explicit solutions for two-point functions are obtained.

Framework applicable to quantum SDYM theory.

Abstract

We give an explicit construction of the quantum-group generators ---local, semi-local, and global --- in terms of the group-valued quantum fields $\tilde g$ and $\tilde g^{-1}$ in the Wess-Zumino-Novikov-Witten (WZNW) theory. The algebras among the generators and the fields make concrete and clear the bi-module properties of the $\tilde g$ and the $\tilde g^{-1}$ fields. We show that the correlation functions of the $\tilde g$ and $\tilde g^{-1}$ fields in the vacuum state defined through the semi-local quantum-group generator satisfy a set of quantum-group difference equations. We give the explicit solution for the two point function. A similar formulation can also be done for the quantum Self-dual Yang-Mills (SDYM) theory in four dimensions.