The K-Z Equation and the Quantum-Group Difference Equation in Quantum Self-dual Yang-Mills Theory

TL;DR

This paper constructs new quantum fields in self-dual Yang-Mills theory that satisfy exchange algebras, deriving K-Z equations with spatial dependence and quantum-group difference equations, and provides explicit solutions for two-point functions.

Contribution

It introduces new group-valued quantum fields satisfying exchange algebras and derives associated K-Z and quantum-group difference equations in SDYM theory.

Findings

Derived K-Z equations with spatial dependence on ar k.

Constructed quantum-group generators and their algebraic relations.

Provided explicit solution for the two-point correlation function.

Abstract

From the time-independent current $\tcj(\bar y,\bar k)$ in the quantum self-dual Yang-Mills (SDYM) theory, we construct new group-valued quantum fields $\tilde U(\bar y,\bar k)$ and $\bar U^{-1}(\bar y,\bar k)$ which satisfy a set of exchange algebras such that fields of $\tcj(\bar y,\bar k)\sim\tilde U(\bar y,\bar k)~\partial\bar y~\tilde U^{-1}(\bar y,\bar k)$ satisfy the original time-independent current algebras. For the correlation functions of the products of the $\tilde U(\bar y,\bar k)$ and $\tilde U^{-1}(\bar y,\bar k)$ fields defined in the invariant state constructed through the current $\tcj(\bar y,\bar k)$ we can derive the Knizhnik-Zamolodchikov (K-Z) equations with an additional spatial dependence on $\bar k$. From the $\tilde U(\bar y,\bar k)$ and $\tilde U^{-1}(\bar y,\bar k)$ fields we construct the quantum-group generators --- local, global, and semi-local --- and their algebraic relations. For the correlation functions of the products of the $\tilde U$ and $\tilde U^{-1}$ fields defined in the invariant state constructed through the semi-local quantum-group generators we obtain the quantum-group difference equations. We give the explicit solution to the two point function.