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

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.

15 pages, uses phyzzx