Quantum Exchange Algebra and Exact Operator Solution of $A_2$-Toda Field Theory

TL;DR

This paper investigates the quantum $A_2$-Toda field theory, establishing its locality, deriving exponential operators via free field quantization, and extending algebraic methods from Liouville theory to obtain exact solutions.

Contribution

It introduces a novel chiral description ensuring locality, constructs Toda exponential operators satisfying quantum exchange algebra, and extends algebraic methods to the $A_2$-system for exact solutions.

Findings

Locality guarantees canonicity of the classical to quantum mapping.

Explicit Toda exponential operators are derived as bilinear forms of chiral fields.

The algebraic method from Liouville theory is extended to solve the $A_2$-system.

Abstract

Locality is analyzed for Toda field theories by noting novel chiral description in the conventional nonchiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to nontrivial relations among the ${\cal R}$-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for $A_2$-system by extending the algebraic method developed for Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.