Exact operator solution of A2-Toda field theory

Abstract

Quantum A 2 -Toda field theory in two dimensions is investigated based on the method of quantizing canonical free field. Toda exponential operators associated with the fundamental weights are constructed to the fourth order in the cosmological constant. This leads us to a conjecture for the exact operator solution.

Quantum exchange algebra and exact operator solution of A2-Toda field theory

“…In ref. [18] the locality is investigated directly order by order in the cosmological constant µ 2 . This approach suffices to guess the general forms of the expansion coefficients.…”

“…In terms of the ⋆-product the screening charges Y a and Y aā introduced in ref. [18] are simply given by…”

“…In ref. [18] the coefficients C a nm are explicitly given as a conjecture based upon the analysis of the locality condition [e κλ a •ϕ(0,σ) , e κλ b •ϕ(0,σ ′ ) ] = 0 (3.16) up to the fourth order in µ 2 . We will fill the gap and give a complete proof for the conjectured forms of the coefficients in the next section.…”

“…† Three of the present authors have recently extended the approach of ref. [7] to A 2 -Toda system and obtained arbitrary Toda exponential operators to fourth order in the cosmological constant by analyzing directly the locality conditons within vector scheme, giving a conjecture for exact operator solution of A 2 -Toda theory [18]. The method is rather restricted to A 2 case and becomes intractable as one goes to higher orders though the basic ideas undelying the arguments presented there could equally well be applied for general Toda systems.…”

“…As stressed in ref. [18], it is very convenient to consider Toda exponential operators associated with the fundamental weights. They contain screening charges and a free field vertex operator which are mutually commuting, hence no ordering problem arise in the operator products.…”