On the Three-Point Couplings in Toda Field Theory

Correlation functions of Toda field vertices are investigated by applying the method of integrating zero-mode developed for Liouville theory. We generalize the relations among the zero-, twoand three-point couplings known in Liouville case to arbitrary Toda theories. Twoand three-point functions of Toda vertices associated with the simple roots are obtained. Since the impressive success of Goulian and Li [1] that the three-point correlators of the Liouville vertex functions appearing as the dressing factor of minimal model can be obtained by continuing the central charge to a certain value where the functional integral can be carried out exactly via free field technique [2], several authors have pushed forward their method [3, 4, 5] and generalized to define arbitrary three-point couplings in Liouville theory [4, 6, 7]. Since Liouville theory is a Toda field theory associated with the Lie algebra sl2 and as two dimensional field theory Toda theories possess distinguished properties such as the W symmetry [8, 9, 10], it is natural to ask whether the similar functional integral approach can apply for the general Toda theories. In this note we shall investigate correlators of the vertex functions of Toda fields by applying the method of ref. [2, 1]. We see that the zero-modes of the Toda fields can be integrated as in Liouville case and the remaining functional integrations over nonzero-modes can be carried out via free field technique if all the s-parameters are nonnegative integers. We thus arrive at complicated multiple integrals over complex planes. Unlike the Liouville case, we can not give closed forms of the integrals for three-point couplings. This prevent us from getting explicit expressions for general three-point couplings. We find, however, some universality among the three-point couplings for the vertex functions associated to simple roots. The relations among the zero-, twoand three-point couplings found in Liouville case [6] can also be generalized to Toda theory. In particular, we will give closed expressions for twoand three-point functions of the vertices associated to simple roots. We consider the Toda field theory associated with the simple Lie algebra G of rank r. The Toda field φ is an r-component vector in the root space. Let us denote the simple roots by α (a = 1, · · · , r), then it is described by the classical field equations ∂z∂z̄φ− μ 8 r ∑ a=1 αe a·φ = 0 . (1) For G = sl2 this reduces to the Liouville equation. The equation of motion (1) is invariant under the conformal reparametrization z → ξ = f(z) by the shift φ→ φ− ρ ln ∂zf∂z̄f̄ , (2) where ρ is a vector in the root space satisfying ρ · α = 1 , (3) 2 for any simple root. In terms of the fundamental weights λ (a = 1, · · · , r) defined by 2λ · α/(α) = δ, it is given by ρ = r ∑ a=1 2λ (αa)2 (4) In quantum theory we start from the action S[ĝ;φ] = 1 8π ∫ dz √ ĝ ( ĝ∂αφ · ∂βφ+Q · φR̂+ μ γ2 r ∑ a=1 e a·φ ) , (5) where ĝαβ is the fiducial metric on the Riemann surface. We have introduced couplings Q, a vector in the root space, with the curvature and γ in the Toda potential. They are determined by requiring the conformal invarince. The central charge of the Toda theory can be found most easily by applying the DDK’s argument [11]. We note that the stress tensor for μ = 0 and in locally flat coordinates is given by T (z) = − 1 2 (∂zφ) 2 + 1 2 Q · ∂ zφ (6) Using the operator product relation φj(z)φk(w) ∼ δjk ln |z − w|, one can easily obtain T (z)T (w) ∼ 1 2 r + 3Q (z − w)4 . (7) The central charge of the Toda sector is thus given by cφ = r + 3Q 2 . (8) One can also determine the conformal dimension of arbitrary Toda vertex functions. Let β be an arbitrary vector in the root space, then the conformal dimension of the vertex function e is given by ∆(e) = 1 2 β · (Q− β) . (9) In particular the Toda potential in (5) must be a (1,1)-conformal field by the requirement of conformal invariance. This relates Q with the coupling constant γ as γα · (Q− γα) = 2 (10) for any simple root. This is the quantum version of the classical relation (3) and has already been obtained in ref. [12]. Since the r simple roots are linearly independent each other, one can uniquely find the expression for Q as Q = 2 ( 1 γ ρ+ γρ̄ ) , (11)