Deformed Minimal Models and Generalized Toda Theory

We introduce a generalization of Ar-type Toda theory based on a non-abelian group G, which we call the (Ar, G)-Toda theory, and its affine extensions in terms of gauged Wess-Zumino-Witten actions with deformation terms. In particular, the affine (A1, SU(2))Toda theory describes the integrable deformation of the minimal conformal theory for the critical Ising model by the operator Φ(2,1). We derive infinite conserved charges and soliton solutions from the Lax pair of the affine (A1, SU(2))-Toda theory. Another type of integrable deformation which accounts for the Φ(3,1)-deformation of the minimal model is also found in the gauged Wess-Zumino-Witten context and its infinite conserved charges are given. 1 E-mail address; [email protected] 2 E-mail address; [email protected] Recently, using the language of operator algebra, Zamolodchikov has shown that there exist some relevant perturbation around conformal field theory which preserve integrability.[1] In particular, when degenerate fields Φ(1,2),Φ(2,1),Φ(1,3) and Φ(3,1) are taken as the perturbations, he suggested that the resulting field theories may possess non-trivial integrals of motion and worked out explicitly for several examples. In the Lagrangian framework, there have been attempts to explain these particular perturbations in terms of the affine extension of Toda field theories[2][3]. In general, arbitrary coset conformal models can be formulated in terms of the gauged Wess-Zumino-Witten (WZW) action[4]. Using this fact and also generalizing the recent work of Bakas for the parafermion coset model[11], one of us (Q.P.) has recently shown that an integrable deformation of G/H-coset models is possible when the gauged WZW action for the G/H-coset model is added by a potential energy term Tr(gTgT̄ ), where algebra elements T, T̄ belong to the center of the algebra h associated with the subgroup H[6]. In this Letter, we consider two types of integrable deformations, by operators Φ(2,1) and Φ(3,1), of the minimal model corresponding to the coset (SU(2)N ×SU(2)N )/SU(2)2N where N denotes the level. This corresponds to the deformation of the critical Ising model forN = 1 and that of c = 1 theory in the super conformal minimal series for N = 2. We formulate the minimal model in terms of the gauged Wess-Zumino-Witten(WZW) action and show that the integrable deformations by Φ(2,1) and Φ(3,1) can be obtained by adding potential terms; Tr(g 1 g2+g −1 2 g1) and Tr(g −1 1 L g1L b)Tr(g 2 M g2M ) respectively, where g1, g2 are Lie group SU(2)-valued fields and {L}, {M} are two sets of generators of the Lie algebra su(2). In particular, the action for the Φ(2,1)-deformation suggests a natural generalization of the abelian Ar-type Toda theory to the non-abelian (Ar, G)-Toda theory for a non-abelain group G and its affine extensions whereas the action for the Φ(2,1)-deformation itself becomes the affine (A1, SU(2))-Toda theory. We demonstrate the integrability of both deformed models by deriving Lax pairs for them and also from which infinitely many conserved charges. We also derive n-soliton solutions for the affine (A1, SU(2))-Toda theory. Recall that a lagrangian of the G/H-coset model is given in terms of the gauged WZW functional[4], which in light-cone variables is S(g, A, Ā) = SWZW (g) + 1 2π ∫ Tr(−A∂̄gg + Āg∂g + AgĀg −AĀ) (1) 2 where SWZW (g) is the usual WZW action [5] for a map g : M →G on two-dimensional Minkowski space M . The connection A, Ā gauge the anomaly free subgroup H of G. In this Letter, we take the diagonal embedding of H in GL × GR, where GL and GR denote left and right group actions by multiplication (g → gLgg −1 R ), so that Eq.(1) becomes invariant under the vector gauge transformation (g → hgh with h : M →H). The restriction to the coset (SU(2)N × SU(2)N )/SU(2)2N where N denotes the level of the Kac-Moody algebra is defined by the functional ∫ [dg1][dg2][dA][dĀ]exp(iI0(g1, g2, A, Ā)) where I0(g1, g2, A, Ā) = NSWZW (g1, A, Ā) +NSWZW (g2, A, Ā) (2) where A and Ā gauge simultaneously the diagonal subgroups of SU(2)×SU(2). The classical equations of motion for g1 and g2 arise in a form of zero curvature condition, [ ∂ + g 1 ∂g1 + g −1 1 Ag1 , ∂̄ + Ā ] = 0 [ ∂ + g 2 ∂g2 + g −1 2 Ag2 , ∂̄ + Ā ] = 0 (3) whereas variations with respect to A and Ā give the constraint equation, − ∂̄g1g −1 1 + g1Āg −1 1 − ∂̄g2g −1 2 + g2Āg −1 2 − 2Ā = 0 g 1 ∂g1 + g −1 1 Ag1 + g −1 2 ∂g2 + g −1 2 Ag2 − 2A = 0 . (4) Φ(2,1)-deformation and generalized Toda theory Having introduced an action for the coset model, we now assert that an integrable deformation is possible when we add to the action a potential term in the following way: I(g1, g2, A, Ā, κ) = I0(g1, g2, A, Ā)− Nκ 2π ∫ Tr(g 1 g2 + g −1 2 g1), (5) where κ is a coupling constant. The potential term transforms at the classical level as (doublet, singlet) in the convention of coset conformal field theory so that it corresponds to the operator Φ(2,1). This changes Eq.(3) by [ ∂ + g 1 ∂g1 + g −1 1 Ag1 , ∂̄ + Ā ]− κ(g −1 1 g2 − g −1 2 g1) = 0 [ ∂ + g 2 ∂g2 + g −1 2 Ag2 , ∂̄ + Ā ] + κ(g −1 1 g2 − g −1 2 g1) = 0 (6) 3 while leaving the constraint equation unchanged. The main observation in proving the integrability of the model is that Eq.(6) is precisely the integrability condition of the linear 4× 4 matrix equations with a spectral parameter λ, L1(λ)Ψ ≡ [∂ + U0 − λT ]Ψ = 0 ; L2(λ)Ψ ≡ (∂̄ + Ā+ 1 λ V1)Ψ = 0 (7) where U0 = G ∂G + GAG , V1 = G T̄G , T = iκΣ , T̄ = iΣ (8) and