On the Exceptional Gauged WZW Theories

We consider two different versions of gauged WZW theories with the exceptional groups and gauged with any of theirs null subgroups. By constructing suitable automorphism, we establish the equivalence of these two theories. On the other hand our automorphism, relates the two dual irreducible Riemannian globally symmetric spaces with different characters based on the corresponding exceptional Lie groups. e-mail address: [email protected] Introduction In the last few years, duality transformation as an abelian or non-abelian symmetry of conformal field theories has been studied extensively [1, 2, 3]. In WZW models, the duality transformations are given by the automorphisms of the group G of the models. In particular, it has been shown that the target space of the gauged WZW model with the group G and vector gauged abelian subgroup is dual to the corresponding target space of the axial gauged WZW model gauged by the same abelian subgroup. Moreover the duality transformation is implemented by an automorphism of the gauged subgroup [4, 5, 6, 7]. When the gauged subgroup is not semisimple, and in particular when it is null, the target space of vector gauged WZW model reduces to a space with less dimensions [8, 9, 10, 11]. In other words, it was shown that in the usual vector gauging of classical Lie group G WZW model, by its maximal null subgroup H , the effective action of the gauged model reduces to that of a Toda theory which the number of Toda fields is less than DimG-DimH . An explanation for the reduction of the degrees of freedom of the target space of the vector gauged model is related to the obvious dimensional reduction of the corresponding chiral gauged model when the left and right gauge actions are independent and in two different subgroups isomorphic to vector null gauged subgroup. In ref. [12], it was proved that the vector and chiral gauged theories with any classical Lie groups and any null gauged subgroup are equivalent to each other. The equivalence map was constructed for every classical Lie groups and it was shown that these mapps were in fact involutive automorphisms of corresponding Lie algebras. These involutive automorphisms could be used for real form construction of simple Lie algebras and also gave us the pairs of dual Riemannian globally symmetric spaces [13]. In this paper, we consider the vector and chiral gauged WZW models which based on the exceptional groups and gauged with any of theirs null subgroups and prove that these models are equivalent to each other. We will find that the equivalence map which relates the two models, is the involutive automorphism of the algebra and in the context of Riemannian geometry relates the two dual irreducible Riemannian globally symmetric spaces based on the exceptional Lie group. In particular, for the exceptional group G2, we give the calculation in detail, and in the case of other exceptional groups, because of complexity, we present the final results. The work done in this paper completes the equivalence of vector and chiral gauged WZW models with any Lie groups. G2 Gauged WZW Models Let’s recall the structure of gauged WZW models based on the exceptional Lie group G2. The vector gauged action is given by [14, 15] SV (g,A, Ā) = S(g) + k 2π ∫ dz T r(− Āg∂g +A∂̄gg −AĀ+ gĀgA ), (1) S(g) = k 4π ∫ dz T r( g∂gg∂̄g )− k 12π ∫ Tr( gdg ), which is invariant under the gauge transformations g → hg h,A → h (A− ∂)h, Ā → h (Ā− ∂̄)h, (2) 1 where h = h(z, z̄) is a group element in subgroup H of G2 and A and Ā take their values in the algebra L(H) of subgroup H . On the other hand, the chiral WZW action [16] SC(g,A, Ā) = S(g) + k 2π ∫ dz T r(− Āg∂g +A∂̄gg + gĀgA ), (3) is invariant under the following transformations g → hg h̄,A → h (A− ∂)h, Ā → h̄ (Ā− ∂̄) h̄, (4) where h = h(z) belongs to the subgroup H1, and h̄ = h̄(z̄) belongs to another subgroup H2 of G2. A takes its value in L (H1) and Ā in L (H2). Now, we impose following transformations on the g field and the gauge fields of the chiral theory g = gθ, Ā = θĀθ, A = A, (5) and demand that the g and Ā become the corresponding fields in the vector theory. After straight forward but lengthy calculations, one can find that