A New Duality Symmetry in String Theory

We consider the conformal gauging of non-abelian groups. In such cases there are inequivalent ways of gauging (generalizing the axial and vector cases for abelian groups) corresponding to external automorphisms of the group. Different σ-models obtained this way correspond to the same conformal field theory. We use the method of quotients to formulate this equivalence as a new duality symmetry. CERN-TH.7310/94 June 1994 e-mail address: [email protected] Laboratoire Propre du CNRS UPR A.0014 e-mail address: [email protected] Duality symmetries are special to string theory and seem to be very useful in understanding stringy physics. Although discovered in flat torroidal backgrounds [1], they were shown to persist semiclassically in curved backgrounds with abelian [2] or non-abelian [3] symmetries. In [4, 5] it was realized that axial and vector gauging of an abelian chiral symmetry provides with two dual versions of the same σ-model. One on the other hand can use CFT arguments in order to show that axial and vector abelian cosets correspond to the same CFT, [6]. Axial-vector duality was employed in [7] to generate, using the method of quotients, the general abelian duality transformations, [2], and used to generalize the O(d, d, Z) symmetry to curved backgrounds [8]. The exactness of abelian duality symmetries in the compact case is well understood [6, 9]. In the non-compact case, we know that axial vector duality is exact, [10], only for abelian cosets possessing appropriate Weyl symmetries. Concerning non-abelian duality [11, 12, 13, 14] the situation is certainely not clear. It is not obvious if one has an exact symmetry and tools like axial-vector duality are lacking so far. In this note we will study a non-abelian analog of axial-vector duality which is generated by gauging non-abelian groups in a σ-model, not only using the standard vector gauging but also other possible gaugings related to vector gauging by external automorphisms, [15, 16]. In [15] it was conjectured that this will lead to another form of duality for gauged WZW models. As we will see, a look at the CFT construction of these gaugings indicates that the various cosets obtainable that way, are equivalent CFTs. This will provide us then with some new duality transformations in the σ-model picture. It is plausible that this duality (which from now on we will label quasi-axial-vector duality) is related to the standard non-abelian duality. Moreover in general it acts on σ-model backgrounds with no isometries. An easy example of that is G/H with H a maximal subgroup of G. This quasi-axial-vector duality realized partly a conjecture in [17]. One expects all the underlying exact symmetries of current algebra to generate duality symmetries for the σmodel description. Affine external automorphisms however are not obvious to implement in the σ-model. Here we will deal with usual external automorphisms and provide their implications for quasi-axial-vector duality. The first explicit example of such duality was given in [16], where two different gaugings were considered for the (E 2 × E c 2)/E c 2 coset model. The respective σ-model backgrounds were related by a series of abelian duality transformations however. The reason is that if one constructs the E 2 groups by appropriately contracting SU(2)× U(1) then the quasi-axial-vector duality is generated by standard axial-vector duality on the U(1) before contraction. We will start by investigating the CFT point of view. Consider a WZW theory for some (simple) group G. The standard spectrum of representations for the left and right current algebras has the form (R, R̄), modulo the affine truncation. The operator product fusion rules follow again group theory modulo truncations. If we act on the spectrum by