On the Equivalence of Dual Theories

Abstract

We discuss the equivalence of two dual scalar field theories in 2 dimensions. The models are derived though the elimination of different fields in the same Freedman-Townsend model. It is shown that tree S-matrices of these models do not coincide. The 2-loop counterterms are calculated. It turns out that while one of these models is single-charged, the other theory is multi-charged. Thus the dual models considered are non-equivalent on classical and quantum levels. It indicates the possibility of the anomaly leading to non-equivalence of dual models.

Perturbative quantum (in)equivalence of dual σ models in 2 dimensions

“…This last conclusion has also been reached in Ref. [20], we disagree, however, with some of the other claims of that paper. Let us remark that in all of the examples investigated so far we have also checked that our conclusions on the two loop (non)renormalizability of the models in question is independent of the well known ambiguity (or freedom) in the counterterms (corresponding to target space diffeomorphism invariance) [12], [13], [14], [15], [16].…”

“…An alternative way to express the correspondence between the two models is to get rid of the scheme dependence of the β functions by eliminating one of the parameters (g or a) in β λ in favour of a renormalization group invariant pa-rameter (M resp.M ). A straightforward computation yields, that the invariant parameter characterizing the trajectories under the renormalization group equations (20) has the form:…”

Dualised σ-models at the two-loop order

“…Then, as first proven in [13,14], we have checked that : In a purely dimensional scheme (even with non minimal subtractions), the dualised SU(2) σ model is not renormalisable at the two-loop order.…”

“…In the same work, the twoloop renormalisability problem was tackled and the need for extra (non-minimal) one-loop order finite counter-terms was emphasized. Some years ago, it was noted that in the minimal dimensional scheme, two-loop renormalisability does not hold for the SU(2) T-dualised model [13,14] .…”

Quantum equivalence ofσmodels related by non-Abelian duality transformations

“…First of all quantum equivalence among sigma models related by non Abelian duality has some problems even in the conformal invariant case, as there are examples [19] where non Abelian duality is mapping a conformal invariant background to a non conformal dual. The second, 'non conformal' motivation is the discovery [9], [20] that the relation between the SU (2) principal model and its non Abelian dual shows the same features as in the case of Abelian duality: in the one loop order the two models are equivalent while at two loops the dual is not renormalizable in the usual, field theoretic sense. The investigation of this problem is also made urgent by one of the results of [21].…”