Conformal Fixed Points of Unidentified Gauge Theories

Soon after the proposal for gauge/strings correspondence [1] and its spectacular implementation in N = 4 Yang-Mills theory [2] (which was also based on the earlier findings [3]) it has been suggested that some nonsupersymmetric gauge theories may become conformal at a fixed coupling [4,5]. The conjecture was based on the one loop estimate of the β function in the AdSp ⊗ Sq sigma model in the non-critical string (p + q < 10). The effective equations of motion have been shown to have a solution with a particular values of the radii of AdSp and Sq . Unfortunately this regime takes place at the curvatures of the order of the string scale where the one loop approximation can be used only as an order of magnitude estimate. Still, the counting of the parameters in [5] makes this result plausible. More recently this conjecture was discussed in the zero dimensional model [6]. In this letter I will take the next step and give further arguments that the above sigma models are conformal. They are also shown to be completely integrable. The bosonic part of the AdS sigma model is the familiar action for the unit vector field ~n which in this case is hyperbolic, satisfying the relation ~n2 = −1 This hyperboloid is embedded in the p + 1 dimensional flat space with the signature (p, 1) or (p− 1, 2) depending on whether the dual gauge theory is assumed to be in the Euclidean or Minkowskian spaces. It is