Beta function of k deformed AdS5 X S5 string theory

We calculate the one loop beta function for the would-be marginal coupling on the world sheet of the k deformed sigma models associated to a quantum group with q = e(i pi/k). This includes the bosonic principal chiral models and symmetric space sigma models but also the k deformed semi-symmetric space sigma model describing strings in a deformation of AdS(5) x S-5. The world sheet sigma model is a current-current deformation of the gauged WZW model for the supergroup PSU(2; 2 vertical bar 4) with level k. In the string theory context the beta function is shown to vanish because of the vanishing of the Killing form of PSU(2; 2 vertical bar 4) which is another piece of evidence that the k deformed theories define consistent string theories.