  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=exp(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|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|4) which is another piece of evidence that the k deformed theories
define consistent string theories.
