Abstract
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.
Highlights
Deformation parameter q, found by Beisert and Koroteev [5]
It is conjectured that the world sheet theory has a consistent Green Schwarz Lagrangian formulation as a deformation of a gauged WZW theory associated to the supergroup PSU(2, 2|4) [20]
We have shown how to calculate the one loop beta function of a series of integrable sigma models and their so-called k deformations by using a background field method
Summary
As a necessary precursor to considering the deformed theories, we calculate the beta functions of the integrable sigma models. These are the Principal Chiral Models (PCM) and Symmetric Space Sigma Models (SSSM). Our approach is to pick a specific background field. The beta function follows from computing the spectrum of fluctuations about the background field. This is not a new calculation; the novelty of our approach, which will pay dividends later, is to calculate the fluctuations at the level of the currents rather than the fundamental field
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