One-loop partition functions in deformed N=4 SYM theory

We study the thermodynamic behaviour of the real beta- and gamma(i)-deformation of N = 4 Super Yang-Mills theory on R x S-3 in the planar limit. These theories were shown to be the most general asymptotically integrable supersymmetric and non-supersymmetric field-theory deformations of N = 4 Super Yang-Mills theory, respectively. We calculate the first loop correction to their partition functions using an extension of the dilatation-operator and Polya-counting approach. In particular, we account for the one-loop finite-size effects which occur for operators of length one and two. Remarkably, we find that the O(lambda) correction to the Hagedorn temperature is independent of the deformation parameters, although the partition function depends on them in a non-trivial way.