Relative entropy in $2d$ QFT, finite-size corrections and irreversibility of the RG

The UV and IR behavior of the relative entropy for a Euclidean $2d$ QFT is studied for their application as irreversible quantities under the RG, focusing on free-field models. Next we consider the modifications introduced by putting the system in a finite geometry, in particular, the cylinder, interpreted as finite-temperature QFT. We find that the relative entropy is not only finite but analytic in the mass, which acts as coupling constant. In addition, we show that it essentially coincides with the $1d$ quantum thermodynamic entropy. Finally, we find the relation of the previous quantities with the complex components of the stress tensor and thence propose the monotonicity theorem for the relative entropy as the analog of Zamolodchikov's $c$ theorem for the cylinder geometry