I have tried to think about this for some time but could not really go anywhere. Sorry for the sloppy question and thanks for any pointer.
My question is about CFT at finite temperature and nonconformal theories. I have sometimes read that CFTs are not conformal at finite temperature but I could not make clear to myself what happened precisely. What theory does a CFT at finite temperature correspond to? To begin the "temperature state" is not unique but rather an equilibrium/Gibbs distribution over several quantum states -a canonical ensemble. So correlation functions are integrals over 2 measures (vs. vev in the CFT), an average over the correlation functions between each pair of states in the ensemble. Now many theories flow to a given CFT so which one should we pick at finite temperature? For instance what is the finite temperature of a gaussian/free massless field, say in 2D, i.e. the basic free boson c=1 CFT? I could imagine this is a massive boson with $m=1/T$ but am not sure. In general one would consider the reduced temperature, centered at the critical temperature. So is the finite temperature Ising model in the "thermodynamic limit" a sort of massive Wilson-Fisher fixed point?
There are often mentions of finite temperature CFT in the AdS/CFT correspondence. The dual of a schwarzschild black hole in AdS should be a finite temperature CFT. Is this correspondence conjectured to be exact at all couplings? QCD at high temperature is free, what does it mean for the dual strongly coupled string theory on AdS with no black hole? (I would guess this means the closed strings propagate freely in some appropriate
Well sorry for the fuzzy question. But I'll be very grateful if you try to empathize.