Sean Hartnoll from the old Cambridge discussed the phase structure of N=4 gauge theory and the main topic of the debates that followed was how can the apparent existence of branch cuts in field theories at weak coupling agree with the meromorphic nature of the Green's functions at strong coupling derived from the dual bulk gravitational description where the poles - e.g. the poles known from the quasinormal mode spectrum - are the only singularities.
My opinion is that the branch cut nature of the amplitudes is always a perturbative illusion and any background with quantum gravity actually resolves the branch cuts into a sequence of individual discrete poles. The main challenge for this "stringy" point of view is to explain how the continuum of two-particle (double-gluon) states that appear as intermediate states in gauge theory can be accounted for in the gravity picture, or why is the spectrum of the double-particle intermediate states discrete even at weak coupling.
If you have some strong or other opinions about the existence of branch cuts and their transitions at finite values of the couplings, I am curious to hear them.