My question is about the AdS/CFT correspondence's place in calculating Sen's quantum entropy functions. In 2008 Sen gave a way to calculate the exact entropy of certain supersymmetric quantum black holes. In particular extremal RN black holes in 4D. Those are asymptotically isometric to $AdS^2\times S^2$. It is a development on Wald's calculation for classical gravity theories with higher derivatives in their lagrangian. The crux is caculating the expectation of a -regularized- boundary Wilson line in the quantized bulk gravity on the near-horizon $AdS^2\times S^2$. He seems to be using the AdS$^2$/CFT$^1$ correspondence but I cannot really see where.
Later articles by Dabholkar, Gomes, Murthy, et al. used localization to compute the bulk functional integral exactly as a finite dimensional integral, following Banerjee et al. But there the functional integral is over bulk fields -vector multiplets, hypermultiplets, Weyl multiplet. I cannot see where the AdS/CFT correspondence enters. Is it just a matter of taking certain limits on couplings without really considering the boundary CFT?
Furthermore if the AdS/CFT correspondence enters does it make sense to speak of a macroscopic calculation of the entropy, because passing to the boundary CFT is essentially passing (modulo a few "passage to the limit"s in between) to the brane worldvolume theory, which represents the microscopic degrees of freedom of string theory? So the quantum entropy function would better be called a different microscopic calculation of the straightforward original logarithm of the brane partition function. If this were the case would it not be better to speak of entropy computed in different frames (the weakly coupled IIB string frame near the horizon, the weakly coupled large N IIB brane frame, etc.)? It is not always a microscopic calculation if 1 passes by brane theories? I would believe macroscopic refers to spacetime string fields, entering lagrangians with arbitrarily high derivatives.
I am a little puzzled so any comment may enlighten me. Sorry if I wrote nonsense.
Main reference: arXiv:0805.0095, arXiv:0905.2686, arXiv:1012.0265, arXiv:1111.1161.