The N = 4 SU(N) Super-Yang-Mills Index and Dual AdS Black Holes

The main subjects of this dissertation are indices of the N=4 SU(N) Super-Yang-Mills (SYM) theory, namely the topologically twisted index and the superconformal index. These indices have received a lot of attention since they provide microscopic understanding of AdS_5 black strings and black holes respectively through the AdS/CFT correspondence. In this dissertation, we focus on the field theory side and investigate these indices with a goal of improving the current microscopic understanding of AdS_5 black strings and black holes. As a result, we unveil interesting physics of the 4d indices such as modular properties and a relation to the S^3 partition function of effective Chern-Simons theory, and also make suggestions in the gravity side based on the structure of indices through the AdS/CFT correspondence. First, we study the topologically twisted index of the N=4 SU(N) SYM theory on T^2xS^2. We introduce the Bethe Ansatz (BA) formula that gives the twisted index as a sum over solutions to the Bethe Ansatz Equations (BAE) and categorize various solutions into two groups: standard ones that compose the SL(2,Z) family and non-standard ones that denote all the other BAE solutions including continuous families. Focusing on the contribution from standard BAE solutions, we confirm that it behaves as an elliptic genus with certain modular properties and further investigate its asymptotic behaviors in the Cardy-like limit. Lastly, we review how the twisted index counts the microstates associated with the dual AdS_5 black string entropy in the Cardy-like limit, and discuss missing steps that should be taken care of to validate the microstate counting. Next, we study the superconformal index of the N=4 SU(N) SYM theory. We compute the superconformal index first by saddle point evaluation and then by the BA formula. In due process, we establish a direct relation between the 4d superconformal index and the S^3 partition function of Chern-Simons theory. Then we investigate the phase structure of the superconformal index in the large-N after the Cardy-like limit, which contains partially-deconfined phases distinguished from the previously well-known fully-deconfined/confined phases. Finally, we discuss implications of a partially-deconfined phase, based on the counting of microstates associated with the dual AdS_5 black hole entropy by the superconformal index.  [