Abstract
We systematically study various sub-leading structures in the superconformal index of mathcal{N} = 4 supersymmetric Yang-Mills theory with SU(N) gauge group. We concentrate in the superconformal index description as a matrix model of elliptic gamma functions and in the Bethe-Ansatz presentation. Our saddle-point approximation goes beyond the Cardy-like limit and we uncover various saddles governed by a matrix model corresponding to SU(N) Chern-Simons theory. The dominant saddle, however, leads to perfect agreement with the Bethe-Ansatz approach. We also determine the logarithmic correction to the superconformal index to be log N, finding precise agreement between the saddle-point and Bethe-Ansatz approaches in their respective approximations. We generalize the two approaches to cover a large class of 4d mathcal{N} = 1 superconformal theories. We find that also in this case both approximations agree all the way down to a universal contribution of the form log N. The universality of this last result constitutes a robust signature of this ultraviolet description of asymptotically AdS5 black holes and could be tested by low-energy IIB supergravity.
Highlights
Introduction and summaryOne of the most remarkable results in the context of the AdS/CFT correspondence has been the microscopic explanation of the entropy of electrically charged, rotating black holes in AdS5 using the superconformal index (SCI) of N = 4 SYM theory
One of the main results of this paper is the exploration of the N = 4 SCI beyond the leading orders in Cardy-like / large-N limit respectively, which was required for a reliable estimation of logarithmic corrections
We have demonstrated by direct computation that the two main approaches to the SCI, namely the saddle point approach for the Cardy-like limit and the BA approach for the large-N limit, are consistent with each other up to exponentially suppressed terms in the Cardy-like limit
Summary
We demonstrate explicitly that the two main presentations are different approximations schemes to the index which result, in the same answer including sub-leading terms all the way down to a universal logarithmic correction. This process helps us clarify a number of central elements and provides a glimpse into an effective matrix model theory governed by SU(N ) Chern-Simons theory. Appendix B investigates contributions to the SCI from C-center saddle-points and BA-solutions, respectively These C-center solutions describe particular eigenvalue configurations that can be dominant over those studied in the main sections 3 and 4 in certain domain of chemical potentials. Appendix D reviews the partition function of SU(N ) Chern-Simons theory which is quite relevant to our computations
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