Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions

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 AdS${}_5$ black holes and could be tested by low-energy IIB supergravity.

55 pages, 9 figures; v2: minor revisions; v3: minor revisions in p.14,20,23; v4: minor revisions in Appendix A; v5: correct minor typos in Appendix C