The superconformal index of $\mathcal{N}=4$ $USp(2N_c)$ and $SO(N_c)$ SYM as a matrix integral

Abstract: We study the superconformal index of 4d $\mathcal{N}=4$ $USp(2N_c)$ and $SO(N_c)$ SYM from a matrix model perspective. We focus on the Cardy-like limit of the index. Both in the symplectic and orthogonal case the index is dominated by a saddle point solution which we identify, reducing the calculation to a matrix integral of a pure Chern-Simons theory on the three-sphere. We further compute the subleading logarithmic corrections, which are of the order of the center of the gauge group. In the $USp(2N_c)$ case we also study other subleading saddles of the matrix integral. Finally we discuss the case of the Leigh-Strassler fixed point with $SU(N_c)$ gauge group, and we compute the entropy of the dual black hole from the Legendre transform of the entropy function.