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.

arXiv comment29 pages, 1 appendix; v2: typos corrected and references added; v3: corrected a mistake in the appendix, expanded discussion about logarithmic corrections in the introduction and about saddle points for SO groups in section 4

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• A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index
• $SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index
• Superconformal index of low-rank gauge theories via the Bethe Ansatz
• Logarithmic Corrections to the Entropy of Rotating Black Holes and Black Strings in AdS${}_5$
• Supersymmetric Grey Galaxies, Dual Dressed Black Holes and the Superconformal Index
• The large-$N$ limit of 4d superconformal indices for general BPS charges