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High-temperature asymptotics of the 4d superconformal index

The superconformal index of a typical Lagrangian 4d SCFT is given by a special function known as an elliptic hypergeometric integral (EHI). The high-temperature limit of the index corresponds to the hyperbolic limit of the EHI. The hyperbolic limit of certain special EHIs has been analyzed by Eric Rains around 2006; extending Rains's techniques, we discover a surprisingly rich structure in the high-temperature limit of a (rather large) class of EHIs that arise as the superconformal index of unitary Lagrangian 4d SCFTs with non-chiral matter content. Our result has implications for $\mathcal{N}=1$ dualities, the AdS/CFT correspondence, and supersymmetric gauge dynamics on $R^3\times S^1$. We also investigate the high-temperature asymptotics of the large-N limit of the superconformal index of a class of holographic 4d SCFTs (described by toric quiver gauge theories with SU(N) nodes). We show that from this study a rather general solution to the problem of holographic Weyl anomaly in AdS$_5$/CFT$_4$ at the subleading order (in the 1/N expansion) emerges. Most of this dissertation is based on published works by Jim Liu, Phil Szepietowski, and the author. We include here a few previously unpublished results as well, one of which is the high-temperature asymptotics of the superconformal index of puncture-less SU(2) class-$\mathcal{S}$ theories.