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Choi, Kim, Kim, and Nahmgoong have recently pioneered analyzing a Cardy-like limit of the superconformal index of the 4d $\mathcal{N}=4$ theory with complexified fugacities which encodes the entropy of the dual supersymmetric AdS$_5$ blackholes.
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E. M. Rains, Limits of elliptic hypergeometric integrals , Ramanujan J. 18 (3), 257 (2009) [math.CA/0607093]
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E. M. Rains, Transformations of elliptic hypergeometric integrals , Ann. Math. 171, 169 (2010) [arXiv:math/0309252v4 [math.QA]]
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A. A. Ardehali, Hyperbolic asymptotics of elliptic hypergeometric integrals arising in supersymmetric gauge theory , SIGMA 14, 043 (2018)
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