The Cardy limit of the topologically twisted index and black strings in AdS5

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We evaluate the topologically twisted index of a general four-dimensional mathcal{N}=1 gauge theory in the “high-temperature” limit. The index is the partition function for mathcal{N}=1 theories on S2× T2, with a partial topological twist along S2, in the presence of background magnetic fluxes and fugacities for the global symmetries. We show that the logarithm of the index is proportional to the conformal anomaly coefficient of the two-dimensional mathcal{N}=left(0,2right) SCFTs obtained from the compactification on S2. We also present a universal formula for extracting the index from the four-dimensional conformal anomaly coefficient and its derivatives. We give examples based on theories whose holographic duals are black strings in type IIB backgrounds AdS5× SE5, where SE5 are five-dimensional Sasaki-Einstein spaces.