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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 isthe partition function for $\mathcal{N} = 1$ theories on $S^2 \times T^2$, witha partial topological twist along $S^2$, in the presence of background magneticfluxes and fugacities for the global symmetries. We show that the logarithm ofthe index is proportional to the conformal anomaly coefficient of thetwo-dimensional $\mathcal{N} = (0,2)$ SCFTs obtained from the compactificationon $S^2$. We also present a universal formula for extracting the index from thefour-dimensional conformal anomaly coefficient and its derivatives. We giveexamples based on theories whose holographic duals are black strings in typeIIB backgrounds AdS$_5 \times \text{SE}_5$, where SE$_5$ are five-dimensionalSasaki-Einstein spaces.

The Cardy limit of the topologically twisted index and black strings in AdS$_5$