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

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 $S^2\times T^2$, with a partial topological twist along $S^2$, 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 $\mathcal{N}=(0,2)$ SCFTs obtained from the compactification on $S^2$. 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 AdS${}_5\times {\text{SE}}_5$, where SE${}_5$ are five-dimensional Sasaki-Einstein spaces.