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

TL;DR

This paper computes the topologically twisted index of 4D $ ext{N}=1$ theories in the high-temperature limit, revealing its relation to 2D conformal anomalies and applying it to holographic black string backgrounds.

Contribution

It provides a universal formula linking the 4D conformal anomaly to the topologically twisted index and explores its implications for holographic black strings in AdS$_5$ backgrounds.

Findings

Logarithm of the index proportional to 2D conformal anomaly coefficient.

Universal formula for extracting the index from 4D anomaly coefficients.

Application to theories with holographic duals as black strings in AdS$_5$.

Abstract

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.