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

Seyed Morteza Hosseini, Anton Nedelin, Alberto Zaffaroni

TL;DR

The authors compute the topologically twisted index of four-dimensional $\mathcal{N}=1$ gauge theories on $S^2\times T^2$ in the high-temperature (Cardy-like) limit and prove that $\log Z$ is governed by the left-moving central charge of the resulting 2d $\mathcal{N}=(0,2)$ CFT. They establish a universal relation between the index and four-dimensional 't Hooft anomalies via the Bethe potential $\mathcal{V}$ and its derivatives, and show that, at large $N$, $\mathcal{V}$ is proportional to the 4d conformal anomaly coefficient $a$ (and $c$) so that the index can be read off from anomaly data. The results hold for general $\mathcal{N}=1$ quivers, with explicit checks in $\mathcal{N}=4$ SYM and the Klebanov-Witten theory, and connect to holographic black strings in AdS$_5\times SE_5$ through $c$- and $I$-extremization. A universal index theorem is presented: $\log Z(\Delta_I,\mathfrak{n}_I) = \frac{\pi^2}{6\beta} c_l(\Delta_I,\mathfrak{n}_I)$, linking microscopic state counting to 2d CFT data and gravity via the Cardy formula. These results illuminate the microscopic structure of black strings and suggest parallels with IR attractor mechanisms and integrable-system pictures.

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.

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

TL;DR

The authors compute the topologically twisted index of four-dimensional gauge theories on in the high-temperature (Cardy-like) limit and prove that is governed by the left-moving central charge of the resulting 2d CFT. They establish a universal relation between the index and four-dimensional 't Hooft anomalies via the Bethe potential and its derivatives, and show that, at large , is proportional to the 4d conformal anomaly coefficient (and ) so that the index can be read off from anomaly data. The results hold for general quivers, with explicit checks in SYM and the Klebanov-Witten theory, and connect to holographic black strings in AdS through - and -extremization. A universal index theorem is presented: , linking microscopic state counting to 2d CFT data and gravity via the Cardy formula. These results illuminate the microscopic structure of black strings and suggest parallels with IR attractor mechanisms and integrable-system pictures.

Abstract

We evaluate the topologically twisted index of a general four-dimensional gauge theory in the "high-temperature" limit. The index is the partition function for theories on , with a partial topological twist along , 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 SCFTs obtained from the compactification on . 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, where SE are five-dimensional Sasaki-Einstein spaces.

Paper Structure

This paper contains 15 sections, 1 theorem, 122 equations.

Table of Contents

  1. Introduction
  2. The topologically twisted index
  3. $\mathcal{N} = 4$ super Yang-Mills
  4. Bethe potential at high temperature
  5. The topologically twisted index at high temperature
  6. The Klebanov-Witten theory
  7. Bethe potential at high temperature
  8. The topologically twisted index at high temperature
  9. High-temperature limit of the index for a generic theory
  10. Bethe potential at high temperature
  11. The topologically twisted index at high temperature
  12. An index theorem for the twisted matrix model
  13. Future directions
  14. Elliptic functions and their asymptotics
  15. Anomaly cancellation

Key Result

Theorem 1

The topologically twisted index of any $\mathcal{N} = 1$$\mathrm{SU}(N)$ quiver gauge theory placed on $S^2 \times T^2$ to leading order in $1/\beta$ is given by which is Cardy's universal formula for the asymptotic density of states in a CFT$_2$Cardy:1986ie. We can write the extremal value of the Bethe potential $\overline \mathcal{V} \left( \Delta_I \right)$ as For theories of D3-branes at lar

Theorems & Definitions (2)

  • Theorem 1
  • proof