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Large $N$ matrix models for 3d ${\cal N}=2$ theories: twisted index, free energy and black holes

Seyed Morteza Hosseini, Alberto Zaffaroni

TL;DR

The paper develops a comprehensive large-$N$ analysis of the topologically twisted index for 3d ${\cal N}\ge2$ gauge theories with M-theory or massive IIA duals. It shows that the index scales as $N^{3/2}$ in the M-theory phase and as $N^{5/3}$ in the massive IIA phase, derives a universal index extraction formula from the Bethe potential, and uncovers a deep link between the index and the $S^3$ free energy, as well as with the AdS$_4$ black-hole attractor mechanism. A central result is an index theorem that expresses $\mathbb{R}e\log Z$ in terms of the extremal Bethe potential $\overline{\mathcal{V}}$, enabling practical computation from $S^3$ data. The work also outlines precise constraints on quiver content for large-$N$ consistency and provides explicit ABJM and SPP examples, highlighting a remarkable map between field theory functionals and holographic volumes. Overall, the results illuminate microscopic black-hole entropy counting in a broad class of 3d theories and suggest a unifying geometric structure connecting $F$, $\overline{\mathcal{V}}$, and prepotentials.

Abstract

We provide general formulae for the topologically twisted index of a general three-dimensional ${\cal N}\geq 2$ gauge theory with an M-theory or massive type IIA dual in the large $N$ limit. The index is defined as the supersymmetric path integral of the theory on $S^2\times S^1$ in the presence of background magnetic fluxes for the R- and global symmetries and it is conjectured to reproduce the entropy of magnetically charged static BPS AdS$_4$ black holes. For a class of theories with an M-theory dual, we show that the logarithm of the index scales indeed as $N^{3/2}$ (and $N^{5/3}$ in the massive type IIA case). We find an intriguing relation with the (apparently unrelated) large $N$ limit of the partition function on $S^3$. We also provide a universal formula for extracting the index from the large $N$ partition function on $S^3$ and its derivatives and point out its analogy with the attractor mechanism for AdS black holes.

Large $N$ matrix models for 3d ${\cal N}=2$ theories: twisted index, free energy and black holes

TL;DR

The paper develops a comprehensive large- analysis of the topologically twisted index for 3d gauge theories with M-theory or massive IIA duals. It shows that the index scales as in the M-theory phase and as in the massive IIA phase, derives a universal index extraction formula from the Bethe potential, and uncovers a deep link between the index and the free energy, as well as with the AdS black-hole attractor mechanism. A central result is an index theorem that expresses in terms of the extremal Bethe potential , enabling practical computation from data. The work also outlines precise constraints on quiver content for large- consistency and provides explicit ABJM and SPP examples, highlighting a remarkable map between field theory functionals and holographic volumes. Overall, the results illuminate microscopic black-hole entropy counting in a broad class of 3d theories and suggest a unifying geometric structure connecting , , and prepotentials.

Abstract

We provide general formulae for the topologically twisted index of a general three-dimensional gauge theory with an M-theory or massive type IIA dual in the large limit. The index is defined as the supersymmetric path integral of the theory on in the presence of background magnetic fluxes for the R- and global symmetries and it is conjectured to reproduce the entropy of magnetically charged static BPS AdS black holes. For a class of theories with an M-theory dual, we show that the logarithm of the index scales indeed as (and in the massive type IIA case). We find an intriguing relation with the (apparently unrelated) large limit of the partition function on . We also provide a universal formula for extracting the index from the large partition function on and its derivatives and point out its analogy with the attractor mechanism for AdS black holes.

Paper Structure

This paper contains 31 sections, 2 theorems, 151 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The topologically twisted index
  3. The large $N$ limit of the index
  4. Cancellation of long-range forces
  5. Bethe potential at large $N$
  6. The ABJM example
  7. The index at large $N$
  8. Bethe potential versus free energy on $S^3$
  9. An index theorem for the twisted matrix model
  10. Theories with $N^{5/3}$ scaling of the index
  11. Long-range forces
  12. Bethe potential at large $N$
  13. The index at large $N$
  14. Discussion and Conclusions
  15. Derivation of general rules for theories with $N^{3/2}$ scaling of the index
  16. ...and 16 more sections

Key Result

Theorem 1

The index of any $\mathcal{N} \geq 2$ quiver gauge theory which respects the constraints no long-range-forces Bethe0 and no long-range forces, and satisfies in addition superpotential0, can be written as where $\overline{\mathcal{V}}$ is the extremal value of the functional bethefunctional and $\mu$ is the Lagrange multiplier appearing in bethefunctional.The second identity in virial theorem is a

Figures (1)

  • Figure 1: The SPP Chern-Simons-matter quiver.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 1
  • proof