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The Large N Limit of Toric Chern-Simons Matter Theories and Their Duals

Antonio Amariti, Claudius Klare, Massimo Siani

TL;DR

This work tests the AdS_4/CFT_3 correspondence by computing the large N limit of the localized S^3 free energy for a broad class of 3d ${ m N}=2$ CS-matter theories with toric CY_4 duals. It demonstrates that vector-like theories reproduce the expected $N^{3/2}$ scaling and comply with the $F$-theorem, while Seiberg duality leaves the free energy invariant at large N. For chiral-like models, it introduces a symmetry-restoration (vectorialization) procedure that enables approximate matching to geometry in specific cases, and it proposes an alternative geometric (quartic) expression for the free energy that often aligns with holographic volumes after symmetry constraints. Overall, the paper strengthens the bridge between field-theoretic localization results and geometric data, and it outlines promising directions for extending these methods to broader classes of theories and dual geometries.

Abstract

We compute the large N limit of the localized three dimensional free energy of various field theories with known proposed AdS duals. We show that vector-like theories agree with the expected supergravity results, and with the conjectured F-theorem. We also check that the large N free energy is preserved by the three dimensional Seiberg duality for general classes of vector like theories. Then we analyze the behavior of the free energy of chiral-like theories by applying a new proposal. The proposal is based on the restoration of a discrete symmetry on the free energy before the extremization. We apply this procedure at strong coupling in some examples and we discuss the results. We conclude the paper by proposing an alternative geometrical expression for the free energy.

The Large N Limit of Toric Chern-Simons Matter Theories and Their Duals

TL;DR

This work tests the AdS_4/CFT_3 correspondence by computing the large N limit of the localized S^3 free energy for a broad class of 3d CS-matter theories with toric CY_4 duals. It demonstrates that vector-like theories reproduce the expected scaling and comply with the -theorem, while Seiberg duality leaves the free energy invariant at large N. For chiral-like models, it introduces a symmetry-restoration (vectorialization) procedure that enables approximate matching to geometry in specific cases, and it proposes an alternative geometric (quartic) expression for the free energy that often aligns with holographic volumes after symmetry constraints. Overall, the paper strengthens the bridge between field-theoretic localization results and geometric data, and it outlines promising directions for extending these methods to broader classes of theories and dual geometries.

Abstract

We compute the large N limit of the localized three dimensional free energy of various field theories with known proposed AdS duals. We show that vector-like theories agree with the expected supergravity results, and with the conjectured F-theorem. We also check that the large N free energy is preserved by the three dimensional Seiberg duality for general classes of vector like theories. Then we analyze the behavior of the free energy of chiral-like theories by applying a new proposal. The proposal is based on the restoration of a discrete symmetry on the free energy before the extremization. We apply this procedure at strong coupling in some examples and we discuss the results. We conclude the paper by proposing an alternative geometrical expression for the free energy.

Paper Structure

This paper contains 29 sections, 105 equations, 7 figures.

Table of Contents

  1. Introduction
  2. $\mathcal{N}=2$ CS toric quivers and their moduli space
  3. The field theory description
  4. The toric description
  5. The Hilbert series
  6. Vector-Like models
  7. The free energy of large $N$ vector-like quivers
  8. Relation with the geometry
  9. Examples
  10. $\mathbb{C} \times \mathcal{C}$.
  11. $\widetilde{SPP}$.
  12. $\widetilde{\mathcal{C}/\mathbb{Z}_2}$.
  13. ABJM$/\mathbb{Z}_2$.
  14. Seiberg duality in vector-like theories
  15. Matching the free energy
  16. ...and 14 more sections

Figures (7)

  • Figure 1: Quiver and toric diagram for $\mathbb{C} \times \cal C$.
  • Figure 2: Quiver and toric diagram for $\widetilde{SPP}$. The diagram is plotted for CS levels $(2,-1,-1)$.
  • Figure 3: Quiver and toric diagram for $\widetilde{\mathcal{C}/\mathbb{Z}_2}$.
  • Figure 4: (Left) Quiver for $\widetilde{\mathbb{F}_0}$. According to the choice of the CS levels, it gives several theories studied in the paper. (Right) Toric diagram for ABJM$/\mathbb{Z}_2$, which corresponds to CS levels $(k,-k,k,-k)$.
  • Figure 5: Quiver diagram for the first type of the Seiberg duality
  • ...and 2 more figures