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From Necklace Quivers to the F-theorem, Operator Counting, and T(U(N))

Daniel R. Gulotta, Christopher P. Herzog, Silviu S. Pufu

TL;DR

This work uses the Kapustin–Willett–Yaakov matrix model to study large-$N$ ${\cal N}=3$ necklace quivers $U(N)^d$, deriving the $S^3$ free energy $F$ and validating the $F$-theorem for certain RG flows. It establishes a deep AdS/CFT dictionary by relating the saddle-point eigenvalue distribution to operator counting and to volumes of seven-dimensional tri-Sasaki Einstein manifolds $Y$, with ${\rm Vol}(Y)$ computable from toric hyperkähler quotients and polygonal data ${\cal P}$. The paper derives a comprehensive matrix model for $(p,q)$-brane constructions, showing $SL(2,\mathbb{Z})$ invariance and providing the finite-$N$ extension that yields the $T(U(N))$ theory partition function. It also links the density $\rho(x)$ to the chiral ring via precise derivatives of the operator-counting function $\psi(r,m)$, connecting geometric volumes to operator counting in the large-$N$ limit. Overall, the results advance exact, controllable tests of 3d SCFT dualities and provide new tools for extracting holographic data from matrix models.

Abstract

The matrix model of Kapustin, Willett, and Yaakov is a powerful tool for exploring the properties of strongly interacting superconformal Chern-Simons theories in 2+1 dimensions. In this paper, we use this matrix model to study necklace quiver gauge theories with {\cal N}=3 supersymmetry and U(N)^d gauge groups in the limit of large N. In its simplest application, the matrix model computes the free energy of the gauge theory on S^3. The conjectured F-theorem states that this quantity should decrease under renormalization group flow. We show that for a simple class of such flows, the F-theorem holds for our necklace theories. We also provide a relationship between matrix model eigenvalue distributions and numbers of chiral operators that we conjecture holds more generally. Through the AdS/CFT correspondence, there is therefore a natural dual geometric interpretation of the matrix model saddle point in terms of volumes of 7-d tri-Sasaki Einstein spaces and some of their 5-d submanifolds. As a final bonus, our analysis gives us the partition function of the T(U(N)) theory on S^3.

From Necklace Quivers to the F-theorem, Operator Counting, and T(U(N))

TL;DR

This work uses the Kapustin–Willett–Yaakov matrix model to study large- necklace quivers , deriving the free energy and validating the -theorem for certain RG flows. It establishes a deep AdS/CFT dictionary by relating the saddle-point eigenvalue distribution to operator counting and to volumes of seven-dimensional tri-Sasaki Einstein manifolds , with computable from toric hyperkähler quotients and polygonal data . The paper derives a comprehensive matrix model for -brane constructions, showing invariance and providing the finite- extension that yields the theory partition function. It also links the density to the chiral ring via precise derivatives of the operator-counting function , connecting geometric volumes to operator counting in the large- limit. Overall, the results advance exact, controllable tests of 3d SCFT dualities and provide new tools for extracting holographic data from matrix models.

Abstract

The matrix model of Kapustin, Willett, and Yaakov is a powerful tool for exploring the properties of strongly interacting superconformal Chern-Simons theories in 2+1 dimensions. In this paper, we use this matrix model to study necklace quiver gauge theories with {\cal N}=3 supersymmetry and U(N)^d gauge groups in the limit of large N. In its simplest application, the matrix model computes the free energy of the gauge theory on S^3. The conjectured F-theorem states that this quantity should decrease under renormalization group flow. We show that for a simple class of such flows, the F-theorem holds for our necklace theories. We also provide a relationship between matrix model eigenvalue distributions and numbers of chiral operators that we conjecture holds more generally. Through the AdS/CFT correspondence, there is therefore a natural dual geometric interpretation of the matrix model saddle point in terms of volumes of 7-d tri-Sasaki Einstein spaces and some of their 5-d submanifolds. As a final bonus, our analysis gives us the partition function of the T(U(N)) theory on S^3.

Paper Structure

This paper contains 16 sections, 3 theorems, 107 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Volumes of Toric tri-Sasaki Einstein Spaces
  3. Toric Hyperkähler Cones from a Quotient Construction
  4. The Volume of the tri-Sasaki Einstein Base
  5. Brane Constructions and an $F$-theorem
  6. Field Theory Computation of the Free Energy
  7. ${\mathcal{N}}=3$ Matrix Model
  8. Operator Counting and the Matrix Model
  9. Operator Counting and Volumes
  10. The Matrix Model for $(p, q)$-Branes
  11. $(p,q)$-branes at finite $N$
  12. Discussion
  13. Proof of the tree formula
  14. A virial theorem for matrix models
  15. A slick proof
  16. ...and 1 more sections

Key Result

Corollary 1

i) If a column $\beta_a$ is removed from $\beta$, $\operatorname{Vol}(Y)$ increases. ii) If two columns $\beta_a$ and $\beta_b$ of $\beta$ are combined to form the new column $\beta_a+\beta_b$ of a new $\beta'$ with one fewer columns, $\operatorname{Vol}(Y)$ either increases or stays the same. The v

Figures (3)

  • Figure 1: A schematic picture of the brane construction. The $N$ D3-branes span the $0126$ direction, and the $(p_a, q_a)$ 5-branes span the $012$ directions as well as the lines in the $37$, $48$, and $59$ planes that make angles $\theta_a = \arg (p_a + i q_a)$ with the $3$, $4$, and $5$ axes, respectively. The three-dimensional ${\cal N}=3$ theories considered in this paper live on the $012$ intersection of these branes.
  • Figure 2: A necklace quiver gauge theory where the gauge sector consists of $d$$U(N)$ gauge groups with Chern-Simons coefficients $k_a$. The matter content consists of the bifundamental fields $A_a$ and $B_a$, as well as $n_a$ pairs of fundamentals and anti-fundamentals transforming under the $a$th gauge group.
  • Figure 3: a) The polygon for ABJM theory which can be built from a $(1,0)$ and $(1,k)$ brane; b) the S-dual configuration involving a $(0,1)$ and $(-k,1)$ brane; c) an $SL(2,\mathbb{Z})$ transform to a $(1,1)$ and $(1-k, 1)$ brane.

Theorems & Definitions (6)

  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof : Proof of Corollary \ref{['cor:n2ordered']}