Operator Counting and Eigenvalue Distributions for 3D Supersymmetric Gauge Theories

TL;DR

The paper strengthens the proposed link between the large-$N$ eigenvalue distributions of the Kapustin–Willett–Yaakov matrix model and operator counts in the chiral ring for 3D supersymmetric gauge theories, extending the tests to non-critical $R$-charges and to ${ m N}=2$ theories with nonchiral bifundamentals. It establishes an equivalence between a matrix-model volume relation and Sasaki–Einstein geometric volumes, connecting $F$ on $S^3$ with $ ext{Vol}(Y)$ and proving the correspondence for a broad class of quivers, including necklace and flavored cases. The work also develops a consistent operator-counting dictionary (via Hilbert-Poincaré and related counts) that matches the matrix-model data through derivatives of operator counts, and discusses both non-chiral and chiral bifundamental theories, highlighting successes and limitations (notably missing operators in chiral theories). These results bolster the AdS$_4$/CFT$_3$ correspondence by linking microscopic operator data to geometric volumes and saddle-point eigenvalue distributions, with practical implications for computing dimensions and volumes in strongly coupled 3D theories. The framework provides a unified view of how $F$-maximization, toric geometry, and monopole sectors encode the same information across varied supersymmetry and flavor content.

Abstract

We give further support for our conjecture relating eigenvalue distributions of the Kapustin-Willett-Yaakov matrix model in the large N limit to numbers of operators in the chiral ring of the corresponding supersymmetric three-dimensional gauge theory. We show that the relation holds for non-critical R-charges and for examples with {\mathcal N}=2 instead of {\mathcal N}=3 supersymmetry where the bifundamental matter fields are nonchiral. We prove that, for non-critical R-charges, the conjecture is equivalent to a relation between the free energy of the gauge theory on a three sphere and the volume of a Sasaki manifold that is part of the moduli space of the gauge theory. We also investigate the consequences of our conjecture for chiral theories where the matrix model is not well understood.

Figures (5)

• Figure 1: A necklace quiver gauge theory where the gauge sector consists of $d$$U(N)$ gauge groups with Chern-Simons coefficients $k_a$ and the matter content consists of the bifundamental fields $A_a$ and $B_a$.
• Figure 2: The quiver for the $\mathbb{C}^3/\mathbb{Z}_3$ theory. When the CS levels are $(2k, -k, -k)$ this field theory is believed to be dual to $AdS_4 \times M^{1, 1,1} / \mathbb{Z}_k$.
• Figure 3: The quiver for $\mathbb{C}^3/(\mathbb{Z}_2 \times \mathbb{Z}_2)$. There are four $U(N)$ gauge groups with Chern-Simons coefficients $k_a$. The matter content consists of the 12 bifundamental fields $A_{ab}$ for $a \ne b$, transforming under the fundamental of the $b$th gauge group and the antifundamental of the $a$th gauge group.
• Figure 4: The area of the polygonal regions $ABC$ and $ABCD$ is proportional to $\partial^2 \psi / \partial r \partial m$ for the $\mathbb{C}^3 / \mathbb{Z}_2 \times \mathbb{Z}_2$ quiver: a) $r - \Delta m - 2 k |m| \Delta_y <0$; b) $r - \Delta m - 2 k |m| \Delta_y >0$
• Figure 5: Quiver gauge theory believed to be dual to $AdS_4 \times Q^{2, 2,2}/\mathbb{Z}_k$.