Matrix Models for Supersymmetric Chern-Simons Theories with an ADE Classification

TL;DR

The authors identify an ADE-type (specifically simply laced affine Dynkin) classification for ${\mathcal N}=3$ Chern-Simons theories with bifundamental matter by enforcing long-range force cancellation in the large-$N$ matrix-model for $Z_{S^3}$. They prove the balancing condition $2 n_a = \sum_b n_b$ selects extended Dynkin diagrams and illustrate the framework with a detailed $D_4$ example, including explicit saddle-point regions and connections to the chiral ring via the function $\psi(r,m)$. They demonstrate generalized Seiberg duality invariance of the partition function and discuss how the free energy can depend only on average eigenvalue data, aligning with AdS$_4$/M-theory expectations through the volume of the dual tri-Sasaki Einstein space $Y$. The work also outlines operator counting from the chiral ring that matches the matrix-model predictions and lays the groundwork for simple ADE solutions and potential extensions to non-simply-laced quivers and $SO/Sp$ cases, connecting to holographic duals and universal scaling properties.

Abstract

We consider N=3 supersymmetric Chern-Simons (CS) theories that contain product U(N) gauge groups and bifundamental matter fields. Using the matrix model of Kapustin, Willett and Yaakov, we examine the Euclidean partition function of these theories on an S^3 in the large N limit. We show that the only such CS theories for which the long range forces between the eigenvalues cancel have quivers which are in one-to-one correspondence with the simply laced affine Dynkin diagrams. As the A_n series was studied in detail before, in this paper we compute the partition function for the D_4 quiver. The D_4 example gives further evidence for a conjecture that the saddle point eigenvalue distribution is determined by the distribution of gauge invariant chiral operators. We also see that the partition function is invariant under a generalized Seiberg duality for CS theories.

Figures (3)

• Figure 1: The simply laced affine Dynkin diagrams of ADE type.
• Figure 2: The eigenvalue distribution for two Seiberg dual $D_4$ theories with $N=30$ and $(k_1, k_2, k_3, k_4)=(2,0,0,0)$ (left) or $(1,-1,-1,-1)$ (right). The small black points correspond to the eigenvalues of the $U(2N)$ node. The large red points correspond to the $k_1>0$ node. The eigenvalues for the remaining gauge groups -- indicated by the blue, green and purple points -- are coincident. The thin black lines are the large $N$ analytic prediction with the additional input $\overline y_0 \approx -x/3$ (left) and $\overline y_0 \approx x/6$ (right).
• Figure 3: We conjecture that these are the six families of quivers involving both unitary and orthogonal/symplectic gauge groups that have $\mathcal{N}=3$ supersymmetry. The circular nodes correspond to unitary groups, and the square nodes correspond to orthogonal or symplectic groups. These quivers can be interpreted as non-simply-laced extended Dynkin diagrams. An edge connecting a unitary gauge group to an orthogonal/symplectic gauge group corresponds to a double bond in the Dynkin diagram, with the arrow pointing toward the unitary gauge group. We are not sure of the precise rules for deciding which gauge groups should be orthogonal and which should be symplectic. There does not appear to be a way to realize the extended Dynkin diagrams $G_2$, $I_1$, $A_{2n-1}^{(2)}$, or $D_4^{(3)}$ as quivers, since we do not have a way of interpreting triple or $\infty$ bonds, or a pair of double bonds pointed in the same direction.