R-charges, Chiral Rings and RG Flows in Supersymmetric Chern-Simons-Matter Theories

TL;DR

This work analyzes the non-perturbative behavior of the $U(1)_R$ symmetry in $N=2$ Chern-Simons-matter theories with gauge group $U(N)$, deriving inequalities from spontaneous SUSY breaking and Seiberg duality to reveal a network of RG flows among 3D SCFTs. It identifies an ADE-classified subset of fixed points, proposes new Seiberg-dual pairs, and draws parallels to 4D ${ m N}=1$ adjoint SQCD where $a$-maximization guides the exact R-symmetry. The study uses brane constructions and dualities to illuminate the chiral rings and operator relevance across deformations, including theories with one or two adjoints and various mesonic/quartic/sextic superpotentials. It also discusses holographic perspectives for certain cases and outlines open questions about an exact 3D analogue of $a$- or $ au_{RR}$-minimization, aiming to deepen the understanding of AdS$_4$/CFT$_3$ and M-theory realizations. Overall, the paper advances the understanding of fixed-point structure, RG flows, and dualities in three-dimensional ${ m N}=2$ CS-matter theories, with potential applications to string/M-theory realizations of M2-branes and holographic duals.

Abstract

We discuss the non-perturbative behavior of the U(1)_R symmetry in N=2 superconformal Chern-Simons theories coupled to matter in the (anti)fundamental and adjoint representations of the gauge group, which we take to be U(N). Inequalities constraining this behavior are obtained as consequences of spontaneous breaking of supersymmetry and Seiberg duality. This information reveals a web of RG flows connecting different interacting superconformal field theories in three dimensions. We observe that a subclass of these theories admits an ADE classification. In addition, we postulate new examples of Seiberg duality in N=2 and N=3 Chern-Simons-matter theories and point out interesting parallels with familiar non-perturbative properties in N=1 (adjoint) SQCD theories in four dimensions where the exact U(1)_R symmetry can be determined using a-maximization.

Figures (4)

• Figure 1: Configuration (a) engineers the electric version of the ${\cal N}=2$ CS-SQCD theory. Configuration (b) engineers the magnetic version.
• Figure 2: Phases of CS-SQCD in a $(\lambda,x)$ diagram. The blue region at the bottom is the perturbative region of the electric theory. The orange region at the right corner is where the topological ${\cal N}=2$ CS theory is recovered.
• Figure 3: A plot of $R_Q$ as a function of $\lambda$ for fixed $x>1$. $R_Q$ interpolates between its classical value $\frac{1}{2}$ and the value $\frac{1}{4}$ at the critical point $\frac{x}{x-1}$ beyond which the theory exhibits spontaneous breaking of supersymmetry. The exact behavior of the function in this interval is currently unknown.
• Figure 4: A plot of $R_X$ in the $\hat{\sf A}$ theory as a function of $\lambda$ for fixed $x$. $R_X$ interpolates between its classical value $\frac{1}{2}$ and a limiting value $R_{X,\rm lim}$ in the range $\frac{1}{2([x]+2)}<R_{X,\rm lim}<\frac{2}{[x]+1}$. $\lambda^*_{n+1}$ is the point where the chiral operator $\mathop{{\rm Tr}} X^{n+1}$ becomes marginal. The critical point $\frac{nx}{x-n}$ is where the ${\sf A}_{n+1}$ theory exhibits spontaneous breaking of supersymmetry.