An N=1 Lagrangian for an N=3 SCFT

TL;DR

The paper constructs an explicit $ N=1$ Lagrangian, based on an $SU(2)\times SU(2)$ gauge theory, that flows in the IR to a rank-1 $ N=3$ SCFT plus a free chiral field. It tests the proposal using anomalies, the superconformal index, and the Schur limit, showing consistency with the $ N=3$ SCFT data (notably for the moduli space $\mathbb{C}^3/\mathbb{Z}_3$ and central charges $a=c$) and matching the index up to order $(pq)^{5/3}$ after removing the decoupled chiral. Mass-deformation analysis further supports the flow to an $ N=2$ IR fixed point and then to an $ N=1$ SCFT, compatible with the proposed IR endpoint. The authors extend the program to a generalized model with a richer class S-based structure, including a rank-3 candidate with moduli space $$(\mathbb{C}^3)^3/G(3,3,3)$$, and analyze its index and Schur limit, suggesting the existence of a common conformal manifold for certain $ N=3$ theories. Overall, the work provides a concrete Lagrangian route to at least some $ N=3$ SCFTs, with nontrivial checks via RG invariants and marginal deformation structure, and opens a path to exploring higher-rank examples within the same framework.

Abstract

We propose that a certain $4d$ $\mathcal{N}=1$ $SU(2)\times SU(2)$ gauge theory flows in the IR to an $\mathcal{N}=3$ SCFT plus a single free chiral field. The specific $\mathcal{N}=3$ SCFT has rank $1$ and a dimension three Coulomb branch operator. The flow is generically expected to land at the $\mathcal{N}=3$ SCFT deformed by the marginal deformation associated with said Coulomb branch operator. We also present a discussion about the properties expected of various RG invariant quantities from $\mathcal{N}=3$ superconformal symmetry, and use these to test our proposal. Finally, we discuss a generalization to another $\mathcal{N}=1$ model that we propose is related to a certain rank $3$ $\mathcal{N}=3$ SCFT through the turning of certain marginal deformations.

Figures (5)

• Figure 1: The quiver diagram of the proposed $\mathcal{N}=1$ model. As usual in quiver diagrams, circles represent gauge symmetries while boxes represent flavor ones. Next to each matter field is written its symbol as well as its charges under the global non-R symmetries. Finally we note that the $(\bold{2},\bold{3})$ below the line associated with the field $C$ gives it representations under the two $SU(2)$ groups.
• Figure 2: A pictorial summary of the proposed relation between the Lagrangian gauge theory studied in this section and the $\mathcal{N}=3$ SCFT. Here the arrows represent RG flows, with the initial one being triggered by the asymptotically free gauge couplings, and the others by superpotential terms. Here at the end point of the flow there is also a decoupled free chiral field, as indicated by the text there.
• Figure 3: The effect of the mass deformation given by the flip field $M$. On the left we have the initial quiver with the flip field $M$ flipping $B^2$, here represented by the $X$ on the field $B$. The mass deformation causes the field $B^2$ to acquire an expectation value, leading to the Higgsing of the two $SU(2)$ groups down to the diagonal $SU(2)$. This leads to the quiver on the right. The fields $A$ and $F$ there come from the same ones on the right side, while the fields $C_Q$ and $C_F$ come from the field $C$. Here the field $C_Q$ is in the $\bold{4}$ of the $SU(2)$.
• Figure 4: A modification of the in figure \ref{['quiverN3']} by the addition of an extra bifundamentl.
• Figure 5: The model proposed to be dual to the $\mathcal{N}=3$ SCFT with moduli space $(\mathbb{C}^3)^3/G(3,3,3)$ with the charges under the gobal symmetries written in terms of fugacities. We also note that $U(1)_x$ acts on the $T_3$ theory, where it acts as $U(1)_g$.