Nilpotent Networks and 4D RG Flows

TL;DR

This paper explores networks of 4D RG flows connecting $ ext{N}=2$ SCFTs to $ ext{N}=1$ SCFTs via nilpotent deformations, revealing structural patterns, numerical coincidences, and providing a comprehensive dataset for further study.

Contribution

It introduces a systematic study of nilpotent deformations in 4D SCFTs, establishing a network structure of RG flows and providing explicit classifications and data for various theories.

Findings

Fixed points connected by a network of RG flows.

Numerical coincidences in conformal anomalies.

Nearly constant ratio of $a_{IR}$ to $c_{IR}$ across flows.

Abstract

Starting from a general $\mathcal{N} = 2$ SCFT, we study the network of $\mathcal{N} = 1$ SCFTs obtained from relevant deformations by nilpotent mass parameters. We also study the case of flipper field deformations where the mass parameters are promoted to a chiral superfield, with nilpotent vev. Nilpotent elements of semi-simple algebras admit a partial ordering connected by a corresponding directed graph. We find strong evidence that the resulting fixed points are connected by a similar network of 4D RG flows. To illustrate these general concepts, we also present a full list of nilpotent deformations in the case of explicit $\mathcal{N} = 2$ SCFTs, including the case of a single D3-brane probing a $D$- or $E$-type F-theory 7-brane, and 6D $(G,G)$ conformal matter compactified on a $T^2$, as described by a single M5-brane probing a $D$- or $E$-type singularity. We also observe a number of numerical coincidences of independent interest, including a collection of theories with rational values for their conformal anomalies, as well as a surprisingly nearly constant value for the ratio $a_{\mathrm{IR}} / c_{\mathrm{IR}}$ for the entire network of flows associated with a given UV $\mathcal{N} = 2$ SCFT. The $\texttt{arXiv}$ submission also includes the full dataset of theories which can be accessed with a companion $\texttt{Mathematica}$ script.