Nilpotent Networks and 4D RG Flows

TL;DR

The paper studies how nilpotent mass deformations of 4D ${ m N}=2$ SCFTs generate networks of 4D ${ m N}=1$ fixed points, with a partially ordered structure guided by nilpotent orbits. It extends the analysis to flipper-field deformations, develops an a-maximization framework to compute IR R-symmetries and anomalies, and demonstrates monotonic flow behavior along the nilpotent cone. Through explicit data for D3-brane probes and 4D conformal matter, the authors provide strong evidence that the orbit partial order governs RG flows, while uncovering rational theories and near-constant ratios ${a_{ m IR}}/{c_{ m IR}}$. The work delivers a rich, machine-readable dataset and highlights connections to geometric constructions in F-theory and M-theory, with implications for understanding IR fixed points and possible SUSY enhancements.

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.

Figures (12)

• Figure 1: Depiction of the network of 4D RG flows generated by elements of the nilpotent cone. Starting from a UV $\mathcal{N} = 2$ fixed point, each nilpotent orbit in the flavor symmetry algebra determines a candidate $\mathcal{N} = 1$ fixed point. Additionally, the network of connections between nilpotent orbits also motivates the existence of additional flows between these $\mathcal{N} = 1$ fixed points.
• Figure 2: Depiction of the deformations from one nilpotent orbit to another. Here, we label a theory by a choice of nilpotent orbit $\mathcal{T}[\mu]$, and subsequent deformations deeper down in the nilpotent cone to theories $\mathcal{T}[\nu]$, $\mathcal{T}[\nu^{\prime}]$ and $\mathcal{T}[\nu^{\prime \prime}]$. These physical paths to new orbits are parameterized by the remnants of the original mesonic operators. An important subtlety with this picture is that as we proceed from the UV to the IR, various mesonic operators may decouple, severing some of the candidate links between theories. In explicit examples, however, we have not observed this pathological behavior.
• Figure 3: The network of RG flows induced by nilpotent plain mass deformations for $\mathcal{N} = 2$ Super Yang-Mills with $SU(2)$ gauge group and four flavors. This theory has an $SO(8)$ flavor symmetry in the UV. This network is identical to the Hasse diagram of the Lie algebra $\mathfrak{so}(8)$. The parameter $r = 2 \mathrm{Tr}_{\mathfrak{so}(8)} (T_3 T_3)$ is the embedding index for the homomorphism $\mathfrak{su}(2) \rightarrow \mathfrak{so}(8)$ defined by a nilpotent orbit. The value of the conformal anomaly $a_{\mathrm{IR}}$ decreases, as expected. These flows are determined using the method described in sections \ref{['sec:INHERIT']} and \ref{['sec:EMERGE']}.
• Figure 4: Plots of $a_{\mathrm{IR}}$ (blue stars) and $c_{\mathrm{IR}}$ (green triangles) vs embedding index $r$ for the different probe D3-brane theories. The red vertical dashed line denotes the largest value of $r$ before the Coulomb branch operator $Z$ decouples. Anything to the right of this line has a single emergent $U(1)$ to rescue the Coulomb branch operator. The plots are log-scaled on the x-axis for presentation purposes due to the fact that the region of deformed theories is denser around lower values of $r$ and becomes more sparse as $r$ increases.
• Figure 5: Plots of $c_{\mathrm{IR}}$ vs. $a_{\mathrm{IR}}$ for plain nilpotent mass deformations of the different probe D3-brane theories.