6D RG Flows and Nilpotent Hierarchies

TL;DR

The paper addresses the problem of classifying RG flows between 6D SCFTs by vevs of conformal matter, organizing flows through a nilpotent orbit hierarchy in the complexified flavor algebra $\mathfrak{g}_{\mathbb{C}}$ and mapping these to a hierarchy of IR fixed points. It develops a systematic, algebraic approach to determine the tensor branch endpoints and to extract the IR flavor symmetry as the commutant of the $\mathfrak{sl}(2,\mathbb{C})$ embedding $\rho$ provided by the Jacobson–Morozov correspondence, i.e., $\text{Comm}_{\mathfrak{g}_{\mathbb{C}}}(\mathrm{Im}\,\rho)$. The analysis covers classical algebras $SU(N)$ and $SO(2N)$ with explicit brane and Higgsing descriptions, and extends to non-simply laced cases and exceptional groups, including the role of conformal matter and spinor representations in SO-type flows; outer-automorphism equivalences are shown to identify distinct nilpotent orbits that yield the same IR theory. The work provides a coherent framework linking algebraic data from flavor symmetries to geometric constructions in F-theory/M-theory, enabling predictive characterization of IR fixed points and residual flavor content across a wide class of 6D SCFTs.

Abstract

With the eventual aim of classifying renormalization group flows between 6D superconformal field theories (SCFTs), we study flows generated by the vevs of "conformal matter," a generalization of conventional hypermultiplets which naturally appear in the F-theory classification of 6D SCFTs. We consider flows in which the parent UV theory is (on its partial tensor branch) a linear chain of gauge groups connected by conformal matter, with one flavor group G at each end of the chain, and in which the symmetry breaking of the conformal matter at each end is parameterized by the orbit of a nilpotent element, i.e. T-brane data, of one of these flavor symmetries. Such nilpotent orbits admit a partial ordering, which is reflected in a hierarchy of IR fixed points. For each such nilpotent orbit, we determine the corresponding tensor branch for the resulting SCFT. An important feature of this algebraic approach is that it also allows us to systematically compute the unbroken flavor symmetries inherited from the parent UV theory.

Figures (5)

• Figure 1: Example of a transposition of a partition.
• Figure 2: Depiction of the IIA suspended brane configuration for a 6D SCFT with $\mathfrak{su}_4$ flavor symmetry. The partitioning of the branes is specified by taking the transpose of the corresponding partition. In the figure, the vertical lines indicate D8-branes, and the horizontal lines denote D6-branes which attach on the left to D8-branes and on the right to NS5-branes.
• Figure 3: IIA realizations of $SO(10)$ nilpotent orbits. TOP: $(1^{10})$, MIDDLE: $(2^2,1^6)$, BOTTOM: $(2^4,1^2)$. Vertical lines indicate the presence of D8-branes, horizontal lines indicate D6-branes, and $\times$'s indicate NS5-branes. The numbers of D6's and D8's are displayed. A $^+$ superscript indicates an $O6^+$ while a $^-$ indicates an $O6^-$. The $(2^4, 1^2)$ theory formally requires a negative number of D6-branes, indicating a breakdown of the IIA description and the presence of $Spin(10)$ spinor representations.
• Figure 4: Flows for $SO(8)$ nilpotent orbits. Blue arrows indicate flows where one or more free tensors appears in the IR.
• Figure 5: Flows for $SO(10)$ nilpotent orbits. Blue arrows indicate flows where one or more free tensors appears in the IR.