Atomic Higgsings of 6D SCFTs II: Induced Flows

TL;DR

This work develops a unified framework for induced RG flows in 6d $\mathcal{N}=(1,0)$ theories, connecting atomic Higgsings driven by induced nilpotent orbits and induced discrete $E_8$ holonomies to flows among conformal matters, orbi-instantons, and little string theories. Using M-/F-theory pictures and magnetic quivers, it identifies elementary transverse slices of quaternionic dimension 1 and classifies plateau, combo, and endpoint-changing flows across A-, D-, and E-type sectors, including non-simply-laced flavors. It also proves $a$-monotonicity for key families (notably A- and D-type conformal matters) and demonstrates a dimension-function conservation for discrete homomorphisms into $E_8$, providing a robust, geometry-grounded map of the 6d RG-flow landscape and its symmetry/anomaly structure. The results yield precise inductions among nilpotent orbits and discrete homomorphisms, offer explicit magnetic-quiver realizations, and have implications for the classification of 6d SCFTs and their string-theoretic realizations.

Abstract

We study a specific type of atomic Higgsings of the 6d $\mathcal{N}=(1,0)$ theories, which we call the induced flows. For the conformal matter theory associated with a pair of nilpotent orbits, the induced flows are given by the inductions of the orbits. We also consider the induced flows for the orbi-instanton theories (as well as some little string theories) that are associated with the homomorphisms from the discrete subgroups of $\mathrm{SU}(2)$ to $E_8$. This gives a physical definition of the inductions among these discrete homomorphisms, analogous to the inductions of the nilpotent orbits. We analyze the Higgs branch dimensions, the monotonicity of the Weyl anomalies (or the 2-group structure constants for LSTs) and the brane pictures under the induced flows.

Figures (6)

• Figure 2.1: Illustration of the M-theory interpretation of an induced flow. On the left, we have the strong coupling point of an induced flow. The IR theory is a disjoint union of two theories, constructed by M5-branes probing $A_3$ and $A_1$ Kleinian singularities. The UV theory takes all the M5-branes probing a D-type Kleinian singularity, with the nilpotent VEVs induced from the trivial nilpotent orbits in the IR to get an atomic RG flow. On the right, we have the full tensor branch description of this atomic RG flow. We remark that, in the UV theory, each $\mathbb{C}^2/\Gamma_{D_4}$ singularity terminates at the full M5-branes on the left and right end, corresponding to the absence of flavour symmetry on both ends of the quiver in the F-theory description. Only in the UV theory, the M5-branes are fractionated into half M5-branes. As explained in Heckman:2016ssk, the effect of $\mathcal{O} = [3^2, 1^4]$ on each side is to shrink the outermost pair of half M5-branes and reduce the singularity type in the next segment from $\mathbb{C}^2/\Gamma_{D_5}$ to $\mathbb{C}^2/\Gamma_{D_4}$.
• Figure 2.2: An illustration of the 7-brane picture. We have used different colours to indicate the stacks the 7-branes belong to in the IR theory. The flow from the leftmost one with $[1^5]$ to the rightmost one with $[1^3]\oplus[1^2]$ is not atomic. The atomic Higgsings would go through $[2,1^3]$ and then $[2^2,1]$ before reaching the atomic induced flow. See Hassler:2019eso for more explanations on the notation that describes nilpotent VEVs by 7-branes and string junctions.
• Figure 2.3: A Type IIA brane picture illustration of the same atomic Higgsing example. The horizontal and vertical lines denote the D6- and D8-branes respectively, and the crosses are the NS5-branes. We have omitted any possible configurations right to the NS5-brane in each brane system. An atomic Higgsing corresponds to separating the whole configuration into two pieces along a direction within the D8s but perpendicular to the D6s.
• Figure 2.4: In the starting orbit $[3,2,1]$, we have three buckets with 7-branes of numbers 3, 2 and 1 respectively. The first (second, resp. third) 7-brane in each bucket is coloured blue (red, resp. orange). In the top case, the blue ones are in the same component while the red and orange ones are in the other component. As there was a red 7-brane and an orange 7-brane in the same bucket, we connect them with an open string. Therefore, we have the orbit $[1^3]\oplus[2,1]$. In the middle case, for the first component, there was a blue 7-brane and an orange 7-brane in the same bucket, so we connect them with an open string as the red one is now in the other component. This yields the orbit $[2,1^2]\oplus[1^2]$. In the bottom case, the two open strings again come from connecting the 7-branes that were in the same buckets. We have omitted the component with only one single 7-brane in the picture. The resulting orbit is $[2^2,1]$.
• Figure 3.1: Left: The M-theoretic picture of an induced flow among orbi-instanton theories, which is parallel to that of conformal matter theories except for the extra M9-brane. Here, the $E_8$ holonomy data is specified by an "induced discrete homomorphism" from $\text{Dic}_3$ to $E_8$, which will be defined in §\ref{['DEtypeorbins']} motivated by the physics of the induced flows. Right: An induced flow among heterotic LSTs, where we put M9-branes on both ends, thus obtaining the Hořava-Witten configuration Horava:1995qa.