T-Branes, String Junctions, and 6D SCFTs

TL;DR

This work develops a string-junction based framework to map Higgs branch deformations of 6D SCFTs to nilpotent orbits of flavor algebras, implemented via multi-pronged [p,q]7-brane junctions in F-theory. The authors reconstruct the full nilpotent cone for ABCDEFG-type algebras through combinatorial string data and derive propagation rules that track brane recombination along quiver-like theories with classical gauge groups, including correlated flows in short quivers. They extend the analysis to IIA realizations, anomaly matching, and detailed short-quiver machinery, including formal SO theories and correction terms to the anomaly polynomial, with explicit examples for SU, SO, and Sp quivers and illustrations of nilpotent hierarchies. The results yield a systematic method to determine IR fixed points after T-brane deformations, clarify how left and right flavor deformations influence interior gauge nodes, and reveal flavor-symmetry enhancements unique to short quivers. The framework sets the stage for extending to exceptional algebras and pursuing a comprehensive classification of RG flows among 6D SCFT fixed points engineered in F-theory.

Abstract

Recent work on 6D superconformal field theories (SCFTs) has established an intricate correspondence between certain Higgs branch deformations and nilpotent orbits of flavor symmetry algebras associated with T-branes. In this paper, we return to the stringy origin of these theories and show that many aspects of these deformations can be understood in terms of simple combinatorial data associated with multi-pronged strings stretched between stacks of intersecting 7-branes in F-theory. This data lets us determine the full structure of the nilpotent cone for each semi-simple flavor symmetry algebra, and it further allows us to characterize symmetry breaking patterns in quiver-like theories with classical gauge groups. An especially helpful feature of this analysis is that it extends to "short quivers" in which the breaking patterns from different flavor symmetry factors are correlated.

Figures (48)

• Figure 1: Separating a collection of A-type branes leads to a deformation of $\mathfrak{su}(N)$ to the Cartan subalgebra. Open strings stretched between distinct branes are associated with specific generators in the complexified Lie algebra. In the figure, this is shown for a string stretched from brane $A_i$ to brane $A_j$.
• Figure 2: Brane diagram of strings/roots stretching between the $A$-branes yielding an $SU(N)$ flavor symmetry (see DeWolfe:1998zf). The dashed lines represent the position of branch cuts. Since they do not contribute to our analysis, they are not drawn in subsequent pictures.
• Figure 3: Three equivalent ways of describing the partition $[3,2,1]$ in the set of nilpotent orbits of $SU(6)$. To each picture is associated a different matrix, but they all have the same Jordan block decomposition and thus belong to the same equivalence class.
• Figure 4: Hasse diagram of $SU(6)$ nilpotent deformations going from top (UV) to bottom (IR) where all simple roots are turned on and all corresponding "simple strings" connect the $A$-branes.
• Figure 5: One way of flowing from the $[2^{2},1^{2}]$ nilpotent orbit (top) to the $[3,1^{3}]$ orbit (bottom). In the top figure we have the matrix representation $M_{1}=E_{1,2}+E_{3,4}$. The flow is then induced by adding an extra string stretching between the $2^{nd}$ and $3^{rd}$ branes, as illustrated in the bottom left figure. This corresponds to the matrix $M_{2}=E_{1,2}+E_{3,4}+\epsilon\cdot E_{2,4}$. This matrix is similar to $M_{2}^{\prime}=E_{1,2}+E_{2,3}$ corresponding to the bottom right diagram. Thus, both bottom string junctions belong to the same nilpotent orbit $[3,1^{3}]$.