Webs of five-branes and N=2 superconformal field theories

TL;DR

This work shows that 4d N=2 SCFTs of Gaiotto type can be realized as circle compactifications of five-dimensional webs of 5-branes terminated by 7-branes. A diagrammatic approach, via the generalized s-rule and dot diagrams, yields precise counts of Coulomb and Higgs moduli and clarifies how general punctures (labeled by Young tableaux) control flavor symmetries, including uniform treatment of E6, E7, and E8 theories. Dualities relate these 5d webs to M-theory on non-toric Calabi–Yau geometries and yield Seiberg-Witten curves consistent with Gaiotto’s class S construction; rank-N generalizations reproduce higher-rank E_n theories with correct operator spectra and central charges. The results provide a cohesive 5d/4d brane picture of isolated SCFTs, illuminate Higgs-branch embeddings of lower-puncture theories, and suggest broad avenues for extending to other gauge groups and higher-genus constructions. Overall, the paper unifies brane-web techniques with class S data to systematically realize and analyze SCFTs with exceptional flavor symmetries.

Abstract

We describe configurations of 5-branes and 7-branes which realize, when compactified on a circle, new isolated four-dimensional N=2 superconformal field theories recently constructed by Gaiotto. Our diagrammatic method allows to easily count the dimensions of Coulomb and Higgs branches, with the help of a generalized s-rule. We furthermore show that superconformal field theories with E6, E7, E8 flavor symmetry can be analyzed in a uniform manner in this framework; in particular we realize these theories at infinitely strongly-coupled limits of quiver theories with SU gauge groups.

Figures (20)

• Figure 1: Left: a brane configuration in type IIA. Vertical lines are NS5-branes and horizontal lines are D4-branes. Right: its lift to M-theory showing the M-theory circle.
• Figure 2: Left: the system in Fig. \ref{['tac:IIA']} as the compactification of M5-branes on a sphere with defects. The symbol $\bullet$ marks the simple punctures, and $\odot$ the full punctures. Right: the compactification of M5-branes corresponding to the $T[A_{N-1}]$ theory, with three full punctures. It has no obvious type IIA realization.
• Figure 3: Left: single junction of a D5, an NS5 and a $(1,1)$ 5-brane. Center: multi-junction of three bunches of $N=3$ 5-branes -- this realizes the $E_6$ theory. Right: the dual toric diagram of $\mathbb{C}^3/\mathbb{Z}_N \times \mathbb{Z}_N$, with a particular triangulation.
• Figure 4: Upper line: examples of minimal polygons with the dual 5-brane web. White dots separate consecutive segments which act as a single edge. They represent s-rules, and come from either multiple 5-branes ending on the same 7-brane, or propagation of the s-rule inside. In the web, some 5-branes jump over another 5-brane, meaning that they cross it without ending. Lower line: two examples of different allowed tessellations of a $2\times 2$ square, given the same constraints on the external edges. The webs of five-branes are related by a "flop transition".
• Figure 5: brane creation mechanism for 5-branes. On the left an NS5 becomes a $(1,1)$ 5-brane because it meets a D5, which comes from a D7-brane shown by a $\otimes$ sign. The dotted line shows the cut associated to the monodromy. On the right the D5 has disappeared because of the brane creation/annihilation mechanism, but the boundary conditions are the same as before because of the cut, along which $\tau \to \tau - 1$.