Two 6d origins of 4d SCFTs: class $\mathcal{S}$ and 6d (1,0) on a torus

TL;DR

This work establishes a dual-origin framework for 4d $ ext{N}=2$ SCFTs: theories from class $ ext{S}$ built from 6d $(2,0)$ on a three-punctured sphere with a simple puncture are shown to have a second origin as torus-compactified 6d $(1,0)$ SCFTs, specifically minimal conformal matter deformations. Central charges and flavor symmetries are matched across origins, with partial puncture closures in class $ ext{S}$ corresponding to Higgs branch deformations in the $(1,0)$ theory; the matching persists under Higgs flows along nilpotent hierarchies for $rak e_6,rak e_7,rak e_8$. The anomaly polynomial machinery in 6d, including one-loop and Green–Schwarz contributions, yields 4d data after $T^2$ compactification, providing a robust cross-check of the dual origins. The results reveal enhanced flavor symmetries and whether a 4d theory is a product SCFT can be inferred from the $(1,0)$ origin, and they suggest new AGT-like connections and a geometric mirror-symmetry dictionary linking the two higher-dimensional pictures. Overall, the paper uncovers a rich web linking class $ ext{S}$ theories, 6d $(1,0)$ conformal matter, and torus compactifications, with concrete computational routes to compare central charges, flavor symmetries, and anomaly data across origins.

Abstract

We consider all 4d $\mathcal{N}=2$ theories of class $\mathcal{S}$ arising from the compactification of exceptional 6d $(2,0)$ SCFTs on a three-punctured sphere with a simple puncture. We find that each of these 4d theories has another origin as a 6d $(1,0)$ SCFT compactified on a torus, which we check by identifying and comparing the central charges and the flavor symmetry. Each 6d theory is identified with a complex structure deformation of $(\mathfrak{e}_n,\mathfrak{e}_n)$ minimal conformal matter, which corresponds to a Higgs branch renormalization group flow. We find that this structure is precisely replicated by the partial closure of the punctures in the class $\mathcal{S}$ construction. We explain how the plurality of origins makes manifest some aspects of 4d SCFTs, including flavor symmetry enhancements and determining if it is a product SCFT. We further highlight the string theoretic basis for this identification of 4d theories from different origins via mirror symmetry.

Figures (5)

• Figure 1.1: The 6d $(2,0)$ SCFTs compactified on a punctured Riemann surface $C_{g,n}$ give rise to some 4d $\mathcal{N}=2$ SCFTs, as depicted in green. On the other hand, the 6d $(1,0)$ SCFTs compactified on a $T^2$ also give rise to 4d $\mathcal{N}=2$ SCFTs, which is depicted in blue. These two sets of 4d $\mathcal{N}=2$ theories via 6d compactifications may have overlaps, and we shade these 4d theories with two different origins. The shaded area is the core interest of this paper.
• Figure 1.2: The 4d SCFTs arising from 6d $(2,0)$ SCFTs of type $\mathfrak{g}$ on a Riemann surface $C_{0,3}$, which corresponds to a 3-punctured sphere where one of the punctures is a simple puncture, have another origin from 6d $(1,0)$ SCFTs via compactifying the nilpotent Higgs branch deformations of minimal conformal matter on a $T^2$.
• Figure 3.1: A depiction of the four steps describing the Higgs branch flow that moves between the two 6d SCFTs, $1\overset{\mathfrak{su}_2}{2}\overset{\mathfrak{so}_7}{3}\overset{\mathfrak{su}_2}{2}1 \rightarrow 1\overset{\mathfrak{su}_2}{2}\overset{\mathfrak{so}_7}{3}\overset{\mathfrak{su}_2}{1} \,,$ where we have written the generic tensor branch configurations in (a) and (d) and the singular configuration when all the compact curves are shrunk in (b) and (c). The black curves are compact and the blue curves are non-compact; the Lie algebra denotes the type of singular fiber tuned over that curve, and the number beneath the compact curves is the self-intersection.
• Figure 4.1: Three examples of quivers involving baryonic $\mathfrak{su}_2$ global symmetries. The circular nodes are the gauge nodes associated to the compact curves, and the square nodes are the flavor symmetries added following Section \ref{['sec:flavor']}. The numbers in blue below each gauge node are the self-intersection numbers of that compact curve. The trivalent vertices indicate the presence of matter in the trifundamental representation and the dashed lines indicate matter transforming in the fundamental representation of the gauge node and the adjoint representation of the flavor node.
• Figure 7.1: Here we depict the Hasse diagram for both the 6d $(1,0)$ theories $\mathcal{T}_{\mathfrak{e}_6}\{Y_1, Y_2\}$ and the 4d $\mathcal{N}=2$ theories $\mathcal{S}_{\mathfrak{e}_6}\langle C_{0,3}\rangle\{Y_1, Y_2, Y_\text{simple}\}$. In each box we write the geometric configuration for $\mathcal{T}_{\mathfrak{e}_6}\{Y_1, Y_2\}$ and $[Y_1, Y_2]_A$, where $Y_1$, $Y_2$ are the Bala--Carter labels for the nilpotent orbits and $A = I\text{(nteracting)}, M\text{(ixed)}$ indicates whether the class $\mathcal{S}$ fixture $\mathcal{S}_{\mathfrak{e}_6}\langle C_{0,3}\rangle\{Y_1, Y_2, Y_\text{simple}\}$ contains a free sector. A sequence of arrows relates boxes associated to $[Y_1, Y_2]$ and $[\widetilde{Y}_1, Y_2]$ if $Y_1 > \widetilde{Y}_1$ in the partial ordering of nilpotent orbits. The central charges and flavor symmetries of the 4d $\mathcal{N}=2$ theories shown here appear in Table \ref{['tbl:E6E6']}.