Compactifications of 6d $N = (1, 0)$ SCFTs with non-trivial Stiefel-Whitney classes

TL;DR

The authors develop a framework to obtain 4d $N=2$ SCFTs by compactifying very Higgsable 6d $N=(1,0)$ theories on $T^2$ with non-trivial Stiefel-Whitney twists for flavor symmetries, viewed also as twisted reductions from 5d. They demonstrate a uniform construction of all 4d rank-1 theories with a dimension-6 Coulomb branch operator and flavor symmetries $E_8$, $ ext{USP}(10)$, $ ext{SU}(4)$, and $ ext{SU}(3)$ from a single 6d single-tensor theory, with explicit 4d spectra and central charges computed via a $u$-plane analysis and GS couplings. The work connects to twisted class S theories, 5d twisted compactifications, and brane web constructions, and extends the approach to minimal conformal matter and other 6d SCFT lifts, including $Z_2$, $Z_3$, $Z_4$, $Z_5$, and $Z_6$ twists. Central charges $a,c,k$ are derived by relating UV 6d data to the IR 4d SCFTs, with detailed consistency checks against known rank-1 classifications and class S results. The results provide a broad, unified picture of how non-trivial global structure and discrete holonomies shape the space of 4d $ obreak N=2$ SCFTs arising from higher-dimensional theories.

Abstract

We consider compactifications of very Higgsable 6d $N =(1,0)$ SCFTs on $T^2$ with non-trivial Stiefel-Whitney classes (or equivalently 't Hooft magnetic fluxes) introduced for their flavor symmetry groups. These systems can also be studied as twisted $S^1$ compactifications of the corresponding 5d theories. We deduce various properties of the resulting 4d $N=2$ SCFTs by combining these two viewpoints. In particular, we find that all 4d rank-1 $N =2$ SCFTs with a dimension-6 Coulomb branch operator with flavor symmetry $\mathfrak{e}_8$, $\mathfrak{usp}(10)$, $\mathfrak{su}(4) $ and $\mathfrak{su}(3)$ can be uniformly obtained by starting from a single-tensor theory in 6d. We also have a mostly independent appendix where we propose a rule to determine the Coulomb branch dimensions of 4d $N =2$ theories obtained by $T^2$ compactifications of 6d very Higgsable theories with and without Stiefel-Whitney twist.

Figures (12)

• Figure 1: The brane webs describing the effective $5d$ theory that results from the compactification of the previously discussed $6d$ SCFTs. Specifically, (a) is for the $\mathfrak{su}(2)+10F$ SCFT, (b) for the $\mathfrak{su}(3)+12F$ SCFT and (c) for the $\mathfrak{su}(4)+1AS+12F$ SCFT. Here we use black dots for D$7$-branes, black $+$ for $(0,1)$$7$-branes and black $X$ for $(1,1)$$7$-branes. The configuration are drawn such that they show the discrete symmetry permuting the flavor symmetry, here manifested by a rotation in the plane of the web accompanied with the appropriate $SL(2,\mathbb{Z})$ transformation.
• Figure 2: The brane web one gets by gauging part of the global symmetry of the $5d$$\mathfrak{su}(10)$ SCFT we get through the compactification of the $6d$$\mathfrak{su}(2)+ 10F$ theory with the appropriate holonomy. Specifically, we gauge both of the $\mathfrak{su}(5)$ subgroups inside the $\mathfrak{su}(10)$ global symmetry group, with an hypermultiplet in the antisymmetric for both $\mathfrak{su}(5)$ groups. In (a), we show the resulting brane web in the limit where both the $\mathfrak{su}(5)$ gauge groups are weakly coupled. In (b), we show the resulting brane web at the SCFT point. In (c) we show the web, after an S-duality, in the limit of negative but weak (in absolute value) coupling. At this point the system is described by a dual $\mathfrak{su}(6)\times \mathfrak{su}(6)$ gauge theory, with a bifundamental hypermultiplet, and a fundamental hyper plus a half-hyper in the $\bold{20}$ for each $\mathfrak{su}(6)$ gauge group.
• Figure 3: Quivers of $6d$ gauge theories which are the low-energy tensor branch descriptions of certain $6d$ SCFTs.
• Figure 4: Quivers of the $4d$ gauge theory that is expected to exist, on a generic point on the Coulomb branch spanned by the operators descending from the tensor multiplets, for theories resulting from the Stiefel-Whitney compactification of the theories in figure \ref{['6dQuiversZ2']}.
• Figure 5: Quivers of $6d$ gauge theories which are the low-energy tensor branch descriptions of certain $6d$ SCFTs. Here all gauge algebras are of type $\mathfrak{su}$.