Global Form of Flavor Symmetry Groups in 4d N=2 Theories of Class S

TL;DR

The paper presents a systematic framework to determine the global form of flavor symmetry groups for $4d$ $\mathcal{N}=2$ theories in the Class S program, arising from compactifications of $6d$ $\mathcal{N}=(2,0)$ theories on Riemann surfaces with regular punctures and outer-automorphism twists. It anchors the construction in the interplay between centers and representations, introducing manifest and true flavor groups via lattices of central charges and their Pontryagin duals, and it incorporates contributions from punctures and wrapped 6d surface defects. A wide range of consistency checks across free-field fixtures, twisted and untwisted punctures, and duality frames—along with comparisons to superconformal indices—support the method, including detailed results for Minahan-Nemeschansky theories. The framework further reveals that distinct global forms of flavor symmetry can occur for interacting SCFTs with identical local data, underscoring the global-group data as a discriminant among closely related theories and highlighting directions for extending the analysis to irregular punctures in future work.

Abstract

We provide a systematic method to deduce the global form of flavor symmetry groups in 4d N=2 theories obtained by compactifying 6d N=(2,0) superconformal field theories (SCFTs) on a Riemann surface carrying regular punctures and possibly outer-automorphism twist lines. Apriori, this method only determines the group associated to the manifest part of the flavor symmetry algebra, but often this information is enough to determine the group associated to the full enhanced flavor symmetry algebra. Such cases include some interesting and well-studied 4d N=2 SCFTs like the Minahan-Nemeschansky theories. The symmetry groups obtained via this method match with the symmetry groups obtained using a Lagrangian description if such a description arises in some duality frame. Moreover, we check that the proposed symmetry groups are consistent with the superconformal indices available in the literature. As another application, our method finds distinct global forms of flavor symmetry group for pairs of interacting 4d N=2 SCFTs (recently pointed out in the literature) whose Coulomb branch dimensions, flavor algebras and levels coincide (along with other invariants), but nonetheless are distinct SCFTs.

Figures (6)

• Figure 1: Compactifying a surface defect of $6d$$(2,0)$ theory of type $\mathfrak{g}$ along $S^1$ direction in spacetime leads to a gauge Wilson line defect in $5d$$\mathcal{N}=2$$\mathfrak{g}$ SYM. Equivalently, a gauge Wilson line in the $5d$ theory lifts to a circle compactified surface defect of the $6d$ theory. If the Wilson line transforms in representation $R$ of $\mathfrak{g}$, then we label the the corresponding surface defect by the representation $R$.
• Figure 2: A surface defect ending at a non-genuine line defect in the $6d$ theory leads to, upon circle compactification, to a gauge Wilson line defect ending at a non-gauge-invariant local operator in the $5d$ theory. Equivalently, a gauge Wilson line defect ending at a non-gauge-invariant local operator in the $5d$ theory lifts in the $6d$ theory to a surface defect ending at a non-genuine line defect compactified along the circle.
• Figure 3: An untwisted puncture vs. a twisted puncture. An untwisted puncture is a genuine codimension-2 defect in the $6d$ theory that lives at a point on the Riemann surface used to compactify the $6d$ theory down to $4d$. An $o$-twisted puncture is a non-genuine codimension-2 defect that is constrained to live at the end of a codimension-1 topological defect corresponding to an element $o$ of the $\mathcal{O}_\mathfrak{g}$ 0-form symmetry group of the $6d$ theory. The codimension-2 defect lives at a point on the Riemann surface which lies at the end of an open line on the Riemann surface along which the codimension-1 topological operator is inserted. We refer to such a line where a codimension-1 topological operator (corresponding to an element of $\mathcal{O}_\mathfrak{g}$) is inserted as an outer-automorphism twist line.
• Figure 4: Compactifying $6d$$\mathcal{N}=(2,0)$ theory of type $\mathfrak{g}$ on a cigar-like non-compact surface with an $o$-twisted puncture $\mathcal{P}$ placed at the tip of the cigar is equivalent to $5d$$\mathcal{N}=2$ SYM theory on a half-line with gauge algebra $\mathfrak{h}_o^\vee$ and a boundary condition associated to $\mathcal{P}$ (which, by an abuse of notation, we label by $\mathcal{P}$ in the figure) placed at the end of the half-line.
• Figure 5: Compactifying $6d$$\mathcal{N}=(2,0)$ theory of type $\mathfrak{g}$ on a sphere-like surface with an $o$-twisted puncture $\mathcal{P}_1$ and $o^{-1}$-twisted puncture is equivalent to compactifying $5d$$\mathcal{N}=2$ SYM theory with gauge algebra $\mathfrak{h}_o^\vee$ on an interval with boundary conditions associated to $\mathcal{P}_1$ and $\mathcal{P}_2$ inserted at the two ends of the interval.