1-form symmetries of 4d N=2 class S theories

Abstract

We determine the 1-form symmetry group for any 4d4d\mathcal{N}=2𝒩=2 class S theory constructed by compactifying a 6d6d\mathcal{N}=(2,0)𝒩=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d6d theory has a group of mutually non-local dimension-2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually non-local line operators in 4d4d, modulo screening and flavor charges. Complete specification of a 4d4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1-form symmetry group of the chosen 4d4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1-form symmetry from a Type IIB realization of class S theories.

1-form symmetry, isolated $$ \mathcal{N} $$ = 2 SCFTs, and Calabi-Yau threefolds

“…• Does our main claim hold for more general N = 2 theories? As we will see in section 4, the arguments in [36][37][38] suggest that many isolated (as N = 2 SCFTs) class S theories (here we mean isolated theories coming from the twisted compactification of the 6D (2, 0) theory on surfaces that do not have irregular punctures) also have no 1-form symmetry. On the other hand, recent arguments suggest that the N = 3 theory related to the G(3, 3, 3) complex reflection group [39] might have Z 3 1-form symmetry [40].…”

“…Next let us focus on other constructions of N = 2 theories. In particular, let us briefly discuss how some of the class S results of [36,38] fit in with our discussion. Of the theories we have checked in these references, all have trivial one-form symmetry when they are isolated.…”

“…Of the theories we have checked in these references, all have trivial one-form symmetry when they are isolated. For example, consider (4.6) of [38], which describes the abelian group of line operators, L, (including mutually non-local lines) of the (2, 0) theory compactified on a genus g surface with n regular twisted punctures…”

“…Having non-trivial genus leads in turn to a theory with an exactly marginal gauge coupling. Somewhat more generally, consider (4.21) in [38] L…”

Liberating confinement from Lagrangians: 1-form symmetries and lines in 4d N=1 from 6d N=(2,0)

“…The 1-form symmetry group O of a Class S theory is encoded in the 1-cycles of the punctured Riemann surface as discussed in detail in the recent work [29] (which is based on [30], also see [31,32] and the study of line operators in [33]), which we review in our context in section 3.2. In a similar spirit, we argue in section 3.5 that the preserved 1-form symmetry group O r in a vacuum r is encoded in the 1-cycles of the N = 1 curve Σ r associated to the vacuum r. Our work can thus be viewed as a part of the recent surge of activity in the study of generalized symmetries of QFTs via compactifications of string theory and higher-dimensional QFTs [29,31,32,[34][35][36][37][38][39][40][41][42][43].…”