1-form Symmetries of 4d N=2 Class S Theories

Abstract

We determine the 1-form symmetry group for any 4d N = 2 class S theory constructed by compactifying a 6d N=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d 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 4d, modulo screening and flavor charges. Complete specification of a 4d 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 4d 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.

Figures (43)

• Figure 1: A closed ${\mathbb Z}_2$ twist line $o$ is inserted along the B-cycle of a torus. An element $\alpha\in\widehat{Z}$ inserted along the A-cycle is acted upon by $o$ as it crosses the closed twist line. Since the A-cycle closes back to itself we deduce that only the elements $\alpha$ left invariant by the action of $o$ can be inserted along the A-cycle.
• Figure 2: A closed ${\mathbb Z}_2$ twist line $o$ is inserted along the B-cycle of a torus. An element $\alpha\in\widehat{Z}$ inserted along the B-cycle can be moved around and converted to the element $o\cdot \alpha$ inserted along the B-cycle.
• Figure 3: A Riemann surface of genus $g$ with a closed ${\mathbb Z}_2$ twist line $o$ wrapped along the $B_1$ cycle.
• Figure 4: Resolving various $S_3$ lines into $a$-lines and $b$-lines.
• Figure 5: An example of resolving a trivalent $S_3$ vertex into an $a$-vertex and a $b$-vertex. Notice that two $b$ lines meet to form a trivial line (since $b^2=1$), which has not been displayed. The vertex formed by $b$ lines can now be smoothened out.