Relative defects in relative theories: Trapped higher-form symmetries and irregular punctures in class S

Abstract

A relative theory is a boundary condition of a higher-dimensional topological quantum field theory (TQFT), and carries a non-trivial defect group formed by mutually non-local defects living in the relative theory. Prime examples are 6d6d\mathcal{N}=(2,0)𝒩=(2,0) theories that are boundary conditions of 7d7d TQFTs, with the defect group arising from surface defects. In this paper, we study codimension-two defects in 6d6d\mathcal{N}=(2,0)𝒩=(2,0) theories, and find that the line defects living inside these codimension-two defects are mutually non-local and hence also form a defect group. Thus, codimension-two defects in a 6d6d\mathcal{N}=(2,0)𝒩=(2,0) theory are relative defects living inside a relative theory. These relative defects provide boundary conditions for topological defects of the 7d7d bulk TQFT. A codimension-two defect carrying a non-trivial defect group acts as an irregular puncture when used in the construction of 4d4d\mathcal{N}=2𝒩=2 Class S theories. The defect group associated to such an irregular puncture provides extra “trapped” contributions to the 1-form symmetries of the resulting Class S theories. We determine the defect groups associated to large classes of both conformal and non-conformal irregular punctures. Along the way, we discover many new classes of irregular punctures. A key role in the analysis of defect groups is played by two different geometric descriptions of the punctures in Type IIB string theory: one provided by isolated hypersurface singularities in Calabi-Yau threefolds, and the other provided by ALE fibrations with monodromies.

The Branes Behind Generalized Symmetry Operators

“…This perspective has by now been generalized in a number of directions, and has reached the stage where there are explicit algorithms for reading off generalized symmetries for a large number of geometries. [ 6,20,23,25,26,29,34,42,46,53,63,70,72,76–78,94,128–130 ]…”

Trinions for the $3d$ compactification of the $5d$ rank 1 $E_{N_f+1}$ SCFTs

“…In the 5d/3d setup, even though there are no anomalies for continues symmetries, there are anomalies involving discrete symmetries. Specifically, certain anomalies of 5d SCFTs are known [55,56,[58][59][60][61] and we can try to match their dimensional reduction with those computed directly in the 3d models, along the lines of the field theory analysis of [61][62][63][64][65][66][67]. 7 This would provide an additional check of the proposals both of this paper and of [38,39], and possibly have new predictions for the anomalies of certain 5d and 3d theories.…”

Generalized Symmetries and Anomalies of 3d N=4 SCFTs

“…In three space-time dimensions, such anomalies are necessarily discrete. The general structure of such anomalies for 3d gauge theories was explored in our previous work [12]. This captured the information about the anomaly in terms of flavor charges carried by mixed flavor-gauge monopole operators, which are in general non-genuine local operators that arise at the end points of flavor-gauge vortex line defects.…”

“…During the flow the flavor charges of these non-genuine local operators get mixed, and the mixing can be deduced using the methods of [11]. Finally, reversing the logic of [12], one can use the information about the charges of these non-genuine local operators to deduce the 't Hooft anomaly of the IR SCFT. This provides a concrete and unambiguous way of implementing 't Hooft anomaly matching from UV 3d N = 4 gauge theories to IR 3d N = 4 SCFTs.…”

“…In a similar fashion, we also determine the 't Hooft anomalies of the IR Coulomb 0form symmetry with 1-form symmetries. The requisite operators are now what were dubbed fractional gauge monopole operators in [12], which are non-genuine local operators living at the ends of topological line operators generating the 1-form symmetry. For 3d N = 4 gauge theories, there exist BPS configurations of fractional gauge monopole operators living at the ends of topological line operators 1 .…”