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 6d N = (2, 0) theories that are boundary conditions of 7d TQFTs, with the defect group arising from surface defects. In this paper, we study codimension-two defects in 6d 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 6d N = (2, 0) theory are relative defects living inside a relative theory. These relative defects provide boundary conditions for topological defects of the 7d bulk TQFT. A codimension-two defect carrying a non-trivial defect group acts as an irregular puncture when used in the construction of 4d N = 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.

Dirac pairings, one-form symmetries and Seiberg-Witten geometries

“…In particular, an immediate implication of the known classification of rank-1 Coulomb branch geometries [21][22][23][24] is that since -with one exception -all have only principally polarized homology lattices, it follows that all rank-1 SCFTs have principal Dirac pairings, their charge and line lattices coincide, and they have no 1form symmetries. This agrees with the predictions from string constructions and BPS quivers [11][12][13][14][15][16][17][18][19][20]. The one exception is the Coulomb branch geometry of the N =2 * su(2) sYM theory which also has a model with non-principally polarized homology lattice, which is consistent with the field theory expectation.…”

“…In this work we only apply our analysis to re-derive the global structures and one-form symmetries of N = 4 sYM theories as examples, but our techniques apply more generally to N = 2 theories for which many results are by now known [11][12][13][14][15][16][17][18][19][20]. In particular, an immediate implication of the known classification of rank-1 Coulomb branch geometries [21][22][23][24] is that since -with one exception -all have only principally polarized homology lattices, it follows that all rank-1 SCFTs have principal Dirac pairings, their charge and line lattices coincide, and they have no 1form symmetries.…”

Higher Symmetries of 5d Orbifold SCFTs

“…As shown in [109], the base B is always of the form C 2 /Γ U (2) for Γ U (2) a particular set of finite subgroups of U (2), and in all these cases, ∂B = S 3 /Γ. In this case, the corresponding defect group is associated with a two-form symmetry, as specified by string-like defects of the 6d SCFT [13] (see also [16,22,23,30,41,44]). While we leave a more complete analysis for future work, in this section we observe that in situations where the geometry faithfully reproduces the 0-form symmetry of the system, Ab[Γ] is closely correlated with the 2-group symmetry of the 5d SCFT.…”

Higher symmetries of 5D orbifold SCFTs

“…Γ Uð2Þ a particular set of finite subgroups of Uð2Þ, and in all these cases, ∂B ¼ S 3 =Γ. In this case, the corresponding defect group is associated with a two-form symmetry, as specified by string-like defects of the 6d SCFT [13] (see also [16,22,23,30,41,44]). While we leave a more complete analysis for future work, in this section we observe that in situations where the geometry faithfully reproduces the 0-form symmetry of the system, Ab½Γ is closely correlated with the 2-group symmetry of the 5D SCFT.…”