Relative Defects in Relative Theories: Trapped Higher-Form Symmetries and Irregular Punctures in Class S

TL;DR

We develop a comprehensive framework for relative defects inside relative theories, focusing on codimension-two defects in $6d\mathcal{N}=(2,0)$ theories and their realization as irregular punctures in $4d\mathcal{N}=2$ Class S. The paper constructs and analyzes defect groups that encode mutually non-local line/surface defects, revealing trapped 1-form symmetries localized at irregular punctures. By combining Higgs-field (spectral cover) monodromies, Type IIB on isolated hypersurface singularities, and ALE fibrations, the authors compute full defect groups for broad families of punctures across $A$, $D$, and $E$-type theories, including many new irregular punctures. A central theme is relating puncture defect data to generalized quivers (and their 3d reductions) to extract 1-form symmetry information and to validate these results against IHS/ALE descriptions. The outcome is a detailed map from puncture data to trapped and non-trapped components of defect groups, enabling precise determinations of 1-form symmetries for wide classes of Class S theories and deepening the bridge between string-theoretic realizations and field-theoretic observables.

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.

Figures (26)

• Figure 1: Moving a defect $\mathfrak{D}_{d-p-2}$ of the relative theory $\mathfrak{T}_d$ around another defect $\mathfrak{D}_{p}$ of the relative theory $\mathfrak{T}_d$ creates a braiding between the topological defects $\alpha_p\in\mathcal{L}_{p+1}$ and $\alpha_{d-p-2}\in\mathcal{L}_{d-p-1}$ of the TQFT $\mathfrak{S}_{d+1}$.
• Figure 2: An equivalence relation in which two defects $\mathfrak{D}_p$ and $\mathfrak{D}'_p$ are equivalent if there exists a non-trivial junction $\mathfrak{J}_{p-1}$ between them.
• Figure 3: A polarization $\Lambda$ is associated to a topological interface (that we refer to as a "boundary condition") between the TQFT $\mathfrak{S}_{d+1}$ and an SPT phase $\mathcal{A}^{\widehat{\Lambda}}_{d+1}$. A compactification of the TQFT $\mathfrak{S}_{d+1}$ on a segment with the relative theory $\mathfrak{T}_d$ at one end and $\mathfrak{B}^\Lambda_d$ at the other end, leads to the absolute theory $\mathfrak{T}^\Lambda_d$, which comes attached to the SPT phase $\mathcal{A}^{\widehat{\Lambda}}_{d+1}$. The SPT phase captures the 't Hooft anomaly of the higher-form symmetry $\widehat{\Lambda}$ of the absolute theory $\mathfrak{T}^\Lambda_d$.
• Figure 4: A defect $\mathfrak{D}_p$ of the relative theory $\mathfrak{T}_d$ attached to a topological defect $\alpha_p\in\Lambda_{p+1}$ of the TQFT $\mathfrak{S}_{d+1}$ becomes a genuine defect of the absolute theory $\mathfrak{T}^\Lambda_d$, as $\alpha_p$ can end on the topological boundary $\mathfrak{B}_d^\Lambda$.
• Figure 5: A topological defect $\alpha_p\in\mathcal{L}_{p+1}$ of the TQFT $\mathfrak{S}_{d+1}$ descends to a $(p+1)$-dimensional topological defect $[\alpha_p]\in\mathcal{L}_{p+1}/\Lambda_{p+1}$ of the absolute theory $\mathfrak{T}_d^\Lambda$.