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

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