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

Lakshya Bhardwaj, Simone Giacomelli, Max Hübner, Sakura Schäfer-Nameki

SciPost Phys. 13, 101 (2022) · published 26 October 2022


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 $6d$ $\mathcal{N}=(2,0)$ 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 $6d$ $\mathcal{N}=(2,0)$ 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 $6d$ $\mathcal{N}=(2,0)$ 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 $4d$ $\mathcal{N}=2$ 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.

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