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

by Lakshya Bhardwaj, Simone Giacomelli, Max Hubner, Sakura Schafer-Nameki

Submission summary

As Contributors: Lakshya Bhardwaj · Simone Giacomelli
Arxiv Link: (pdf)
Date submitted: 2022-02-01 17:37
Submitted by: Giacomelli, Simone
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


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.

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Submission & Refereeing History

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Submission 2201.00018v1 on 1 February 2022

Reports on this Submission

Anonymous Report 1 on 2022-6-10 (Invited Report)


This paper studied the line defect groups of large classes of irregular punctures in 6d $\mathcal{N}=(2,0)$ theories as well as the one-form symmetry and the line defect group of the Class S theories constructed using these punctures . These irregular punctures are interesting and important because they provide extra contributions to the one-form symmetry and the line defect group of the Class S theories. Conceptually, since the 6d $\mathcal{N}=(2,0)$ theories is a relative theory, these irregular punctures with nontrivial line defect group should be viewed as relative defects inside relative theories where the line defect group on the irregular punctures and the surface defect group of the 6d theory can have interesting interplay. The authors developed a general formalism for these relative defects in relative theories. Combining various techniques, they computed the line defects groups of large classes of punctures. Along the way, they also studied various new punctures that have not discussed in the literature.

The paper contained many interesting, useful and concrete results. I thus recommend the publication of this paper.

  • validity: high
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

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