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3d $\mathcal{N}=4$ mirror symmetry with 1-form symmetry

by Satoshi Nawata, Marcus Sperling, Hao Ellery Wang, Zhenghao Zhong

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Submission summary

Authors (as registered SciPost users): Marcus Sperling
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-05-30
Date submitted: 2023-04-14 13:34
Submitted by: Sperling, Marcus
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


The study of 3d mirror symmetry has greatly enhanced our understanding of various aspects of 3d $\mathcal{N}=4$ theories. In this paper, starting with known mirror pairs of 3d $\mathcal{N}=4$ quiver gauge theories and gauging discrete subgroups of the flavour or topological symmetry, we construct new mirror pairs with non-trivial 1-form symmetry. By providing explicit quiver descriptions of these theories, we thoroughly specify their symmetries (0-form, 1-form, and 2-group) and the mirror maps between them.

List of changes

1) We have clarified the considered groups $\Gamma$ in the introduction, p. 2, 3rd paragraph, 4th sentence. Moreover, in Section 2.3, p.12, 1st paragraph we have commented that the restriction to k|q and q|k is for convenience. In the sense that this choice offers straightforward quiver descriptions, while more general choices lack thereof, even though these cases are well-defined.
Lastly, we have included some comments on future generalisation in the “open questions” paragraph on p.37. Such as gauging a diagonal $Z_q$ in two Cartan factors.

2) We have added necessary and sufficient conditions for the existence of a 2-group in the discussion on p.5, below eq. (2.5). Since all unitary quiver type examples consider in this paper are of this type, the considerations are analogous in later sections. We have mentioned the non-triviality of the Postnikov class in several places for the convenience of the reader.

3) We have supplemented the results of Section 2.1.2 by explicit Hilbert Series calculations in Appendix D, specifically eqs. (D.7)-(D.11).

4) We have elaborated on the equivalence of (2.25) and (2.26) in the first paragraph of p.16. The Higgs branch side offers a clear perspective, as the discrete $Z_6$ action can be moved from one set of fundamental flavours to the other by a global rotation. Similarly, the Coulomb branch side allows to reach the same conclusion by an analysis of the set of balanced nodes and the position (and balance) of the overbalanced node.

5) We have added clarifications on multiplicity vs higher charge in quiver pictures in the text below (2.14) and in similar equations.

6) We have commented on the choice of how the discrete $Z_q$ symmetry acts on the fundamentals flavours in the captions of Figure 2 and 7. The reason behind is the freedom to apply an overall U(1) global rotation. This may have been somehow hidden in Appendix C.

7) We have clarified the use of arrows for higher charge for 3d N=4 multiplets vs the arrow for 3d N=2 chiral multiplets whenever appropriate.

8) Some typos fixed.

Published as SciPost Phys. 15, 033 (2023)

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