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On the 6d Origin of Non-invertible Symmetries in 4d

Vladimir Bashmakov, Michele Del Zotto, Azeem Hasan

TL;DR

This work advances the program of realizing non-invertible symmetries in 4d by leveraging six-dimensional (2,0) theories. By compactifying on Riemann surfaces and carefully tracking global structures via the 6d partition vector, the authors connect fixed points of the mapping class group to emergent 0-form symmetries and detect mixed anomalies with one-form symmetries, which after gauging yield non-invertible M-ality defects. They reproduce known 4d N=4 duality defects and, crucially, construct infinite families of defects of order $M=p^k$ through higher-genus constructions and prime-field techniques, illustrating the rich 6d–4d interplay. The results lay groundwork for a broader classification and fusion-algebra analysis of non-invertible defects across class-S theories, with potential extensions to other gauge algebras and punctured geometries.

Abstract

It is well-known that six-dimensional superconformal field theories can be exploited to unravel interesting features of lower-dimensional theories obtained via compactifications. In this short note we discuss a new application of 6d (2,0) theories in constructing 4d theories with Kramers-Wannier-like non-invertible symmetries. Our methods allow to recover previously known results, as well as to exhibit infinitely many new examples of four dimensional theories with "M-ality" defects (arising from operations of order $M$ generalizing dualities). In particular, we obtain examples of order $M=p^k$, where $p>1$ is a prime number and $k$ is a positive integer.

On the 6d Origin of Non-invertible Symmetries in 4d

TL;DR

This work advances the program of realizing non-invertible symmetries in 4d by leveraging six-dimensional (2,0) theories. By compactifying on Riemann surfaces and carefully tracking global structures via the 6d partition vector, the authors connect fixed points of the mapping class group to emergent 0-form symmetries and detect mixed anomalies with one-form symmetries, which after gauging yield non-invertible M-ality defects. They reproduce known 4d N=4 duality defects and, crucially, construct infinite families of defects of order through higher-genus constructions and prime-field techniques, illustrating the rich 6d–4d interplay. The results lay groundwork for a broader classification and fusion-algebra analysis of non-invertible defects across class-S theories, with potential extensions to other gauge algebras and punctured geometries.

Abstract

It is well-known that six-dimensional superconformal field theories can be exploited to unravel interesting features of lower-dimensional theories obtained via compactifications. In this short note we discuss a new application of 6d (2,0) theories in constructing 4d theories with Kramers-Wannier-like non-invertible symmetries. Our methods allow to recover previously known results, as well as to exhibit infinitely many new examples of four dimensional theories with "M-ality" defects (arising from operations of order generalizing dualities). In particular, we obtain examples of order , where is a prime number and is a positive integer.
Paper Structure (18 sections, 67 equations, 2 figures)

This paper contains 18 sections, 67 equations, 2 figures.

Table of Contents

  1. Abstract
  2. Introduction
  3. "M-ality" from $\mathbb Z_M^{(0)} \times \mathbb Z_N^{(1)}$ mixed anomalies
  4. Global structures from 6d -- a quick review
  5. The 6d partition vector
  6. 6d origin of global structure of class $\mathcal{S}$ theories
  7. Non-invertible symmetries from 6d
  8. 0-form symmetry from mapping class group fixed points
  9. Reading off the mixed anomaly from 6d
  10. Examples of applications
  11. The $\mathbb Z_2^{(0)} \times \mathbb Z_2^{(1)}$ mixed anomaly in $SO(3)_{-}$ at $\tau = i$ from 6d
  12. More $\mathcal{N}=4$ SYM examples from class $\mathcal{S}$
  13. Higher genus examples with non-invertible symmetries
  14. Geometry of Riemann surfaces $\Sigma_g^*$ with $\mathbb Z_{4g+2}$ isometry
  15. Infinitely many examples of $p^k$-ality defects
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Schematic picture of 6d SCFTs as relative field theories to a 7d TFT
  • Figure 2: Decagon in the hyperbolic space, giving rise to a $g=2$ surface upon identification of the opposite sides. Coloured lines indicate homology cycles on the surface, among which any four are linearly independent.