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Non-invertible Symmetries of Class $\mathcal{S}$ Theories

Vladimir Bashmakov, Michele Del Zotto, Azeem Hasan, Justin Kaidi

Abstract

We study the non-invertible symmetries of class $\mathcal{S}$ theories obtained by compactifying the type $\mathfrak{a}_{p-1}$ 6d (2,0) theory on a genus $g$ Riemann surface with no punctures. After setting up the general framework, we describe how such symmetries can be classified up to genus 5. Of central interest to us is the question of whether a non-invertible symmetry is "intrinsic," i.e. whether it can be related to an invertible symmetry by discrete gauging. We then describe the higher-dimensional origin of our results, and explain how the Anomaly and Symmetry TFTs, as well as $N$-ality defects, of class $\mathcal{S}$ theories can be obtained from compactification of a 7d Chern-Simons theory. Interestingly, we find that the Symmetry TFT for theories with intrinsically non-invertible symmetries can only be obtained by coupling the 7d Chern-Simons theory to topological gravity.

Non-invertible Symmetries of Class $\mathcal{S}$ Theories

Abstract

We study the non-invertible symmetries of class theories obtained by compactifying the type 6d (2,0) theory on a genus Riemann surface with no punctures. After setting up the general framework, we describe how such symmetries can be classified up to genus 5. Of central interest to us is the question of whether a non-invertible symmetry is "intrinsic," i.e. whether it can be related to an invertible symmetry by discrete gauging. We then describe the higher-dimensional origin of our results, and explain how the Anomaly and Symmetry TFTs, as well as -ality defects, of class theories can be obtained from compactification of a 7d Chern-Simons theory. Interestingly, we find that the Symmetry TFT for theories with intrinsically non-invertible symmetries can only be obtained by coupling the 7d Chern-Simons theory to topological gravity.
Paper Structure (56 sections, 4 theorems, 205 equations, 9 figures, 9 tables)

This paper contains 56 sections, 4 theorems, 205 equations, 9 figures, 9 tables.

Table of Contents

  1. Abstract
  2. Introduction
  3. Global variants of $\mathfrak{su}(p)$ SYM
  4. Modular orbits of global variants
  5. Invariant global forms
  6. Higher-dimensional point of view
  7. Goals and organization
  8. Non-invertible symmetries for class ${\cal S}$ theories
  9. A review of class ${\cal S}$
  10. Non-invertible symmetries and the modular group
  11. Global variants of theories of class ${\cal S}$
  12. An example
  13. Modular orbits of global variants
  14. Non-invertible symmetries for genus-2 class ${\cal S}$
  15. Genus 2 and the modular group $Sp(4, {\mathbb Z}_p)$
  16. ...and 41 more sections

Key Result

Theorem 1

Gauging the ${\mathbb Z}_2^{\mathrm{EM}}$ electro-magnetic duality symmetry acting as $F={\mathsf S}$ in $(4+1)$d ${\mathbb Z}_p^{(2)}$ gauge theory gives a spin Dijkgraaf-Witten theory if and only if $p \in 4 {\mathbb N}+1$.

Figures (9)

  • Figure 1: At $\tau_{\mathrm{YM}}=i$, the $SU(2)$ theory has a non-invertible defect ${\cal N}_{\mathsf{S}}$, which can be understood as the composition of a defect $\sigma$ implementing gauging of the ${\mathbb Z}_2^{(1)}$ one-form symmetry, together with an invertible ${\mathsf S}$ defect.
  • Figure 2: Labelling of global variants of $\mathfrak{su}(2)$ SYM by matrices in $SL(2, {\mathbb Z}_2) / {\mathbb Z}_2^\times = SL(2, {\mathbb Z}_2)$. They are defined such that modular transformations $\mathsf{S}$ and $\mathsf{T}$ act from the left. The subscript $0,1$ denotes the number of copies of the invertible phase ${\pi \over 2}\int {\cal P}(B)$ that we stack with.
  • Figure 3: The 6d (2,0) theory of type $\mathfrak{a}_{N-1}$ should be thought of as a state $|\mathfrak{a}_{N-1}\rangle$ in the Hilbert space of a non-trivial 7d TQFT. This TQFT does not admit topological boundary conditions.
  • Figure 4: When $GF(L)=L$ and $F(\Omega)=\Omega$ admit solutions, then the composite ${\cal N}_F:= F G$ gives a non-invertible symmetry of the class ${\cal S}$ theory.
  • Figure 5: Two weakly-coupled limits for class ${\cal S}$ on $\Sigma_{2,0}$, with the corresponding generalized quivers. The trivalent junctions represent $T_p$ theories, while the circle represent $\mathfrak{a}_{p-1}$ gauge nodes.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1