Universal Non‐Invertible Symmetries

Abstract

It is well-known that gauging a finite 0-form symmetry in a quantum field theory leads to a dual symmetry generated by topological Wilson line defects. These are described by the representations of the 0-form symmetry group which form a 1-category. We argue that for a d-dimensional quantum field theory the full set of dual symmetries one obtains is in fact much larger and is described by a (d − 1)-category, which is formed out of lower-dimensional topological quantum field theories with the same 0-form symmetry. We study in detail a 2-categorical piece of this (d − 1)-category described by 2d topological quantum field theories with 0-form symmetry. We further show that the objects of this 2-category are the recently discussed 2d condensation defects constructed from higher-gauging of Wilson lines. Similarly, dual symmetries obtained by gauging any higher-form or higher-group symmetry also form a (d − 1)-category formed out of lower-dimensional topological quantum field theories with that higher-form or higher-group symmetry. A particularly interesting case is that of the 2-category of dual symmetries associated to gauging of finite 2-group symmetries, as it describes non-invertible symmetries arising from gauging 0-form symmetries that act on (d − 3)-form symmetries. Such non-invertible symmetries were studied recently in the literature via other methods, and our results not only agree with previous results, but our approach also provides a much simpler way of computing various properties of these non-invertible symmetries. We describe how our results can be applied to compute non-invertible symmetries of various classes of gauge theories with continuous disconnected gauge groups in various spacetime dimensions. We also discuss the 2-category formed by 2d condensation defects in any arbitrary quantum field theory.

Branes and Non‐Invertible Symmetries

“…[ 5–8 ] A number of recent works have shown that symmetry operators without inverses (and which are therefore not elements of any group, but should rather be thought of in categorical terms) are also very common in higher dimensional theories. [ 9–26 ] In this paper we will focus on one class of theories where such non‐invertible symmetries appear: $$\mathcal {N}=4$$ theories with gauge group $$\mathrm{Pin}^+(4N)$$, $$Sc(4N)$$ and $$PO(4N)$$. [ 14 ] The details are a little different in the three cases, so in this introduction we will focus on the $$Sc(4N)$$ case for concreteness.…”

“…[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] In this paper we will focus on one class of theories where such non-invertible symmetries appear:  = 4 theories with gauge group 1 Pin + (4N), Sc(4N) and PO(4N). [14] The details are a little different in the three cases, so in this introduction we will focus on the Sc(4N) case for concreteness. This theory has three 2-surface symmetry generators, which we will call D 𝖼,e 2 (Σ 2 ), D 𝗌,m 2 (Σ 2 ) and their product D 𝖼,e 2 (Σ 2 )D 𝗌,m 2 (Σ 2 ).…”

“…In order to explain how this is possible, it is useful to review briefly how the fusion relations (1) are derived in [14] (see also [12]). We start with the SO(4N) theory, which has a ℤ 2 outer automorphism 0-form symmetry and a ℤ 2 × ℤ 2 1-form symmetry.…”

Decomposition, Condensation Defects, and Fusion

“…As this paper was nearing publication, we were informed that related results will also appear in [45, 46].…”

Symmetry TFTs for Non-Invertible Defects

“…• Gauging a diagonal symmetry between the quantum field theory and a lower dimensional topological theory on a defect [37].…”