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.

Figures (10)

• Figure 1: On the left hand side $\mathfrak{T}$ is a $d$-dimensional theory with $\Gamma^{(0)}$ 0-form global symmetry stacked with a 1d TQFT $\mathsf{T}_R$ with $\Gamma^{(0)}$ 0-form symmetry, which is specified by a representation $R$ of $\Gamma^{(0)}$. After gauging the diagonal $\Gamma^{(0)}$ these two formerly decoupled theories become coupled, and the 1d TQFT becomes a topological line defect $D_1^{(R)}$ (topological Wilson line) of the $d$-dimensional gauged theory $\mathfrak{T}/\Gamma^{(0)}$. The topological line defect $D_1^{(R)}$ describes a (dual) symmetry of $\mathfrak{T}/\Gamma^{(0)}$.
• Figure 2: On the left hand side $\mathfrak{T}$ is a $d$-dimensional theory with $\Gamma^{(p)}$$p$-form global symmetry stacked with a $(p+1)$-dimensional TQFT $\mathsf{SPT}_\chi$ which is an SPT phase protected by $\Gamma^{(p)}$$p$-form symmetry associated to a character $\chi$ of $\Gamma^{(p)}$. After gauging $\Gamma^{(p)}$, these two formerly decoupled theories become coupled, and the $p+1$ dimensional SPT becomes a $p+1$ dimensional topological defect $D_{p+1}^{(\chi)}$ (Wilson surface) in the $d$-dimensional gauged theory $\mathfrak{T}/\Gamma^{(p)}$, which generates a dual $(d-p-2)$-form symmetry of $\mathfrak{T}/\Gamma^{(p)}$.
• Figure 3: On the left hand side, we have an $\mathcal{S}$-symmetric $d$-dimensional QFT $\mathfrak{T}$ stacked with decoupled $\mathcal{S}$-symmetric $(d-1)$-dimensional TQFTs $\mathsf{T}^{(i)}$, along with an $\mathcal{S}$-symmetric $(d-2)$-dimensional topological interface $\mathcal{I}^{(1)(2)}$ between $\mathsf{T}^{(i)}$, where $\mathcal{S}$ is either a higher-form or a higher-group symmetry. Upon gauging $\mathcal{S}$ diagonally, $\mathsf{T}^{(i)}$ become $(d-1)$-dimensional topological defects $D_{d-1}^{(i)}$ of the gauged $d$-dimensional theory $\mathfrak{T}/\mathcal{S}$ and $\mathcal{I}^{(1)(2)}$ becomes a topological $(d-2)$-dimensional defect $D_{d-2}^{(1)(2)}$ of $\mathfrak{T}/\mathcal{S}$ living at the interface between defects $D_{d-1}^{(i)}$.
• Figure 4: Associativity condition satisfied by line operators $L_\gamma$ and local operators $O_{\gamma,\gamma'}$.
• Figure 5: The operators $O_{\gamma_i}$ for $\gamma_i\in{\Gamma^{(0)}}'$ acting on the vector space carried by the boundary condition need to describe a projective representation of ${\Gamma^{(0)}}'$, because the junction local operator where lines $L_{\gamma_i}$ meet contributes the phase $\alpha_{\gamma_1,\gamma_2}\in U(1)$.