Disconnected 0-Form and 2-Group Symmetries

TL;DR

This work characterizes how quantum field theories with both continuous and finite 0-form symmetries organize their global symmetry content. It introduces disconnected 0-form symmetry groups and disconnected 2-group symmetries, where finite 0-form data acts on 1-form and flavor sectors, enriching the standard connected structures. The authors develop a cohesive framework using twisted cohomology, Bockstein and twisted Bockstein maps, and Postnikov data to classify backgrounds and symmetry actions, with gauge theories as explicit realizations. They further relate these disconnected higher-form structures to mixed ’t Hooft anomalies, and show that such features persist in arbitrary spacetime dimensions, offering a path toward understanding non-Lagrangian theories through generalized symmetry data.

Abstract

Quantum field theories can have both continuous and finite 0-form symmetries. We study global symmetry structures that arise when both kinds of 0-form symmetries are present. The global structure associated to continuous 0-form symmetries is described by a connected Lie group, which captures the possible backgrounds of the continuous 0-form symmetries the theory can be coupled to. Finite 0-form symmetries can act as outer-automorphisms of this connected Lie group. Consequently, possible background couplings to both continuous and finite 0-form symmetries are described by a disconnected Lie group, and we call the resulting symmetry structure a disconnected 0-form symmetry. Additionally, finite 0-form symmetries may act on the 1-form symmetry group. The 1-form symmetries and continuous 0-form symmetries may combine to form a 2-group, which when combined with finite 0-form symmetries leads to another type of 2-group, that we call a disconnected 2-group and the resulting symmetry structure a disconnected 2-group symmetry. Examples of arbitrarily complex disconnected 0-form and 2-group symmetries in any spacetime dimension are furnished by gauge theories: with 1-form symmetries arising from the center of the gauge group, continuous 0-form symmetries arising as flavor symmetries acting on matter content, and finite 0-form symmetries arising from outer-automorphisms of gauge and flavor Lie algebras.

Figures (5)

• Figure 1: Two ways of understanding the action of $\Gamma^{(0)}$ on $\Gamma^{(1)}$. Top: A codimension-two topological defect $\gamma\in\Gamma^{(1)}$ travelling through a codimension-one topological defect $o\in\Gamma^{(0)}$ emerges as another codimension-two topological defect $o\cdot\gamma\in\Gamma^{(1)}$. Bottom: Passing a codimension-one topological defect $o\in\Gamma^{(0)}$ across a codimension-two topological defect $\gamma\in\Gamma^{(1)}$ changes it into another codimension-two topological defect $o\cdot\gamma\in\Gamma^{(1)}$.
• Figure 2: All three correlation functions shown in the figure involve a line defect with charge $\widehat{\gamma}\in\Gamma^{(1)}$ passing through a topological codimension-one defect $o\in\Gamma^{(0)}$, which converts it into another line defect with charge $o\cdot\widehat{\gamma}\in\Gamma^{(1)}$. In addition, the correlation function (1) involves a topological codimension-two defect $o^{-1}\cdot\gamma\in\Gamma^{(1)}$ linking the line defect with charge $\widehat{\gamma}$, and the correlation function (2) involves a topological codimension-two defect $\gamma\in\Gamma^{(1)}$ linking the line defect with charge $o\cdot\widehat{\gamma}$. The correlations functions (1) and (2) are the same as we discussed above. Now the correlation function (1) can be related to the correlation function (3) by contracting $o^{-1}\cdot\gamma$ on top of $\widehat{\gamma}$, leading to an additional phase factor $\widehat{\gamma}(o^{-1}\gamma)\in U(1)$. Similarly, the correlation function (2) can be related to the correlation function (3) by contracting $\gamma$ on top of $o\cdot\widehat{\gamma}$, leading to an additional phase factor $o\cdot\widehat{\gamma}(\gamma)\in U(1)$. Consistency leads to the equation (\ref{['eq:chargeconsistency']}).
• Figure 3: The local operators corresponding to matter fields $\phi_i$ are non-gauge invariant local operators, but can be made gauge invariant by inserting them at the end of a Wilson line defect $W_q$ of charge $q$ under the $U(1)$ gauge group. Thus $\phi_i$ give rise to well-defined gauge-invariant non-genuine local operators.
• Figure 4: Suppose there exists a local operator $O$ between two line defects $L_1$ and $L_2$. Consider the correlation function (1), in which we have linked $L_1$ by a topological codimension-two defect $\gamma\in\Gamma^{(1)}$. Squeezing $\gamma$ on top of $L_1$, we obtain correlation function (2) with an extra phase factor $\widehat{\pi}(q_{L_1})(\gamma)\in U(1)$. The correlation function (1) is equal to the correlation function (3), in which $\gamma$ now links $L_2$. Squeezing $\gamma$ on top of $L_2$, we obtain correlation function (4) with an extra phase factor $\widehat{\pi}(q_{L_2})(\gamma)\in U(1)$. For consistency we must have $\widehat{\pi}(q_{L_1})=\widehat{\pi}(q_{L_2})$.
• Figure 5: Pictorial representation of the homomorphism (\ref{['hom']}). In blue and teal are shown pieces of $(d-2)$-cycle Poincare dual to the $\mathcal{E}$ valued 2-cocycle $B_w$. The blue piece is labeled by an $\widetilde{z}\in\mathcal{E}$ which is a lift of $z\in\mathcal{Z}$. Upon passing the blue piece through codimension-one topological defect $o\in\Gamma^{(0)}$, we obtain an additional piece, shown in teal, labeled by the element $i[o,z]$ living in the $\Gamma^{(1)}$ subgroup of $\mathcal{E}$.