Introduction to Generalized Global Symmetries in QFT and Particle Physics

TL;DR

The notes survey generalized global symmetries in quantum field theory, detailing continuous higher-form symmetries ($G^{(p)}$), discrete symmetries, and their anomalies via background-field methods and symmetry defect operators. They introduce higher-group and 3-group structures to capture mixing between symmetries of different degrees, and develop non-invertible (categorical) symmetries through condensation defects and half-space gauging, with concrete realizations in Maxwell theory, BF theory, Yang–Mills, and axion–gauge systems. The framework clarifies how symmetries constrain RG flows and IR dynamics, including anomaly inflow mechanisms and fractional instanton phenomena, with direct implications for phenomenology and model-building. Collectively, the work provides a pedagogical bridge between formal symmetry structures and their applications to particle physics, offering tools to diagnose, constrain, and engineer IR behavior in gauge theories and beyond.

Abstract

Generalized symmetries (also known as categorical symmetries) is a newly developing technique for studying quantum field theories. It has given us new insights into the structure of QFT and many new powerful tools that can be applied to the study of particle phenomenology. In these notes we give an exposition to the topic of generalized/categorical symmetries for high energy phenomenologists although the topics covered may be useful to the broader physics community. Here we describe generalized symmetries without the use of category theory and pay particular attention to the introduction of discrete symmetries and their gauging.

Figures (8)

• Figure 1: This figure shows the deformation of a symmetry defect operator $U_g$ wrapped on a surface $\Sigma$ to a homotopically equivalent $\Sigma'$. Due to the conservation of the associated current, the symmetry operators are equivalent $U_g(\Sigma)\cong U_g(\Sigma')$.
• Figure 2: This figure illustrates how symmetry defect operators $U_g$ can act on a local operator $O(x)$ in $3d$. (a) is an initial configuration in which $U_g(\Sigma)$ wraps an $S^2$ that links $O_R(x)$. (b) shows the action of $U_g$ on $O_R(x)$ by contracting the $S^2$ through the point $x$ to $U_g(\Sigma')$.
• Figure 3: In this figure we illustrate the idea of linking manifolds in $3d$. Here, $U_1$ and $V_1$ are linking manifolds which are described by summing over the signs of the intersection points of $W_2$, which fills in $V_1$, with $U_1$.
• Figure 4: This figure illustrates how 1-form symmetry defect operators $U_g$ act on charged line operators $L(\gamma)$. (a) shows a 1-form symmetry defect operator $U_g$ wrapping a charged line operator $L(\gamma)$. (b) then shows the effect of contracting $U_g$ through the line operator.
• Figure 5: This figure illustrates why 1-form symmetries must be abelian. (a) shows a configuration of parallel 1-form symmetry defect operators $U_{g_1},U_{g_2}$ wrapping a charged line operator $L(\gamma)$. The symmetry defect operators here have a natural radial ordering. (b)-(d) show a series of topological deformations of the symmetry defect operators that allows one to exchange the order of these operators. Hence, consistency with topological invariance implies that the product of 1-form symmetry defect operators must be abelian.