Lectures on Generalized Symmetries

TL;DR

These lecture notes provide a concise, technically detailed introduction to generalized global symmetries, focusing on invertible higher-form and higher-group structures in quantum field theory. They develop a formalism based on topological operators, background fields, and screening arguments to classify and characterize higher-form symmetries across Abelian and non-Abelian gauge theories, including discrete theta terms and symmetry-related anomalies. Central concepts include SymTFT, higher-group symmetries, and the interplay between gauging, spontaneous symmetry breaking, and topological operators, with gauge theories serving as a unifying backbone. The text also sketches advanced topics such as holographic realizations and geometric engineering in string theory, illustrating how generalized symmetries appear in broader high-energy theory contexts and their potential phenomenological implications.

Abstract

These are a set of lecture notes on generalized global symmetries in quantum field theory. The focus is on invertible symmetries with a few comments regarding non-invertible symmetries. The main topics covered are the basics of higher-form symmetries and their properties including 't Hooft anomalies, gauging and spontaneous symmetry breaking. We also introduce the useful notion of symmetry topological field theories (SymTFTs). Furthermore, an introduction to higher-group symmetries describing mixings of higher-form symmetries is provided. Some advanced topics covered include the encoding of higher-form symmetries in holography and geometric engineering constructions in string theory. Throughout the text, all concepts are consistently illustrated using gauge theories as examples.

Figures (28)

• Figure 1: Two $(d-1)$-dimensional manifolds $\Sigma_{d-1}, \Sigma'_{d-1}$ which are related by a topological deformation. $\Sigma_d$ is a $d$-dimensional manifold whose boundary is formed by $\Sigma'_{d-1}$ and an orientation-reversed copy of $\Sigma_{d-1}$. Placing a topological operator $U$ along $\Sigma_{d-1}$ is equivalent to placing it along $\Sigma'_{d-1}$.
• Figure 2: Left: A topological operator $U$ is inserted on a sphere $S^{d-1}$ which links with the point $x$ where local operator $\mathcal{O}$ is placed. Middle: $U(S^{d-1})$ has been topologically deformed such that it now does not link with $x$. Due to (\ref{['a2']}), this process modifies the local operator placed at $x$ to be $\mathcal{O}'$. Right: As $S^{d-1}$ is now the boundary of a $d$-dimensional ball, we can contract away $U(S^{d-1})$.
• Figure 3: Beginning with the top configuration, we can either take the left route in which we first fuse the topological operators and then act on the local operator, or we can take the right route in which we sequentially act on the local operator $\mathcal{O}$. Both routes must yield the same result, and so we obtain equation (\ref{['actfus']}).
• Figure 4: As shown in the figure, for $p\ge1$, one can move a topological operator associated to a $p$-form symmetry around a topological operator associated to another $p$-form symmetry. This changes the ordering of the two topological operators, implying that upon fusion the resulting operators are same $U_{gg'}=U_{g'g}$, and hence $p\ge1$-form symmetry groups have to be abelian.
• Figure 5: Moving a $p$-form symmetry topological operator $U_g$ across a $p$-dimensional operator $\mathcal{O}(M_p)$ leaves behind a topological local operator $\mathcal{O}(x)$ living on $\mathcal{O}(M_p)$ which evaluates to a number $\phi(g)\in{\mathbb C}^\times$.