ICTP Lectures on (Non-)Invertible Generalized Symmetries

TL;DR

This work surveys generalized global symmetries with emphasis on non-invertible structures, presenting a unifying framework built around topological operators and the stacking-with-TQFT construction. It develops the theta-defect paradigm to generate universal non-invertible symmetries across 2d, 3d, and 4d, and shows how gauging and anomalies yield intricate fusion rules described by fusion higher-categories and 2-groups. The Symmetry TFT (SymTFT) is introduced as a coherent device to encode symmetries, gauging data, and generalized charges, with the Drinfeld center capturing the full space of charges. The notes connect these ideas to concrete QFT examples (including disconnected gauge groups, Ising-like RCFTs, and outer-automorphism gauging) and outline implications for confinement, dualities, and quantum gravity constraints. The overarching message is that modern symmetry theory resides in a rich categorical landscape where generalized charges and defects organize into higher-fusion structures, with the SymTFT providing a practical computational and conceptual toolkit across high-energy, condensed matter, and holographic contexts.

Abstract

What comprises a global symmetry of a Quantum Field Theory (QFT) has been vastly expanded in the past 10 years to include not only symmetries acting on higher-dimensional defects, but also most recently symmetries which do not have an inverse. The principle that enables this generalization is the identification of symmetries with topological defects in the QFT. In these lectures, we provide an introduction to generalized symmetries, with a focus on non-invertible symmetries. We begin with a brief overview of invertible generalized symmetries, including higher-form and higher-group symmetries, and then move on to non-invertible symmetries. The main idea that underlies many constructions of non-invertible symmetries is that of stacking a QFT with topological QFTs (TQFTs) and then gauging a diagonal non-anomalous global symmetry. The TQFTs become topological defects in the gauged theory called (twisted) theta defects and comprise a large class of non-invertible symmetries including condensation defects, self-duality defects, and non-invertible symmetries of gauge theories with disconnected gauge groups. We will explain the general principle and provide numerous concrete examples. Following this extensive characterization of symmetry generators, we then discuss their action on higher-charges, i.e. extended physical operators. As we will explain, even for invertible higher-form symmetries these are not only representations of the $p$-form symmetry group, but more generally what are called higher-representations. Finally, we give an introduction to the Symmetry Topological Field Theory (SymTFT) and its utility in characterizing symmetries, their gauging and generalized charges. Lectures prepared for the ICTP Trieste Spring School, April 2023.

Figures (18)

• Figure 1: The co-dimension 1 (i.e. $d-1$-dimensional) topological operator $D_{d-1}^{(g)}(M_{d-1})$ is a generator of the 0-form symmetry group $G$. The charge of the local operator (physical, not necessarily topological) $\mathcal{O}_0$ located at the point ${\bf p}$ is computed in terms of the linking of the manifold $M_{d-1}$ and the point ${\bf p}$. The charge is invariant under deformations of the manifold $M_{d-1}$, which do not change the linking, i.e. as long as the operator insertion is contained within the volume that $M_{d-1}$ bounds.
• Figure 2: Non-Invertible fusion of two $q$-dimensional topological defects $D_q^{(a)}$ and $D_q^{(b)}$ into a sum of $q$-dimensional topological defects $D_q^{(c_k)}$. In the group-like case, $a, b \in G$, the right hand side would be $D_q^{(ab)}$ with $ab \in G$.
• Figure 3: Action of the Kramers-Wannier duality defect $D$ on the local operator $\sigma$. As we move this through the topological defect, via the figure in the middle, we can use the fusion $D^2 = \text{id} \oplus \sigma$, which then results in the right most figure. The operator becomes a disorder operator $\sigma'$, i.e. an operator with the same conformal weights, but now attached to an $\eta$-line.
• Figure 4: The layer structure of topological defects of dimensions $q=2, 1, 0$, denoted by $D_q^{(a)}$ is shown on the LHS. The two surfaces $D_2^{(a)}$ and $D_2^{(b)}$ can have topological line defects as an interfaces (or junctions) $D_1^{(a,b)}$ and $D_1^{(a,b)'}$, which in turn can have a 0-dimensional interface (junction) $D_0$. This is precisely the setting for a fusion 2-category (RHS): the objects are the surfaces, and 1-morphisms between objects are the line operators. In turn the morphisms between 1-morphisms, i.e. points, are called 2-morphisms.
• Figure 5: Screening of a $p$-form symmetry: $p$-dimensional defects $\mathcal{O}_{p}$ are charged under the $p$-form symmetry generators $D_{d-(p+1)}^{(g)}$. However, if $\mathcal{O}_p$ can end on $\mathcal{O}_{p-1}$, these $p$-form symmetries can get screened, by following the equality in the figure.