What's Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries

TL;DR

The work surveys non-invertible generalized global symmetries across spacetime dimensions, emphasizing the operator/defect perspective and the rich fusion-category structure that generalizes ordinary group-like symmetries. It develops concrete Ising-model constructions of the non-invertible Kramers–Wannier duality defect, then situates these within higher- and half-gauging frameworks to produce new non-invertible defects in 2+1d and 3+1d theories, including Maxwell, QED, QCD, and N=4 SYM. The authors demonstrate how these symmetries constrain RG flows, anomalies, and dynamics, and provide novel interpretations for phenomena such as neutral-pion decay, axion physics, and monopole/string bounds, highlighting potential implications for quantum gravity and emergent phenomena. By connecting lattice realizations, CFT machinery, and continuum gauge theories, the paper presents a unifying Picture where non-invertible symmetries offer powerful, broadly applicable tools for understanding strong coupling, dualities, and beyond-Standard-Model scenarios.

Abstract

We survey recent developments in a novel kind of generalized global symmetry, the non-invertible symmetry, in diverse spacetime dimensions. We start with several different but related constructions of the non-invertible Kramers-Wannier duality symmetry in the Ising model, and conclude with a new interpretation for the neutral pion decay and other applications. These notes are based on lectures given at the TASI 2023 summer school "Aspects of Symmetry."

Figures (21)

• Figure 1: Via the operator-state map, the action of a topological line ${\cal L}$ on a state $\ket{\cal O}$ in the Hilbert space $\cal H$ is mapped to the Euclidean configuration of the line encircling the corresponding local operator ${\cal O}(x)$.
• Figure 2: Under the operator-state map, a state $\ket{\psi}$ in the Hilbert space ${\cal H}_{\cal L}$ twisted by the topological line $\cal L$ is mapped to a point operator $\psi(x)$ attached to the topological line ${\cal L}$. Here $\psi(x)$ need not have integer spin $h-\bar{h}$.
• Figure 3: As we sweep the $\mathbb{Z}_2$ line $\eta$ past a local operator, we produce a sign for the $\mathbb{Z}_2$ odd operator $\sigma$, while the correlation function is invariant for the $\mathbb{Z}_2$ even local operator $\varepsilon$.
• Figure 4: Left: The line ${\cal D}$ flips the sign of the thermal operator $\varepsilon$. Right: As we sweep the non-invertible duality line ${\cal D}$ past the local, order operator $\sigma$, it becomes the non-local, disorder operator $\mu$ attached to the $\mathbb{Z}_2$ line $\eta$.
• Figure 5: Shrinking the ${\cal D}$ loop in this tadpole diagram creates a $(h,\bar{h})=(0,0)$ operator attached to the $\eta$ line. However, there is no such a state in ${\cal H}_\eta$, and hence this Euclidean configuration must vanish.