In this lecture note, we give a basic introduction to the rapidly develo** concepts of
generalized symmetries, from the perspectives of both high energy physics and condensed …

The most well-known mechanism for fermions to acquire a mass is the Nambu–Goldstone–
Anderson–Higgs mechanism, ie, after a spontaneous symmetry breaking, a bosonic field …

A global symmetry (0-symmetry) in an n-dimensional space acts on the whole space. A
higher symmetry acts on closed submanifolds (ie, loops and membranes, etc.), and those …

We investigate mixed states exhibiting nontrivial topological features, focusing on symmetry-
protected topological (SPT) phases under various types of decoherence. Our findings …

A large class of gapped phases of matter can be described by topological finite group gauge
theories. In this paper, we show how such gauge theories possess a higher-group global …

A system with charge conservation and lattice translation symmetry has a well-defined filling
ν, which is a real number representing the average charge per unit cell. We show that if ν is …

We generalize the notion of an anomaly for a symmetry to a noninvertible symmetry enacted
by surface operators using the framework of condensation in 2-categories. Given a …