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

What are topological phases of matter? First, they are phases of matter at zero temperature.
Second, they have a nonzero energy gap for the excitations above the ground state. Third …

A bstract It is known that the't Hooft anomalies of invertible global symmetries can be
characterized by an invertible TQFT in one higher dimension. The analogous statement …

Anomalies of global symmetries are important tools for understanding the dynamics of
quantum systems. We investigate anomalies of non-invertible symmetries in 3+ 1d using 4+ …

Topological materials have become the focus of intense research in recent years, since they
exhibit fundamentally new physical phenomena with potential applications for novel devices …

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 …

Gauging is a ubiquitous tool in many-body physics. It allows one to construct highly
entangled topological phases of matter from relatively simple phases and to relate certain …

A bstract In many gauge theories, the existence of particles in every representation of the
gauge group (also known as completeness of the spectrum) is equivalent to the absence of …

We present simple lattice realizations of symmetry-protected topological phases with q-form
global symmetries where charged excitations have q spatial dimensions. Specifically, we …

String and particle braiding statistics are examined in a class of topological orders described
by discrete gauge theories with a gauge group G and a 4-cocycle twist ω 4 of G's …