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 …

We identify infinitely many noninvertible generalized global symmetries in QED and QCD for
the real world in the massless limit. In QED, while there is no conserved Noether current for …

For any quantum system invariant under gauging a higher-form global symmetry, we
construct a noninvertible topological defect by gauging in only half of the spacetime. This …

Symmetry plays a central role in quantum field theory. Recent developments include
symmetries that act on defects and other subsystems, and symmetries that are categorical …

A bstract We study generalized symmetries of quantum field theories in 1+ 1D generated by
topological defect lines with no inverse. This paper follows our companion paper on gapped …

A bstract In general quantum field theories (QFTs), ordinary (0-form) global symmetries and
1-form symmetries can combine into 2-group global symmetries. We describe this …

A bstract We analyze four-dimensional quantum field theories with continuous 2-group
global symmetries. At the level of their charges, such symmetries are identical to a product of …

A bstract Geometric engineering is a collection of tools developed to establish dictionaries
between local singularities in string theory and (supersymmetric) quantum fields. Extended …

A bstract We present a class of three-dimensional quantum field theories whose ordinary
global symmetries mix with higher-form symmetries to form a continuous 2-group. All these …

Symmetry is usually defined via transformations described by a (higher) group. But a
symmetry really corresponds to an algebra of local symmetric operators, which directly …