# How to identify higher-form symmetries?

A $q$-form symmetry is a symmetry that naturally acts on objects whose support is a $q$-dimensional surface (ref.1). For example, what we usually call a "regular" symmetry, is actually a $0$-form symmetry, because it acts on local operators (that is, operators supported on a point). Symmetries that act on line operators are $1$-form symmetries, etc.

Is there any systematic way to identify these $q$-form symmetries in a given theory? For example, according to the reference, there is a $1$-form symmetry whenever the gauge group has a non-trivial centre and the matter fields do not transform under it. Does this prescription exhaust all $1$-form symmetries? Or can there be $1$-form symmetries that do not arise this way? A glance at the literature seems to suggest that people usually only look at the centre to find $1$-form symmetries; but it seems unreasonable to expect that such a "simple" prescription, specially when $0$-form symmetries are very erratic and hard to identify. But I haven't been able to find a counter-example either, so maybe $q$-form symmetries (for $q\ge1$) are in a sense simpler (cf. they are always abelian) than $0$-form symmetries.

In the same vein, is there any systematic prescription for higher-form symmetries similar to that of $q=1$? I would be interested in such a prescription even it it is partial.

References.

1. Gaiotto, Kapustin, Seiberg, Willett - Generalized Global Symmetries, arXiv:1412.5148.

1-form symmetries correspond to codimension 2 topological defects. In 3+1D these are surface operators, which in gauge theories are often center symmetries, like you say, but there is also a dual possibility. Consider Maxwell electrodynamics without matter for instance. It has two $U(1)$ 1-form symmetries, with conserved charges $F$ and $\star F$, whose integrals over surfaces can be shown to give rise to topological operators. One can also construct arbitrary examples of higher SPT phases, although they are finely tuned.