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.


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

The most general way to phrase what a symmetry of a field theory is (which led us to understand these higher symmetries) is as a subalgebra of topological operators in that theory (so the most general higher symmetry is itself a TQFT). These act on the other operators by fusion, braiding, etc.

For instance, operators which are codimension 1 in spacetime can be measured along a spatial slice, and the topological invariance implies that this measurement is unchanged by time evolution, so it is a conserved charge. Turning the operator to be transverse to the spatial slice results in a domain wall for the corresponding 0-form symmetry.

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.

I would say that 1-form group symmetries are in some ways simpler than 0-form symmetries, because they are related to the 2nd homotopy group, which is always abelian, while the 1st homotopy group can be nonabelian. The real nonbelian higher symmetries are as complicated as TQFTs though.

Forgot to answer your question: to find the higher symmetries of a theory, just identify all its topological operators!


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