### Physics and 2-Groups

#### Posted by David Corfield

The founding vision of this blog, higher gauge theory, progresses. Three physicists have just brought out Exploring 2-Group Global Symmetries . Two of the authors have worked with Nathan Seiberg, a highly influential physicist, who, along with three others, had proposed in 2014 a program to study higher form field symmetries in QFT (Generalized Global Symmetries), without apparently being aware that this idea was considered before.

Eric Sharpe then published an article the following year, Notes on generalized global symmetries in QFT, explaining how much work had already been done along these lines:

The recent paper [1] proposed a more general class of symmetries that should be studied in quantum field theories: in addition to actions of ordinary groups, it proposed that we should also consider ‘groups’ of gauge fields and higher-form analogues. For example, Wilson lines can act as charged objects under such symmetries. By using various defects, the paper [1] described new characterizations of gauge theory phases.

Now, one can ask to what extent it is natural for n-forms as above to form a group. In particular, because of gauge symmetries, the group multiplication will not be associative in general, unless perhaps one restricts to suitable equivalence classes, which does not seem natural in general. A more natural understanding of such symmetries is in terms of weaker structures known as 2-groups and higher groups, in which associativity is weakened to hold only up to isomorphisms.

There are more 2-groups and higher groups than merely, ‘groups’ of gauge fields and higher-form tensor potentials (connections on bundles and gerbes), and in this paper we will give examples of actions of such more general higher groups in quantum field theory and string theory. We will also propose an understanding of certain anomalies as transmutations of symmetry groups of classical theories into higher group actions on quantum theories.

In the new paper by Cordova, Dumitrescu, and Intriligator, along with references to some early work by John and Urs, Sharpe’s article is acknowledged, and yet taken to be unusual in considering more than discrete 2-groups:

The continuous 2-group symmetries analyzed in this paper have much in common with their discrete counterparts. Most discussions of 2-groups in the literature have focused on the discrete case (an exception is [17]). (p. 26)

Maybe it’s because they’ve largely encountered people extending Dijkgraaf-Witten theory, such as in Higher symmetry and gapped phases of gauge theories where the authors:

… study topological field theory describing gapped phases of gauge theories where the gauge symmetry is partially Higgsed and partially confined. The TQFT can be formulated both in the continuum and on the lattice and generalizes Dijkgraaf-Witten theory by replacing a finite group by a finite 2-group,

but then Sharpe had clearly stated that

The purpose of this paper is to merely to link the recent work [1] to other work on 2-groups, to review a few highlights, and to provide a few hopefully new results, proposals, and applications,

and he refers to a great many items on higher gauge theory, much of it using continuous, and smooth, 2-groups and higher groups.

Still, even if communication channels aren’t as fast as they might be, things do seem to be moving.

## Re: Physics and 2-Groups

See what happens - they’ve used the term ‘Poincaré 2-group’ for a 2-group made of the Poincaré group, $\mathcal{P}$, $U(1)$ and an element of $H^3(\mathcal{P}, U(1))$, when John took that name over 15 years ago for something else (see here).