Schreiber On 2-group symmetries

Recently, on hep-th, one may find articles revolving around the term “2-group symmetries” (e.g. arXiv:1802.04790, arXiv:2009.00138). These articles are concerned with differential form-expressions that are reminiscent of the curvature 2-form $d A + [A \wedge A]$ of a gauge field $A$ in Yang-Mills theory, but in higher degree, such as the 3-form expresssion $d B + CS(A)$ famously involved in the Green-Schwarz mechanism in the presence of a B-field (where $CS(A)$ is a Chern-Simons form).

Inspection of these articles and discussion with some of their authors suggests that they use the term “2-group symmetry” informally and vaguely, with no input from a theory or even a definition.

Now, such a theory does exist, it has been developed in the last decade under slightly different but clearly similar terminology, such as higher gauge theory for Lie 2-groups or more generally Lie n-groups. A key result here, non-trivial, is that Chern-Simons terms such as in the GS-mechanism indeed appear as summands of curvature 3-forms of 2-form connections (higher gauge fields), specifically for gauge 2-group the String 2-group.

So apparently some inkling of this result, but not it substance, did percolate into the hep-th folklore. Here are pointers to more of the substance:

The concept of 2-groups is old,

but for a time authors mostly considered geometrically discrete 2-groups: the higher analogue of discrete groups (permutation groups and their infinite cousins). There is no differential form data associated with these. (Though they do appear in infinity-Dijkgraaf-Witten theoryhigher analogues? of finite gouge theory such as Dijkgraaf-Witten theory).

The discussion of actual higher gauge theory for Lie 2-groups, with Lie 2-algebra valued differential forms originates with

….

But these articles left open the question of whether Chern-Simons forms such as in the Green-Schwarz mechanism may arise as summands in the higher curvature forms.

That this is indeed the case as found in

and then developed from there.

(…)