# 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.

(…)

Last revised on September 4, 2020 at 02:55:15. See the history of this page for a list of all contributions to it.