The n-Category Café

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).

It seems to me that “generalized global symmetries” (or “n-form symmetries” – to use Seiberg’s nomenclature) are far more general than 2-groups (a rather special case, involving a particular combination of “0-form” and “1-form” symmetries).

Yes, most of the recent work has focused on the case where $\Gamma$ is discrete since that’s what arises naturally in discussions of gauge theories. Though I have to say that this nomenclature is particularly odd when $\Gamma$ is discrete, since there are no differential forms in evidence.

A better name (which I learned from Dan Freed) is “$B^{n}\Gamma$ symmetry.” When $\Gamma$ is abelian, $B\Gamma$ is again an abelian group (and hence, so is $B^2\Gamma=B(B\Gamma)$, etc.). So, e.g., the $\Gamma$-principal bundles on $M$ form a group and that group acts on a theory with “$\Gamma$ 1-form symmetry.”

David wrote:

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

The internet has made certain forms of communication much more rapid and efficient, but mainly in situations where people are already predisposed to being interested in the message. When the message forces one to rethink a lot of assumptions (for example: “category theory is not something I need to know about”), I’m not sure communication has sped up very much. Most physicists will only get interested in higher groups and higher gauge theory when some very reputable physicists convince them that it’s worth the bother… which will first require that these very reputable physicists themselves be convinced.

And while this slow process may seem frustrating — for example, I’ve long since lost interest in higher gauge theory, except in the way you might pay intermittent attention to an old college friend — it may be fine. When you’re backing what you feel is a winning horse you want everyone to immediately recognize it as such. But there are dozens of research programs out there claiming to be the best thing since sliced bread. Most of them aren’t. So there has to be some sort of filtering process—and it since it requires working through complicated and unfamiliar ideas, it takes a lot of time.

Another 2-group paper

• Francesco Benini, Clay Cordova, Po-Shen Hsin, On 2-Group Global Symmetries and Their Anomalies, (arXiv:1803.09336)

In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related ’t Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the “obstruction to symmetry fractionalization” discussed in some condensed matter literature is really an instance of 2-group global symmetry.

Another physics paper using 2-groups

• Clay Cordova, Thomas T. Dumitrescu, Kenneth Intriligator, 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories, (arXiv:2009.00138)

Urs points out the references to the literature they might have made, some of it 12 years old now.