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I have spent some minutes starting to put some actual expository content into the Idea-section on higher gauge theory. Needs to be much expanded, still, but that’s it for the moment.
added pointer to the preprint
Is that a good sign that physicists are becoming more aware of 2-groups?
Intriligator is a core HEP/string theorist, even more so is his frequent coauthor Seiberg. A while back, Seiberg, who is highly influential in the field, had proposed that there ought to be a program to study higher form field symmetries in QFT (arXiv:1412.5148), without apparently being aware that this idea was considered before. Back then, with Eric Sharpe I was talking about writing a reply kindly pointing this out, which Eric eventually published as arXiv:1508.04770. This is now cited on p. 14 of the new article, so I gather an information flow eventually did happen.
Cordova is at IAS with Seiberg. If at IAS it is becoming the rule to write about higher gauge theory as if it needs no further introduction, as in this article, then I suppose this means that a decade-old communcation problem is showing signs of resolution.
They only get as far in the references to what you and John wrote in 2004 and then you and Waldorf in 2008. All that work you’ve done over the past decade is perhaps mathematically unfamiliar, but they should see the point of it from the physics perspective, no?
To put things the other way around, are they doing something for physics with 2-groups that you haven’t wanted to do?
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)
That’s rather misleading, no?
are they doing something for physics with 2-groups that you haven’t wanted to do?
I need to study the article in more detail. They highlight the Green-Schwarz mechanism. Of course that phenomenon is exactly what got me started in “Twisted differential string structures” (arXiv:0910.4001) with generalizing my 2005-version of higher gauge theory with Baez/Waldorf to the 2013 version that we have been discussing here a lot.
That’s rather misleading, no?
Yes. I had missed that sentence on earlier reading. Curious.
Maybe it’s because they’ve largely encountered people extending Dijkgraaf-Witten theory, such as Higher symmetry and gapped phases of gauge theories:
We 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.
Looks like another article that could do with a dose of higher gauge theory:
Sociologically, it’s quite intriguing this lack of communication. They’ve taken ’Poincaré 2-group’ for something other than John’s version from 2002.
They make much of Ward identities for 2-groups. In that you’ve largely worked on that page for the purposes of that course you just taught, and I don’t recall Ward identities for 2-groups appearing in our discussions before, is there something to the question
are they doing something for physics with 2-groups that you haven’t wanted to do?
The “2-group symmetries” discussed around (1.31) are transformations of 2-form fields $B$ by an exact term (as usual) plus a shift
$B \mapsto B + d \Lambda + shift \,.$The freedom to have such shifts appears in gauge transformations for 2-groups whose corresponding crossed modules $(\mathfrak{g}_1 \overset{\delta}{\to} \mathfrak{g}_0)$ have a non-trivial differential $\delta$. In this case there may be shifts as above in the image of $\delta$.
I suppose this is the origin of the 2-group terminology in the article.
If one assumes that this is a symmetry of the given Lagrangian in the usual sense, then these are gauge transformation in the usual sense and as such have usual Ward identities.
The full higher gauge theory would come to bear once one observes that genuine 2-group symmetries in addition have gauge-of-gauge transformations between them, going between some of the 1-gauge transformations of 2-form fields above. I am not sure if I see this discussed in the present article.
But in principle infinitesimal higher gauge symmetries have been absorbed into the QFT formalism way back, with the BV-BRST formalism. I am not sure if the BRST-complex for general Lie crossed modules has been written down, but it will be straightforward. I was planning to do it in my lectures before I decided that I am too busy already with laying out the more standard material.
I thought I’d take it to the blog.
Now Jacques Distler is suggesting that Seiberg et al. were looking beyond 2-groups:
“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).
Of course, 2-groups are just the first step on the way to $\infty$-groups.
Oh, is that all that’s at stake? He might have mentioned that. Presumably he knows of the full higher gauge theory story.
Presumably he knows of the full
Under this assumptions there wouldn’t be any discussion like this in the first place, I suppose.
I wish there were a forum on the web where these matters could be discussed profitably. Physics Overflow tried to be such a place, but it is not blossoming due to lack of quality input.
Another 2-group paper to be added.
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.
Regarding the papers mentioned in #2 and #16,
As this work was being completed, [89, 90] appeared, which study the first of the phenomena we mention here, operator-valued ’t Hooft anomalies, in much more detail; we direct the reader there for more on this phenomenon.
In those papers the authors introduce new background gauge fields, which are in general higher-form fields, and then modify the definition of “gauge transformation” to include transformations of these new background gauge fields which are designed to cancel the operator-valued anomalies of the type we point out here. They then prefer to use the terminology “$n$-group global symmetry” instead of “operator-valued ’t Hooft anomaly”. In this language, c-number ’t Hooft anomalies in $d$ spacetime dimensions are “$d$-group global symmetries”. We’ll stick with “’t Hooft anomaly” here since we’ve been using it so far, but in the long run getting rid of the word “anomaly” in this context is probably a good idea.
Re-reading the last paragraph of #4 above
If at IAS it is becoming the rule to write about higher gauge theory as if it needs no further introduction, as in this article, then I suppose this means that a decade-old communcation problem is showing signs of resolution.
reminds me of the announcement that Jacob Lurie takes a position at IAS in a few days (here).
Oh wow, I didn’t hear that!
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