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February 15, 2018

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.

Posted at February 15, 2018 9:18 AM UTC

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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, 𝒫\mathcal{P}, U(1)U(1) and an element of H 3(𝒫,U(1))H^3(\mathcal{P}, U(1)), when John took that name over 15 years ago for something else (see here).

Posted by: David Corfield on February 15, 2018 10:30 AM | Permalink | Reply to this

Re: Physics and 2-Groups

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 Γ\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 Γ\Gamma is discrete, since there are no differential forms in evidence.

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

Posted by: Jacques Distler on February 15, 2018 5:54 PM | Permalink | PGP Sig | Reply to this

Re: Physics and 2-Groups

far more general than 2-groups

Is this in line with the previous discussion of whether 2-groups can cope with 6d SCFTs?

In the current paper they write

…six-dimensional gauge theories…can possess 2-group symmetries.

By contrast, six-dimensional SCFTs do not admit 2-group symmetries [4], because they cannot possess a conformal primary 2-form current.

Posted by: David Corfield on February 15, 2018 9:23 PM | Permalink | Reply to this

Re: Physics and 2-Groups

The statement is more general. They’re observing that a 6D SCFT cannot have a continuous “1-form symmetry” (whether or not it fits into a 2-group).

A theory with a continuous 1-form symmetry has a conserved 2-form current. But there’s no superconformal multiplet (of either (1,0) or (2,0) superconformal symmetry) containing a conserved 2-form current.

Posted by: Jacques Distler on February 15, 2018 11:49 PM | Permalink | PGP Sig | Reply to this

Re: Physics and 2-Groups

Sorry to persist, but is this just a matter of 2-groups won’t do, but the resources of the full account with \infty-groups should suffice?

Will the full nLab: higher gauge theory not suffice:

higher gauge fields have a rich global (“topological”) structure, witnessed by the higher analog of their instanton sectors. Namely, while a higher gauge field to which a pp-brane may couple is locally given by a (p+1)(p+1)-form A p+1A_{p+1}, as one moves across coordinate charts this form gauge transforms by a p-form, which then itself, as one passes along two charts, transforms by a (p1)(p−1)-form, and so on.

The global structure for higher gauge fields obtained by carrying out this globalization via higher gauge transformations is the higher analog of that of a fiber bundle with connection on a bundle in higher differential geometry. This is sometimes known as a gerbe or, more generally, a principal infinity-bundle.

Posted by: David Corfield on February 16, 2018 8:46 AM | Permalink | Reply to this

Re: Physics and 2-Groups

Or to put it in other terms, do you disagree with Eric Sharpe when he writes the following in his abstract?:

It was recently argued that quantum field theories possess one-form and higher-form symmetries, labelled ‘generalized global symmetries.’ In this paper, we describe how those higher-form symmetries can be understood mathematically as special cases of more general 2-groups and higher groups, and discuss examples of quantum field theories admitting actions of more general higher groups than merely one-form and higher-form symmetries.

I guess in the context of the thread there are two questions:

Does the \infty-group, higher gauge theory account cover everything Seifert et al. were talking about?

Can either account cope with 6d SCFTs?

Posted by: David Corfield on February 16, 2018 9:38 AM | Permalink | Reply to this

Re: Physics and 2-Groups

If you’re willing to go to nn-groups (whatever those are), then yeah, I think all of these higher global symmetries can be cast in that language.

As to 6D SCFTs, I’m not sure what the word “cope” means. Cordova et al point out that a 6D SCFT cannot have a continuous 1-form global symmetry. To that, I think I can add that an interacting 6D SCFT cannot have a continuous 2-form global symmetry.

Doubtless, there are more things that could be said, if one thought some more …

Posted by: Jacques Distler on February 16, 2018 7:14 PM | Permalink | PGP Sig | Reply to this

Re: Physics and 2-Groups

Jacques - Kapustin claims in http://physics.berkeley.edu/sites/default/files/_/PDF/kapustin.pdf that the (2,0) theory has 2-form symmetries (p.26) - 6d versions of the dual electric and magnetic 1-form symmetries of 4d Maxwell theory, or of the electric 1-form and magnetic 2-form symmetry of 5d SYM. How does this reconcile with your statement?

Posted by: David Ben-Zvi on February 19, 2018 8:28 PM | Permalink | Reply to this

Re: Physics and 2-Groups

Oops sorry never mind, that’s the abelian case, you’re discussing the interacting case.

Posted by: David Ben-Zvi on February 19, 2018 8:40 PM | Permalink | Reply to this

Re: Physics and 2-Groups

Oops sorry never mind, that’s the abelian case, you’re discussing the interacting case.

Indeed, that’s the point.

You can prove, from the superconformal representation theory, that all of the components of the superconformal multiplet, containing a conserved 3-form, obey free-field equations of motion (and hence decouple).

In fact, I am pretty sure you were in the room when we reviewed this in the GST.

Posted by: Jacques Distler on February 20, 2018 2:12 AM | Permalink | PGP Sig | Reply to this

Re: Physics and 2-Groups

Perhaps one shouldn’t resist the nn-, and indeed \infty-, gruppenpest.

We know what happened last time:

It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their “Gruppenpest”: the pest of the group theory… The authors of the “Gruppenpest” wrote papers which were incomprehensible to those like me who had not studied group theory, in which they applied these theoretical results to the study of the many electron problem. The practical consequences appeared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. Yet there were no good texts from which one could learn group theory. It was a frustrating experience, worthy of the name of a pest.

I had what I can only describe as a feeling of outrage at the turn which the subject had taken…

As soon as this [Slaters] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as “Slater has slain the ‘Gruppenpest’”. I believe that no other piece of work I have done was so universally popular.

Posted by: David Corfield on February 17, 2018 7:22 AM | Permalink | Reply to this

Re: Physics and 2-Groups

David Gross tells the story of how, when he was in graduate school, he and his mates spent an entire semester studying the representation theory of SU(3)SU(3) – a subject which might, nowadays, merit a lecture or two in a class on group representation theory.

In practical situations, the symmetry of a QFT is not simply a product G (0)×BΓ (1)×B 2Γ (2)×G_{(0)}\times B\Gamma_{(1)}\times B^2\Gamma_{(2)}\times \dots but is, rather, some complicated extension.

Perhaps that make the 2-group (or nn-group) point of view more natural. But that’s not exactly clear to me at the moment.

I would be happy to hear how any of the results of that paper could be more naturally phrased in the language of 2- (or nn-) groups.

Posted by: Jacques Distler on February 17, 2018 8:17 AM | Permalink | PGP Sig | Reply to this

Re: Physics and 2-Groups

Could someone not organise a two or three week workshop to bring together those theoretical physicists who are reaching for higher groups and those mathematical physicists who have developed higher gauge theory?

Posted by: David Corfield on February 19, 2018 8:25 AM | Permalink | Reply to this

Re: Physics and 2-Groups

That would be nice. There is the Durham workshop later this year…

Posted by: David Roberts on February 19, 2018 10:59 AM | Permalink | Reply to this

Re: Physics and 2-Groups

Jacques wrote:

In practical situations, the symmetry of a QFT is not simply a product G (0)×BΓ (1)×B 2Γ (2)×G_{(0)}\times B\Gamma_{(1)}\times B^2\Gamma_{(2)}\times \dots but is, rather, some complicated extension.

The ‘Postnikov tower’ description of an nn-group (or equivalently, a connected pointed homotopy nn-type) is what mathematicians came up with to work with these iterated extensions.

I described the idea in Section 3 of my paper with Mike Shulman called Lectures on nn-categories and cohomology, though probably not in a very physicist-friendly manner. This section is called “Cohomology: the layer-cake philosophy”. The layers of your layer cake are the Eilenberg–MacLane spaces

K(Γ (n),n)=B n(Γ (n)) K(\Gamma_{(n)}, n) = B^n(\Gamma_{(n)})

The ‘jam’ or ‘frosting’ that holds the layers together is the Postnikov data.

Posted by: John Baez on February 18, 2018 10:18 PM | Permalink | Reply to this

Re: Physics and 2-Groups

I described the idea in Section 3 of my paper with Mike Shulman called Lectures on nn-categories and cohomology, though probably not in a very physicist-friendly manner.

That’s putting it mildly.

To wrap my head around it, it would be helpful (maybe a future blog post?) to spell out the Potsnikov data associated to some of Tachikawa’s examples.

More generally, if one delves a little deeper, one runs into more general relative field theories than your run-of-the-mill anomalous theories (theories relative to a (d+1)(d+1)-dimensional invertible field theory) when studying the “obstruction to gauging” some (possibly higher) symmetry.

Posted by: Jacques Distler on February 19, 2018 4:08 AM | Permalink | PGP Sig | Reply to this

Re: Physics and 2-Groups

Unfortunately, Urs does not write here now. But you can see what he thinks of ‘relative field theories’ at nLab: twisted smooth cohomology in string theory:

Meanwhile in (Freed-Teleman 2012) special cases of the general notion of twisted fields above are being called relative fields. We briefly spell out how the definitions considered in that article are examples of the general notion above.

That addition was reported in this nForum discussion. I’m sure Urs would be only too happy to answer any questions there.

Posted by: David Corfield on February 19, 2018 8:43 AM | Permalink | Reply to this

Re: Physics and 2-Groups

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.

Posted by: John Baez on February 18, 2018 9:52 PM | Permalink | Reply to this

Re: Physics and 2-Groups

I wonder how rapidly the invention of the wheel took hold. Perhaps people were promoting the hexagon, and even better, the octagon, while the proponents of the circle sighed.

Maybe the younger generation will adapt first. Samuel Monnier has a paper Hamiltonian anomalies from extended field theories where the nLab is acknowledged as

a very useful reference for many of the higher categorical concepts appearing in the present paper,

and the author knows of the larger picture:

Our construction generalizes the construction of the classical Dijkgraaf-Witten theory by Freed… and is strongly inspired by this work. Note that such theories have been constructed using elaborate technology under the name of \infty-Chern-Simons theories.

This technology is described at Schreiber: infinity Chern Simons theory.

Whether something is judged elaborate or simple depends on one’s commitments, as Polanyi explained. Lakatos feared that this ‘personal knowledge’ introduced an elitist, unarticulable Fingerspitzengefühl (“finger tips feeling”) into what should be objectively assessable.

Posted by: David Corfield on February 19, 2018 10:48 AM | Permalink | Reply to this

Re: Physics and 2-Groups

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.

Posted by: David Corfield on October 15, 2018 9:17 PM | Permalink | Reply to this

Re: Physics and 2-Groups

Referring to this paper and to the first mentioned in the post, we read in Daniel Harlow, Hirosi Ooguri, Symmetries in quantum field theory and quantum gravity, (arXiv:1810.05338):

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 “nn-group global symmetry” instead of “operator-valued ’t Hooft anomaly”. In this language, c-number ’t Hooft anomalies in dd spacetime dimensions are “dd-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.

Posted by: David Corfield on October 16, 2018 7:35 AM | Permalink | Reply to this

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