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Thanks.
Maybe it’s worth highlighting that this crowd of authors specifically (if maybe implicitly) typically means (generalized) global symmetries as opposed to (generalized) gauge symmetries?
Where the entry says
As emphasized by Freed, Moore & Teleman (2022), these are thought to be controlled by higher algebra.
I have taken the liberty of appending the following comment, let me know if you disagree:
Of course, this is not quite a new observation, ideas broadly in this direction date back to Roberts (1979). Concretly, “non-invertible symmetry insertions” are at least close to what elsewhere has long been known as defects in QFT which have routinely been discribed via higher categorical algebra, cf. eg. Lurie (2009) §4.3; Fuchs, Schweigert & Valentino (2013).
Re #3: in math, “formal” can mean “rigorously formalized”, but it can also have this other sense of superficiality. See Formal calculation which, although there’s “Talk” where this usage is disputed, I’ve certainly heard and seen many times.
For example, here and here, but maybe a better reference is to Terry Tao’s post here.
The right word in the entry would be “hand-wavy” — but “in-formal” may be more diplomatic while still conveying the idea: The main ideas of the article in question are expressed in words. It contains formulas but hardly a computation.
I’m just pointing out FYI that in math, “formal” does not have to mean “rigorously formalized”; it also has this pejorative meaning of surface-level symbol shuffling that, when examined, often has an element of unsoundness to it. I think Terry Tao expresses pretty well what is going on in this possibly confusing tangle of meanings.
Also, the codimension of the submanifolds is allowed to be non-negative, where “codimension 0” defects, it seems, are taken to be “subtheories”. These are referred to as (-1)-form symmetries (following the naming convention established in GKSW14). However, these seem to have a somewhat strange status, as they never appear in the fusion (n-)categories that are supposed to describe all the generalized symmetries of a given QFT, yet in some papers they are supposed to appear as dual symmetries (i.e. the symmetry H of a theory T/G obtained from gauging a symmetry G of another theory T, such that (T/G)/H=T), and as such one is supposed to be able to gauge by these. I’m not sure what to make of this, though. The would-be quantization of the quantomorphism n-group clearly misses there (-1)-form symmetries, so I wonder if there’s anything to be gained, starting from the prequantum perspective, from instead studying something like a prequantum (n+1)-groupoid, with objects being subobjects of the prequantum n-bundle.
Thanks, looking good now.
Regarding the data in codimension-0 and the action of defects on field theories:
It strikes me that this is all rather clear in the defect picture going back to Schweigert et al. from the early 2000s:
Here an -dim QFT will have an -category “of defects” (a category named after its morphisms), where
the objects are the “phases” of the field theory, assigned in codimension 0
the morphsims are the “defects” where two phases meet, assigned in codimension 1
the higher morphisms are correspondingly higher morphisms.
Now given a cobordism stratified by oriented defect hyperplanes, an assignment of phases and defects is an -functor from the Poincaré-dual -graph to that -category.
Concretely, the FRS theorem on rational 2d CFT says that for 2d CFTs the “phases” in codimension 0 are labeled by (Frobenius) algebras in a modular tensor category, defects are labeled by bimodules between these, etc. Invertible defects are the dualities (the actual “symmetries”, e.g. Kramers-Wannier duality may be formalized this way). Analogously for higher dim QFTs.
(Can try to dig out specific references – searching now I got distracted by rediscovering that a note pdf in this direction that I made to myself in 2006 is still online.)
added pointer to:
Jürg Fröhlich, Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Kramers-Wannier duality from conformal defects, Phys.Rev.Lett. 93 (2004) 070601 [doi:10.1103/PhysRevLett.93.070601, arXiv:cond-mat/0404051]
Jürg Fröhlich, Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354-430 [arXiv:hep-th/0607247, doi:10.1016/j.nuclphysb.2006.11.017]
I see now that you had a reference to Schweigert et al. 2002 as a “precursor” to GKSW.
Usually by a “precursor” one means a result or idea that has remained somewhat incomplete or ill-understood and later gets properly refined. But in going from Schweigert et al. to GKSW it seems to go more in the other direction, no?
So I have taken the liberty of rewording to:
Early discussion of “generalized global symmetries” under the name of defects …
but please feel invited to disagree.
In references related to gauging a “generalized symmetry”, say in 2d, one uses the gauging process wrt to a Frobenius algebra A in a fusion category as described in FRS02. For example, for a torus, one “inserts” A in both nontrivial loops. This looks very similar to how one defines a flat connection of a principal G-bundle for G discrete on the 2-torus, by using a functor from the fundamental groupoid of the torus to BG.
If one wants to provide a similar formulation, one should define a functor from the fundamental groupoid to, well, some linear category, such that it is meaningful to assign A to the nontrivial (classes of) loops. But of course, the fundamental groupoid does not have a sufficient structure (it knows nothing about linearity, or being enriched over R-Mod, or related). Since the torus is in particular in Fun[CartSp^op,Set], it makes me wonder whether one should be working on a different category so that there is a more intrinsic notion of linearity to everything. The most naive guess would be something like Fun[CartSp^op,Vec_k]. But I have not seen any reference describing this particular category even though at first sight there should be stuff to say about it. Sure, Vec_k is not Cartesian closed, so I’m not sure Fun[CartSp^op,Vec_k] is even a topos, but it is monoidal closed wrt to the tensor product, and it’s (the canonical example of) a tensor category, so I would expect some of this rich structure to appear.
So is there anything nontrivial to say about Fun[CartSp^op,Vec_k]?
Here I don’t follow why the topos would be changed, don’t you just want a linear object in the ordinary topos?
Say to describe flat vector bundles with connection we could consider the (groupoid core) of the stack of vector bundles over (which is equivalently ) and then flat vector bundles would be morphisms in the ordinary topos (2-topos) over from some fundamental groupoid to that stack of vector bundles.
Once I tried to understand the Fuchs-Runkel-Schweigert construction in this form, as a higher parallel transport over worldsheets analogous to that for a flat 2-vector bundle, but with other coefficients. I still think this should work, but it’s fiddly and back then I didn’t get to wrap it up. My old notes on this are here.
Thanks, Urs. Is Definition 24 from those notes described again in a published paper or preprint? I think in general the content of Sections 4 and 5 answers some of the questions thought to be open in the Physics community, as exemplified in Section 1.2 of arXiv:2301.07112 where they say:
“However, because the notion of background fields for noninvertible symmetries is still poorly understood, a systematic understanding of this AnomTFT is lacking. We will not shed light on this issue here.”
Of course, what they would have in mind would be a slight step away form a Lie algebra-valued differential from, but still this goes in the right direction.
Is Definition 24 from those notes described again in a published paper or preprint?
The closest that I am aware of is the rencet work of Kristel, Ludewig & Waldorf 2022, who take a serious look at the idea of 2-vector bundles understood as Cech-data with coefficients in Algebras+Bimodules+Intertwiners, hence as 2-functors much as in that Def 24.
Maybe more explicitly in the spirit of that Def. 24 is pp. 25 in arXiv:0806.1079.
added older reference emphasizing the linearization process and higher algebras Daniel Freed, Mike Hopkins, Constantin Teleman, Jacob Lurie. Topological Quantum Field Theories from Compact Lie Groups. (arXiv:0905.0731 )
added publication data and link to Topological Quantum Field Theories from Compact Lie Groups
for what it’s worth, right now there’s a Categorical Symmetries in Quantum Field Theory conference.
it can make sense to list conferences among the references of an entry (have now added the pointer here here)
for meetings of substantial interest it can also make sense to open a separate category:reference-page (an example is Strings 2022)
in any case, if you are following the talks and spot anything noteworthy, feel invited to drop a note here!
added couple of additional sentences to Idea section, and added section on the proposal by Moore-Freed-Teleman trying to make sense of this, including a remark regarding the potential relevance of the geometric cobordism hypothesis for their definitions.
3,5,6,7 (about formal/formalized/informal) – comment in general
While it looks inconsistent all these usages of word “formal” are related, and may be viewed from the same point of view, as they arose historically. Formal refers to syntax and syntactic manipulation, as opposed to reasoning about model/interpretation/content. This is in the view of Saussurean sign which consists of signified (object of reference) and signifying (the expression). Working within an axiomatics which is seriously thought to represent the intended reality is formalized rather than only formal. Formal may be a little less than that – it is simply working syntactically with rules assumed for a particular formal calculation. What stays “in-formal” is not the calculation but rather justification for its intended interpretation/semantical conclusions. For example, for set theory people believe that ZF theory is consistent and that it represents (at least a variant of) what we think informally as sets. This belief is quite serious – took the test of several decades, not finding a contradiction in the theory. Thus, we consider ZF as a a formalization of set theory virtually free from paradoxes of the naive set theory. For some weaker theories there is a finitary proof of consistency, which is viewed as even higher level of formalization. Now, people may take (even ad hoc) some other set of common syntactic manipulations with formulas to get some insight, e.g. calculations with Feynman integrals, not seriously defined and rules not justified. In another example, one can use formulas for real integrals and as arguments put complex expressions without full justification. Or use manipulation with infinite sums without checking if they converge, if one can exchange the order in double summations etc. These are all formal calculations as they are syntactic manipulations only. If we try to use them for intended interpretation these calculations are not a priori fully justified, and in this sense they are nonrigorous, or some would put it informal. Some systems of such formal manipulations may be proved to be justified. For example, in Hopf algebra theory we often use calculations with Sweedler notation and we do not doubt those proofs. Unfortunately I am not sure if anybody ever published a paper giving a full formalized set of rules (I was thinking to do it as a student, including a case in an arbitrary monoidal category, the “abstract Sweedler notation”). There are also informal reasonings which can not be named formal, as they are not syntactic but simply non-rigorous or non-complete. Formal sums (like in free Abelian group), formal sums (in another meaning, meaning infinite sums without requirement of convergence), formal power series, calculating with (pre-Robinson and pre-synthetic differential geometry) infinitesimals, Heaviside calculus etc. – all this terminology is about calculations with formal expressions extending (or sometimes mimicking rules for) calculations with similar ordinary expressions.
The fact that formal calculations/manipulation is often not appropriate to get some conclusions in intended interpretation and even if it does, often only scratches the surface as it does not use deeper semantic properties and checks in intended interpretation, is a justified reason to often view formal manipulation as a superficial method. By extension of this, formalization in science in general is often of limited application scope and strength, hence the somewhat pejorative common connotation and expectation that “formal methods” are superficial.
All what I said is about rather standard and widely spread usage in science and mathematics. The wikipedia entry needs here to be seriously reworked.
added reference to recent paper
added pointer to
just because we have a talk about it today (here)
Not sure yet how to cite this in the nlab article but the paper:
is really at the heart of this whole thing. Take for instance the action of the category of -modules for a Hopf algebra mentioned in p.10. If one specializes to semisimple this category is a fusion category, which is what everyone talks about in the 2d “non-invertible symmetries” literature.
There are certainly other papers that had been talking about these things before GKSW14, but it’s just, let’s say interesting, that this paper isn’t cited even with such an explicit title (the same that they nowadays use for conferences).
For whatever value it has (maybe none, for your point) I’ll remark that, pedantically speaking, one might say that Hopf algebras represent “categorical” but not “categorified” symmetries:
Namely Hopf algebras are 1-groups (instead of higher groups) but internalized into a non-classical (non-cartesian monoidal) ambient 1-category (as remarked here).
If we grant the usual tacit understanding of both “group” and “symmetry” as shorthand for “symmetry group” (which in turn is a shorthand for the original “groups of symmetry operations”) then the historical origin of “categorified symmetry” (albeit not related to QFT) goes half a century back:
Solian 1980 spoke of “categorical groups” and previously Sính 1973 said the same, albeit in more jargonese: “Gr-categories”.
A nostalgic aside on the arXiv paper you mentioned:
The observation in §8.2, §9.6.2 that transgression in (group) cohomology is essentially just forming the internal hom out of the circle was finally, finally fully proven and published (with CMP, currently in print) as Cyclification of Orbifolds (Thm. 3.4).
Now that you bring that up, I think in particular that distinction (or rather, connection) between “categorical” vs “categorified” is important to highlight even for the topic of this page. Currently, in particular the “non-invertible symmetries” literature is flooded with articles attempting, roughly speaking, to directly and exlcusively adapt to their arguments these higher linear categorified structures (e.g. fusion -categories), which are nontrivial by themselves. At the same time, these structures usually have a manifestation in terms of more familiar objects (e.g. Hopf algebras as 3-vector spaces, fusion categories as representation categories of Hopf algebras, 2-vector bundles as algebra bundles, etc), and it is this what is often overlooked. This is mainly due to a shallow understanding of these structures, since they’d regard as classical everything that doesn’t obviously and explicitly have (higher) morphisms, even if they’re actually there (albeit in a subtler way).
listed (arXiv:2311.14666) (which has also been included in higher gauge theory of the Green-Schwarz mechanism – references), and added a small comment connecting this to the issue of smoothness the page mentions several times
On early appearances on the terminology “categorical symmetry groups” I added pointer to the original article on 2-groups:
and I took the liberty of also pointing to
following comment #26 above.
added pointer to:
(for the record, not as endorsement)
27 Urs: action of a Hopf algebra on a k-linear category is the action of its category of modules (or comodules) which satisfies certain admissibility condition, namely it lifts the trivial action on vector spaces where Hopf algebra lives (thus the Hopf algebra itself produces a comonad (monad in the other case) on the k-linear category acted upon. This automatically implies distributive laws like the entwining structures of Brzezinski-Majid (and their generalization). Thus it is fully categorified if defined in this geometric way, and produces nontrivial distributive laws which have a major role in this subject.
As a tangential observation, the lack of rigour in the “field” stems not from not knowing how to define an action of a rep category of a (finite-dim’l) Hopf algebra, or a fusion category for that matter, on some other linear category (here I’m just talking about 2d QFT’s, the higher-dimensional case is even more out of bounds), people certainly talk about (or at least state the definition of) module categories and such. The main issue in my estimation is that they don’t really know on what the category is supposed to act. All the papers will circularly “define” action of a fusion -category as “action of -morphisms on the operators of the QFT supported on dimension submanifolds”, always supplemented by some sort of picture of a defect engulfing insertion submanifolds. In 2d one should be talking about actions on some 2-vector space (or 2-Hilbert space), crucially not of “finite-dimensional” character as characteristic to TQFT’s. I think this fact that a finite-dim’l Hopf algebra encodes the same information as a fusion category with a fiber functor is quite handy, since instead of working with module categories one could look into actions of these finite-dim’l Hopf algebras on 2-vector spaces realized as von Neumann algebras as described as Urs pointed out in #15 by Konrad et al here (of course, one also needs this action to be compatible with evolution and so, pretty much one would want to describe actions of a Hopf algebra on a 2-vector bundle with connection).
Alonso, Hopf algebras and fusion categories may be one aspect considered under the heading “generalized symmetries”. But for what I keep looking at (claims about the Green-Schwarz mechanism) it does not seem like they are claimed to play any role.
After a while it dawned on me that these authors say “2-group symmetry” or “categorical symmetry” not for anything explicitly related to (categorified/2-connections with values in) 2-groups, but fully generally for perceived shift symmetries of form fields.
Of course, we know there is a 2-group behind the Green-Schwarz mechanism, bu I am unsure if these authors even care about that. In any case, Hopf algebras nor fusion categories do not seem to be claimed to play a role in these “2-gorup symmetries”
Right, finite-dimensional Hopf algebras are essentially the “non-invertible generalization” of finite (1-)group symmetries. In a sense, the “higher-group of continuous symmetries” is the orthogonal generalization and its literature, as you have noticed, is equally plagued with all sorts of issues. But again, these two notions should find an actual rigorous formulation in terms of actions on higher vector space bundles with connection, or equivalently, and as some authors from the math side have (sort of) started realizing, in terms of FQFT. Surely the bulk of the community wouldn’t really care about this (especially since virtually none of them seem to have a problem with these circular definitions and working by analogy), but the fact that smooth actions of higher algebras play a role in QFT is worth formalizing for its own sake.
pointer
pointer
to record that supergroup-parameterized symmetries are starting to being discussed
pointer
pointers
Lakshya Bhardwaj, Thibault Décoppet, Sakura Schäfer-Nameki, Matthew Yu. Fusion 3-Categories for Duality Defects (2024). (arXiv:2408.13302).
Theo Johnson-Freyd. On the classification of topological orders (2020). (arXiv:2003.06663).
(It is interesting to see they are actually attempting to climb the ladder step-by-step…)
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