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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- Kevin Costello, Owen Gwilliam.
*Factorization algebra*(2023). (arXiv:2310.06137).

pointer

- Federico Ambrosino, Ran Luo, Yi-Nan Wang, Yi Zhang.
*Understanding Fermionic Generalized Symmetries*(2024). (arXiv:2404.12301).

to record that supergroup-parameterized symmetries are starting to being discussed

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- Masaki Okada, Yuji Tachikawa.
*Non-invertible symmetries act locally by quantum operations*(2024). (arXiv:2403.20062).

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.

]]>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”

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 $n$-category as “action of $d$-morphisms on the operators of the QFT supported on dimension $d$ 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).

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.

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- Clay Cordova, Thomas T. Dumitrescu, Kenneth Intriligator, Shu-Heng Shao,
*Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond*[arXiv:2205.09545]

(for the record, not as endorsement)

]]>On early appearances on the terminology “categorical symmetry groups” I added pointer to the original article on 2-groups:

- Alexandru Solian,
*Coherence in categorical groups*, Communications in Algebra**9**10 (1980) 1039-1057 [doi:10.1080/00927878108822631]

and I took the liberty of also pointing to

- Urs Schreiber, Zoran Škoda,
*Categorified symmetries*, 5th Summer School of Modern Mathematical Physics, SFIN XXII Series**A1**(2009) 397-424 [arXiv:1004.2472, inspire:851901]

following comment #26 above.

]]>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

]]>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 $n$-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).

]]>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).

Not sure yet how to cite this in the nlab article but the paper:

- Urs Schreiber, Zoran Škoda.
*Categorified symmetries*. (2010). (arXiv:1004.2472).

is really at the heart of this whole thing. Take for instance the action of the category of $H$-modules for $H$ a Hopf algebra mentioned in p.10. If one specializes to $H$ 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).

]]>added pointer to

- Pavel Putrov, Juven Wang,
*Categorical Symmetry of the Standard Model from Gravitational Anomaly*[arXiv:2302.14862]

just because we have a talk about it today (here)

]]>after skimming a bit more carefully through the paper mentioned above, had to change the way I referenced its content…

]]>added reference to recent paper

- Ben Gripaios, Oscar Randal-Williams, Joseph Tooby-Smith.
*Smooth generalized symmetries of quantum field theories*. (2023). arXiv:2310.16090.

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 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.

]]>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!

]]>for what it’s worth, right now there’s a Categorical Symmetries in Quantum Field Theory conference.

]]>added publication data and link to *Topological Quantum Field Theories from Compact Lie Groups*

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
)

Maybe more explicitly in the spirit of that Def. 24 is pp. 25 in arXiv:0806.1079.

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