Representation theory for categorical symmetries

Important contribution, timely, connects math developments in the study of 2-fusion categories and their representations to QFT.

Could use some more detailed explanations and simple illustrative examples.

The manuscript develops the representation theory of 2-categorical symmetries, extending the well-known relationship between tube algebras of fusion 1-categories and the Drinfeld center of a fusion category to the 2-categorical setting. This is a very timely work, given the interest in generalized symmetries and the fact that generalized finite internal symmetries are naturally described by higher fusion categories.

Overall, the paper presents a significant contribution to the field. The results are of clear conceptual and technical significance. At times, I felt the explanations could be a bit more detailed and supplemented with examples. In any case, I am happy to recommend this work for publication in SciPost Physics, pending minor clarifications on a few points listed below.

1. On Page 44, around Eq. (3.60): One may also gauge an anomalous subgroup K of G by first stacking with a modular tensor category (MTC) carrying the opposite anomaly. Such gaugings are presumably labeled by 2-algebras corresponding to \alpha|_{K}-twisted K-graded fusion categories. Could the authors comment on how the representation theory of the resulting 2-category is organized after such a gauging?

2. It would be helpful if the authors could also describe the twisted sector lines in the 2-representation category of the Ising-like symmetry discussed in section 3.6.

3. Are there spin selection rules for twisted sector lines in 2-categories with anomalous 0-form symmetries, analogous to those for twisted sector operators Vec_G^{omega} in 1+1d?

4. Equations (4.7), (4.8), and (4.15) are somewhat difficult to parse. Including illustrative diagrams or pictures in terms of topological defects would be very helpful. Furthermore, could the authors clarify how one systematically deduces that these are a complete set of coherence relations?

Publish (easily meets expectations and criteria for this Journal; among top 50%)

This manuscript studies the notion of (possibly extended) operators charged under a generalised symmetry. Defining a symmetry in terms of topological defects suggests two extensions of the ordinary notion, whereby topological defects may be higher-codimensional and/or may be non-invertible, respectively. The main focus of this manuscript is on three-dimensional theories where, under various assumptions, such generalised symmetries are encoded into fusion 2-categories. The authors contemplate the analogue of charged operators with respect to such generalisations and the corresponding representation theory. The main tool that the authors use is a three-dimensional generalisation of the so-called tube category. Although such a construction has appeared before in the literature, the authors largely extends it in this manuscript. There are a few other articles dealing with this timely question, which appeared around the same time, but the present manuscript is arguably the most relevant one. Moreover, it is well written, well illustrated and quite rigorous. I have personally enjoyed reading this manuscript. As such, I have no doubt that it should eventually see publications in Scipost. Nonetheless, there are several aspects that could be improved and I would like to encourage the authors to consider the following general remarks:

(a) Generally, additional physical motivation would be appreciated. The authors discuss sophisticated tools allowing them to classify charged operators in many cases, but are there specific physically relevant examples that benefit from such results? Besides, in some concrete cases, the classification of extended charged operators involve various algebraic objects, it would be interesting to shed light on the physical interpretation of the various components.

(b) In most examples, the authors only derive a few aspects of the higher categorical structure encoding the “representation theory” of the generalised symmetries. Although the authors state the fusion 2-category that should describe the charged operators, they effectively only describe the objects of such a 2-category. In particular, 1-morphisms are hardly discussed, even in the simpler setting of gauge or higher gauge theories. It is unfortunate since the formalism set up by the authors would precisely allow them to discuss 2-representations beyond the objects, and such studies have in fact be carried out before in specific cases. Furthermore, the monoidal structure of these fusion 2-categories is also missing. However, the latter point would probably go far beyond the scope of the present manuscript, possibly requiring to endow the tube category with a quasi-hopf category structure.

(c) References to existing literature on tube algebras, which is quite extensive, could be a little bit improved. Starting in the abstract the authors claim that the higher tube algebras they consider provide “higher analogues of twisted Drinfeld doubles of finite groups” while such generalisations have appeared before in a slightly different context, possibly in a less general setting. In general, I think it is important to stress that tube algebras, and higher generalisations thereof, have already been used (sometimes in disguise) to compute (higher) Drinfeld centers. Emphasising these aspects will give the authors the opportunity to better clarify in which way their contribution goes beyond the existing literature.

(1) Page 7: The authors comment that “tube $m$-representations can be recovered from tube $n$-representations”. What is the mechanism? Maybe, the authors could add a reference justifying this claim?

(2) Page 8: The authors start using the notation $G/G$ to refer to the set of conjugacy classes of $G$. This notation seems very unusual to me. The authors also start employing the concept of transgression map. The paper by Willerton should probably be cited here.

(3) Page 9, table 1: In relation to a previous comment, it seems particularly important to me to stress that the statements in the column “Tube representation” are equivalences. Furthermore, it seems to me that these equivalences are typically not demonstrated in the manuscript. Sometimes, only one direction of the equivalence is established. I am not necessarily suggesting to make these results more mathematically precise, but the authors may want to stress in some places that they are only sketching the equivalence. I am wondering whether the notation for the tube category is the most judicious one. In particular, $\mathsf{T}_{S^2}(\mathsf{C})$ and $\mathsf{T}_{S^1}(\mathsf{C})$ strike me as not being on equal footing. Since a tube category can be associated with any open or closed two-dimensional surface, $S^1$ should maybe be replaced by $S^1 \times [0,1]$, or at least the authors could maybe comment on their notation.

(4) Page 10: Some references should be added regarding the description of symmetries in terms of spherical fusion categories. Also some general reference where these mathematical concepts are introduced in detail would be useful.

(5) Page 11: I find the use of $\leftrightarrow$ in figure 7 confusing, what is it supposed to mean?

(6) Page 12: The authors comment that a simple object “cannot be decomposed as a direct-sum of non-trivial sub-objects” and then comment on a consequence of the semi-simplicity of $\mathsf{C}$. But the first statement is already a consequence of semi-simplicity since they are implicitly saying that all indecomposable objects are simple.

(7) Page 14: Placement of the “.” In (2.9) is not ideal. I am not convinced that Figure 13 actually depict the statement the authors have in mind. Indeed, the authors would need to resolve the diagrams on the r.h.s. so as to obtain a single red dot labelled by $\varphi \odot \psi$.

(8) Page 15: The authors comment that the tube category has appeared in “many guises in the mathematical literature” and cite six references. As I commented above, the tube category has actually appeared very explicitly both in the mathematical and physical literature before. Regarding its definition, the authors may want to comment that a more general definition exists that allows for several marked points decorated by objects on the circle (see e.g. Walker’s). This more general definition in turn accommodates Morita equivalent tube algebras. About footnote 5, it is not strictly true that there is a single tube category in two dimensions since a tube category based on $[0,1]$ rather than $S^1$ can also be constructed. As a matter of fact this tube category is the direct analogue of the tube category $\mathsf{T}_{S^1}(\mathsf{C})$ considered in three dimensions.

(9) Page 17: Why is the tube algebra semisimple? Below (2.23), it is the irreducible characters rather than irreducible representations that correspond to primitive central idempotents.

(10) Page 18: As comment above, the authors are here quoting an equivalence between the representation category of the tube category of $\mathsf{C}$ and its Drinfel’d center. But the authors do not quite establish this equivalence. For instance, how do you extract the half-braiding from the representation category? Similar comments hold regarding the higher-dimensional cases.

(11) Page 19: Regarding the statements around (2.31), various references should be referred to, including works by Balsam and Kirillov.

(12) Page 20: Below equation (2.32), the authors could also cite the Dijkgraaf-Witten paper. Equation (2.34), it seems that $1_{\mathbb C_{ghk}}$ is missing after the $\cdot$. Similar typos can be found throughout the manuscript (see e.g. (2.35)).

(13) Page 21: In equation (2.36), the authors do not really justify the appearance of the transgression map here. At least, a reference could be added. I think the action groupoid should be defined more explicitly. Also the paper by Willerton needs to be cited before equation (2.40). References should also accompany the “known equivalence” (2.47).

(14) Page 24: Regarding the case of Ising, it may be worth mentioning (and citing the relevant references) the fact that in the case of an MTC, there is a particularly convenient way of writing the idempotents of the tube algebra.

(15) Page 30: The manuscript where the non-ubiquitous arrow notation employed before equation (3.7) was introduced should probably be cited. Further, the authors refer to inserting networks of lines but the surfaces along which such networks are inserted are not specified.

(16) Page 39: Since the notation $2\mathsf{Vec}_{\mathcal G}$ is slightly ambiguous, the authors could introduce its explicit definition.

(17) Page 40: Below equation (3.42), when referring to 2-group generalisations of four-dimensional Dijkgraaf-Witten theory, references could be added. The notation $1_g$ for the indecomposable $G$-graded 2-vector spaces is not quite consistent with the notation used before for the simple objects in $\mathsf{Vec}_G$.

(18) Page 48: When referring to abelian 3-cocycles, references could be added, including the seminal work by Eilenberg and Maclane.

(19) Page 53: In order to demystify the “formidable condition” given in equation (4.8), it may be worth mentioning that the associahedron axiom of the pentagonator is of this form.

(20) Page 65: As commented before, the equivalence (4.42) is not actually demonstrated, especially not as a braided fusion 2-category.

(21) Page 69: As far as I can tell, statements surrounding (4.54) are largely speculative. Such a correspondence was established in the case Dijkgraaf-Witten theory in works by Bullivant and Delcamp, and similar statements were made by other authors for specific Crane-Yetter theories, but I am not aware of any proof in the general case.

(22) Page 73, there is a typo, “G” should be “$G$”.

(23) Page 75: Around equation (4.75), the notation $C_{b/a}$ should be recalled/introduced.

(24) Page 78: The paragraphs below (4.91) do not really discuss the physical interpretation of the various components. As I mentioned before, in these simpler settings, I would encourage the authors to discuss further the physical aspects.

(24) Page 79: The results presented in section 4.5.5. appeared in several references before, where the discussion goes beyond that of the classification of simple objects.

(25) Page 81: Regarding (4.97), the paper by Douglas and Reutter, or possibly others, could be cited. About (4.98), it may be worth clarifying that $\text{Mod}(\mathsf{Vec}_A)$ and $\text{Mod}(\mathsf{Rep}_A)$ are equivalent, while the Morita equivalence is between $\mathsf{Vec}_A$ and $\mathsf{Rep}(A)$.

Ask for minor revision