Higher representations for extended operators

The manuscript discuss the action of 0-form symmetry on extended operators in terms of higher representations. The construction is based on studying how symmetry acts on the worldvolume of extended operators.

I highly recommend the manuscript for publication once the following questions/comments/suggestions are addressed:

• How does the framework deal with "soft symmetries", i.e. symmetry that does not act by permutation or H^2(G,A) fractionalization with Abelian anyon A, such as https://arxiv.org/abs/2501.03314 and the reference inside. In particular, does the 3-representation need to be modified?

-The support of extended operators can have tangential structures different from the bulk, e.g. a 2d support can be unorientable or spin even when the bulk is only oriented for bosonic theories. This gives new ways of how symmetry acts on the support. Perhaps the authors can comment on how the framework can deal with this.

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This paper investigates how general invertible symmetries act on extended operators in quantum field theory (QFT). The authors develop a representation-theoretic framework for describing this action, presenting multiple complementary perspectives: 1. By analyzing the consistency conditions of topological operators. 2. Through a construction based on the symmetries of the worldvolume theory of the defect, from which the relevant representations are induced. 3. From a categorical viewpoint utilizing auxiliary topological quantum field theories (TQFTs).

The paper first revisits the well-understood case of invertible symmetries acting on local operators, then extends the discussion to line operators, including the case of general 2-group symmetries. Finally, the authors address the action of a 0-form symmetry $G$ on surface operators, explaining the concept of 3-representations of $G$ and their relevance in this context.

The manuscript is clearly written and well-organized, with excellent diagrams that significantly aid the reader in navigating the abstract concepts involved. One of the main contributions is to demonstrate that traditional representation theory of 0-form symmetries must be extended in order to understand their action on extended operators. This insight opens the door to unifying various previous results and charting new directions in constraining the dynamics of QFTs.

I strongly recommend this paper for publication in SciPost, subject to the following minor corrections and clarifications.

1. Page 2: The authors claim that the manner in which extended operators transform under higher group representations is independent of the ’t Hooft anomaly. Naively, one might expect the anomaly to constrain the possible actions. Please clarify this point, perhaps by specifying the precise sense in which the action is said to be independent of the anomaly.

2. Page 5 (related to Comment 1): In Section 2, does the anomaly of the 0-form symmetry $G$ affect its action on local operators? For example, in quantum mechanics, the anomaly of a finite 0-form symmetry is characterized by a class in $H^2(G, U(1))$ . Equation (2.8) introduces a quantity $ c(g,h)$. Is this related to the anomaly? Why is it natural to require the cohomology class $[c]$ to be trivial? A discussion of this would help clarify the assumptions underlying the framework. Similarly, what motivates the requirement that the 3-cocycle in Equation (3.58) must be trivial?

3. Page 9: In Section 2.2.1, auxiliary 1d TQFTs introduced earlier are shown to yield Wilson lines upon gauging $G$ . However, gauging a symmetry requires that its anomaly be trivial. Is there an interpretation for the auxiliary TQFTs when the $G$ symmetry is anomalous? Perhaps when the QFT is considered on the boundary of a $G$-SPT to cancel the anomaly, and when the $G$ symmetry of the bulk-boundary system is gauged, the auxillary TQFTs become defects of the gauged theory?

4. Page 11: Section 3.1 begins by noting that general line operators may support algebras of topological local operators. The section then focuses on line operators hosting only the identity operator and its complex multiples. What about non-topological local operators? Does the action of $G$ on such operators require additional data? Later, on page 19 (before Section 3.3), it is noted that junction operators transform under certain graded projective representations, even when such local operators are non-topological. If this addresses the earlier question, it would be helpful to indicate at the beginning of Section 3.1 that such issues are deferred to Section 3.2.3.

5. Page 48: It is stated that every multifusion category is Morita equivalent to a fusion category, implying that any indecomposable surface operator admits a topological interface to one whose line operators form a fusion category. How does the representation theory change when the category of lines is not itself fusion? Clarifying this point would enhance the reader’s understanding of the generality of the results.

6. Page 49 (related to Comment 4): In Section 5.2.1, the authors consider surface operators without nontrivial topological line defects. However, such surfaces may still support non-topological line defects. Can the $G$ -symmetry act nontrivially on these, and if so, what additional data does this action involve?

7. Page 53: In Section 5.3, the authors claim that all 3-representations can be constructed from 1-dimensional 3-representations by induction. Is this a provable mathematical statement, or is it a conjecture motivated by physical arguments presented in the paper? Clarifying the status of this claim would be beneficial.

8. Page 58: There is a typo below Figure 56, point 2: “ech” → “each”.

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