Fermionic Higher-form Symmetries

Extends the construction of higher-form symmetries to the case of symmetries involving fermionic parameters.

The whole construction is a straight generalization of the case of higher-form symmetries. The work collects several examples of theories exhibiting such symmetries, but the authors do not discuss at length their physical consequences. In this way, the work does not make the case for the eventual power of these symmetries.

In this manuscript the authors extend previous studies on higher-form symmetries to the case of fermionic higher-form symmetries, i.e., symmetries involving fermionic parameters. They present several examples exhibiting this type of symmetry, where the whole construction parallels straightly the case of usual higher-form ones. However, contrarily to the usual case, it is not clear what are the physical consequences of the existence of such fermionic higher symmetries. For example, the expectation value of a Wilson loop in a gauge theory can be interpreted as an order parameter for the 1-form symmetry and its behavior (according to Coulomb, perimeter, or area laws) dictates whether the 1-form symmetry is spontaneously broken or not. In this way, the spontaneous higher-form symmetry breaking leads to Goldstone excitations in the case of continuum symmetries (e.g. photon in QED) and to topological order in the case of discrete symmetries. These types of relations seem to be difficult to establish in the case of fermionic analog of Wilson loop. With the exception of the brief discussion in subsection 5.2, the authors do not explore the physical consequences in a deeper level and the whole construction sounds like a purely formal matter. Therefore, in the present form, I am not sure if the manuscript meets the Scipost acceptance criteria.

Questions

1) Does the fermionic analog of the Wilson loop defined in (2.11) carry physical meaning? It is a meaningful order parameter for the the fermionic symmetry? If this is the case, could you please comment about spontaneous symmetry breaking scenario.

2) Concerning the BF-like theories discussed in Sec. 3.1, I have some questions:

2a) These theories are gapped?

2b) What are the topological indicators? In the usual BF theories in 2+1 for example, the ground state degeneracy depends on the topology of the manifold.

2c) Is there a fermionic analog of the bulk-edge correspondence? In this case, what would be the edge theory?

3) What kind of constraints follow from the existence of 't Hooft anomalies discussed in Sec. 4?

The paper studies a natural generalization of supersymmetry that has not previously been discussed in the literature.

Most of the examples are somewhat artificial. The most interesting class of examples is not studied in very much detail.

This paper discusses the possility of fermionic p-form symmetries in the context of relativistic field theory.
This is a generalization of supersymmetry where the supercharges are supported on codimension >1 loci in spacetime.
It is a possibility that I know has been considered by many people, but I have not previously seen any discussion of it in the literature (in more than two spacetime dimensions).

The authors exhibit several gaussian theories that exhibit such symmetries.

They make the point that such symmetries of these free theories seem to be explicitly broken in coupling to a general curved metric, and therefore do not provide counterexamples to various speculations about quantum gravity. A small comment about the logic: It is true that the usual way of coupling to curved space breaks these symmetries, but the authors didn't seem to show that there is no possible way of writing a covariant action that respects these symmetries.

An interesting and nontrivial class of examples with such symmetry discussed here is fermionic versions of BF theory.
The authors should say a few more words about the canonical quantization and spectrum of these theories (or give a reference) -- as the authors claim in passing, they are topological, in sharp contradistinction to the gapless example of the free Rarita-Schwinger field.
It would be useful to comment on a possible lattice realization of this theory, in the same way the ordinary BF theory (with appropriate coefficient) is a low energy description of the toric code ($Z_n$ gauge theory).
Specifically, it is not clear to me what is the relation between this theory and various fermionic generalizations of the toric code, such as this one:
https://arxiv.org/abs/1309.7032


-- Before equation 2.3:
it says "As another simple example,"
but this is the first example presented.

see above.

Novelty

The paper lies the groundwork to explore fermionic higher-form symmetries in arbitrary spacetime dimensions. The authors define higher-form fermionic fields and their transformations, and find intriguing differences with bosonic higher-form symmetries. The paper is clearly written and technically solid. The construction is detailed in various explicit examples, including the exotic 6d $\mathcal{N}=(4,0)$ theory. The paper is the first exploration along these lines and is potentially of great impact. It meets the criteria of SciPost. I would recommend it for publication, if the authors address a concern about 't Hooft anomalies (see the "Requested Changes" section).

I request one important change, plus some minor corrections/clarifications.

The important aspect to address is the discussion on 't Hooft anomalies. In footnote 1 and throughout the paper, the authors work in flat Minkowski spacetime. In section 4, they encounter an obstruction to gauging, and call it a 't Hooft anomaly. However, anomalies typically describe the non-conservation of a current due to some characteristic class (topological invariant) appearing on the right-hand side of $d \ast j \ne 0$. One should not be able to detect them in flat space, where the bundles trivialize. The authors should clarify the language and how the obstruction they observe is related to the standard notion of anomaly.

A possibly related comment: it is observed in section 4 that the fermionic higher-form symmetry can be gauged only in theories whose Lagrangian involves $d \psi$ but not $\psi$. They provide the examples of $d \psi \wedge \ast d \psi$ (can be gauged) versus $d \psi \wedge \psi \wedge \gamma$ (cannot). This is very reminiscent of what happens with bosonic gauge fields: in Maxwell theory, the Lagrangian depends on $F=dA$ but not on $A$, and the 1-form symmetry is gauged shifting $F$ appropriately. This would not work in e.g. Chern-Simons theory, because the term $dA \wedge A$ cannot be made invariant by a shift of $F$. It would be enlightening if the authors can explain or comment on this analogy.

A list of minor points: 1. page 3 "in section 2 we introduce the notions" --> "notion" 2. page 4. By the definition of $G$ above (2.1), I understand that the fermionic symmetry cannot be a finite group. If this is the case, it would be better to state this fact explicitly, since for now this is only mentioned in the outlook. 3. In (2.11), $\eta$ should be a fermionic analog of the charge of a Wilson loop. It should be defined explicitly 4. In footnote 5, isn't the definition of the authors just the usual one? 5. Below (2.23), the sentence "there is no electromagnetic type dualities between p-form fermionic". Are the authors claiming a no-go theorem, or do they mean that there is no reason to expect such a duality? If the latter, it would be convenient to nuance a bit the statement. 6. In (3.16) and (3.17), state that $\mathcal{C}$ is a line and $\mathcal{S}$ is a surface. 7. In (4.1) and subsequent, the notation $M^{(d)}$ of earlier sections is changed to$M_d$. 8. In (4.5) the authors write the action as $S_{\rm gauged}$, but then they state that there is an anomaly that prevents the gauging. I suggest to replace the notation into a less misleading one, e.g. $S_{\rm gauge~invariant}$