SciPost Submission Page
On a class of selection rules without group actions in field theory and string theory
by Justin Kaidi, Yuji Tachikawa, Hao Y. Zhang
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Hao Zhang |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2402.00105v1 (pdf) |
| Date submitted: | 2024-03-08 15:55 |
| Submitted by: | Zhang, Hao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approach: | Theoretical |
Abstract
We discuss a class of selection rules which i) do not come from group actions on fields, ii) are exact at tree level in perturbation theory, iii) are increasingly violated as the loop order is raised, and iv) eventually reduce to selection rules associated with an ordinary group symmetry. We start from basic field-theoretical examples in which fields are labeled by conjugacy classes rather than representations of a group, and discuss generalizations using fusion algebras or hypergroups. We also discuss how such selection rules arise naturally in string theory, such as for non-Abelian orbifolds or other cases with non-invertible worldsheet symmetries.
Current status:
Reports on this Submission
Report
This manuscript discusses classes of selection rules in quantum field theories that do not come from group actions on fields. In contrast to their group action counterparts, these selection rules need not be preserved at higher-loop levels. While this fact has been readily known in some parts of the literature, the authors make an important contribution by setting up a general and systematic method to deal with these selection rules, illustrated on various helpful examples.
The authors characterize the selection rules through conjugacy classes of groups and fusion algebras. The former serves as a pedagogical introduction that readily suffices for many string theory examples, while the latter connects with the recent literature on generalized symmetries. In both cases they characterize how the selection rules are broken at higher loop orders, and quantify when they converge to the Abelianization of the group/fusion algebra through commutator lengths. To illustrate these general methods, they provide examples of these selection rules that come both from groups and fusion algebras.
In my opinion this is an interesting manuscript that provides new, general insights into selection rules that do not come from group actions. As such selection rules evade traditional conservation laws, it is important to understand precisely how these rules may be broken at higher loops. The authors describe these rules in a general way through conjugacy classes of groups and fusion algebras, and provide a well-rounded set of examples to illustrate their methods. I highly recommend this paper for publication. Below I added some minor points/questions that could help in improving this manuscript even further.
Requested changes
1. The authors appear to assume that the symmetry group has to be finite. Perhaps the authors can clarify whether infinite groups may also be relevant to their study. And if so, whether they expect their methods to extend to this case, e.g. regarding the finiteness of commutator lengths.
2. This manuscript describes classes of selection rules that do not come from group actions on fields, but are given by conjugacy classes of groups and fusion algebras. Perhaps the authors can comment on whether they expect all alternative selection rules to be described by their methods, or if not, if they know of any counterexamples.
3. In section 3.1 the authors could perhaps provide the product table for the binary dihedral groups, as I believe it could be enlightening for this simple example.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report
It is well-known that selection rules in quantum field theory and string theory that are valid at tree level need not remain at loop level. The authors give a formal and systematic way to understand this phenomena in a general context where the tree level selection rules arise from hypergroups (or fusion algebras) that have a natural origin on the string worldsheet from non-invertible symmetries of the worldsheet CFT. The authors explain how the hypergroup symmetry is broken, at which loop level, and what symmetry survives at arbitrary loop level, and give explicit examples to illustrate the main ideas. The paper is very well written and simple to read.
See below for some comments/suggestions for changes.
Requested changes
1. The authors may want to comment the situation where the symmetry is infinite, or even continuous (relatedly it seems that the authors do not specify that G is finite until Section 2).
2. As explained clearly in this paper, the selection rules one derive are weaker at loop level compared to the tree level (whenever Com(A) for the hypegroup A is nontrivial), assuming general loop amplitudes are nonvanishing. Perhaps the authors can comment on when accidentally, in specific theories, the tree-level selection rules could still hold at loop level (at or beyond cl(A)).
3. While it is clear that for the purpose of the paper, it suffices to consider hypergroups, which are more general than fusion rings that are necessary for nonperturbatively defined generalized symmetries, perhaps it's worthwhile for the benefit of readers to give at least one example where the accidental symmetry at tree level is a hypergroup that is not a fusion ring (or something from linear combinations of fusion ring generators).
4. In the paragraph above Sec 3.4, the sentence "note that the Virasoro primary in \epsilon1\epsilon2 has dimension (1, 1)" is somewhat misleading. The descendants of epsilon1\epsilon2 wrt to the Ising^2 chiral algebra contain infinite Virasoro primaries. It should instead just say the Virasoro primary \epsilon1\epsilon2 here.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)