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Massless chiral fields in six dimensions

by Thomas Basile

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Submission summary

Authors (as registered SciPost users): Thomas Basile
Submission information
Preprint Link: https://arxiv.org/abs/2409.12800v1  (pdf)
Date submitted: Nov. 25, 2024, 4:52 p.m.
Submitted by: Thomas Basile
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

Massless chiral fields of arbitrary spin in six spacetime dimensions, also known as higher spin singletons, admit a simple formulation in terms of $SL(2,\mathbb{H})$ tensors. We show that, paralleling the four-dimensional case, these fields can be described using a $0$-form and a gauge $2$-form, taking values in totally symmetric tensors of $SL(2,\mathbb{H})$. We then exhibit an example of interacting theory that couples a tower of singletons of all integer spin to a background of $\mathfrak{g}$-valued higher spin fields, for $\mathfrak{g}$ an arbitrary Lie algebra equipped with an invariant symmetric bilinear form. Finally, we discuss the formulation of these models in arbitrary even dimensions, as well as their partially-massless counterpart.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Has been resubmitted

Reports on this Submission

Report #4 by Anonymous (Referee 4) on 2025-3-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2409.12800v1, delivered 2025-03-12, doi: 10.21468/SciPost.Report.10822

Strengths

This is a high-quality research work on chiral fields in six dimensions. The author is a great expert on the subject, and the paper is written very carefully, showing a level of knowledge in the subject that is hard to match.

The results are genuinely interesting and add a brick in the wall of knowledge on the singletons, the shortest unitary representations of conformal group, that are arguably the simplest field-theoretical objects, yet hard to describe explicitly.

Weaknesses

It is hard to find a weakness in this work as it clearly defines its scope and reaches the declared goals. We would only suggest the author to use notations that do not lead to confusion. The letter H was used both for curvature three-form and for a basis of forms.

Report

The manuscript represents an interesting original work, is well-written, and meets all the criteria for being accepted to SciPost.

Requested changes

The manuscript is extremely well-written and can be published in its current form. As we mentioned above, the notations could be slightly improved, but we leave it to the author to make changes (or not). We also noticed one or two typos, which will hopefully be fixed during typesetting (we did not keep track of them).

Recommendation

Publish (surpasses expectations and criteria for this Journal; among top 10%)

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Report #3 by Anonymous (Referee 3) on 2025-2-24 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2409.12800v1, delivered 2025-02-24, doi: 10.21468/SciPost.Report.10716

Strengths

Very interesting and well-written paper, generalising the higher-spin actions recently discovered in 4D to the case of a 6D space. The actions described are analogous to those in 4D, with the 2-component spinor indices being replaced by 4-component ones in 6D.

Weaknesses

Only the higher-spin YM type theories are described. It would be very interesting to see if also the gravity-type (analogs of higher-spin SDGR) can be extended to 6D.

Report

The paper is very well written, and can be accepted in the present form. The following recommended changes are optional.

Requested changes

My only complaint is the notation. The construction of various differential forms on page 4 is a nice one. However, 2-forms, 3-forms and 4-forms are all denoted by the same symbol H. The author differentiates between then by the position of their spinor indices. However, I have still found this to be confusing. At the same time, I do not have a better suggestion. Putting the degree of the form somewhere will lead to overcrowded notation. Using different letters is an option, if there is a good and well-motivated choice for these letters. A similar complaint is that it would be desirable to develop some notation for fields with a set of indices that are also differential forms. The H-fields are like this, but also the field omega that appears in (1.31) is similar. Again, I do not have a specific suggestion, but it would be desirable to develop notation that makes it clear whether the field is a form of some non-zero degree or not. I will leave these suggestions as optional, to possibly improve presentation and readability of the paper.

Recommendation

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

  • validity: high
  • significance: good
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Report #1 by Anonymous (Referee 2) on 2025-2-9 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2409.12800v1, delivered 2025-02-09, doi: 10.21468/SciPost.Report.10512

Strengths

  1. The paper introduces a general framework for constructing chiral higher-spin theories in six or more dimensions.

  2. The presentation is well-organized, and the connections to related mathematical structures are compelling.

Weaknesses

  1. Only a limited class of interactions is discussed.

Report

The paper investigates massless chiral fields of arbitrary spin in six-dimensional spacetime, referred to as higher-spin singletons, and presents a framework for describing these fields using symmetric $SL(2,\mathbb{H})$ tensors. The author aims to extend concepts from four-dimensional higher-spin theory, where the fields are described using a combination of 0-forms and 1-forms that take values in $SL(2,\mathbb{C})$ tensors. The paper also explores interactions involving these massless fields and extends the formalism to arbitrary even-dimensional spacetimes.

In my opinion, the paper makes an important contribution to the field of higher-spin theories by providing a novel framework for describing massless chiral fields in six dimensions. The results offer promising avenues for further research, particularly in the development of interacting theories and their extension to higher dimensions. The presentation is well-organized, and the connections to related mathematical structures are compelling. Therefore, I am pleased to recommend the paper for publication in SciPost Physics.

Requested changes

Below are some comments and suggestions for improvement, most of which are optional.

  1. The sentence "Such massless fields are known to be conformal, ...'' before Eq. (1.21). It is not quite clear which "such massless fields" are referred to. What is the precise relation between the off-shell strength $F$ and the fields $\Psi$ and $\Phi$ subject to Eqs. (1.21) and (1.22)?

  2. In Eqs. (1.21, 1.22) and below. What is the significance of using the wavy equality sign instead of the standard one? Not all readers may be familiar with the theory of constrained Hamiltonian systems, from which this symbol is apparently borrowed.

  3. At the beginning of the subsection "Reducibility of the gauge symmetries," the author mentions "the correct number of degrees of freedom" for the free equations (1.21, 1.22), but provides no indication of what it might be. It seems that the author's argument here is that the (gauge equivalence classes) of on-shell fields carry a unitary irreducible representation of the conformal group $SO(2,5)$, which is a characteristic property of singletons. If so, it would be nice to state this explicitly.

At the same time, there is a more direct way to convince oneself that equations (1.21, 1.22) do propagate the correct number of physical degrees of freedom, without appealing to the conformal group. To this end, one can use a general formula for the physical degrees of freedom from Ref. [1, Eq.(80)]. When applied to Eq. (1.22), it reduces to

$$ N=1\cdot t-1\cdot r+2\cdot r_1\,, $$
and $r_1$ is the number of redundant gauge symmetries. Using the well-known formulas for the dimension of the space of totally symmetric tensors and tensors with hook symmetry, we readily find
$$ t=\frac{(p+1)(p+2)(p+3)}{6}\,,\qquad r=\frac{(p^2-1)(p+2)}{2}\,,\qquad r_1=\frac{(p^2-1)p}{6}\,, $$
where $p=2s$. Combining these numbers, we get $N = 2(2s+1)$. This is the number of physical degrees of freedom per point in the phase space of fields. The number of physical polarizations of the field $\Phi$ is thus equal to $2s + 1$. It remains to note that $2s + 1$ is exactly the dimension of the irrep of the little group $SU(2) \times SU(2)$ whenever either of the two factors acts trivially. A similar analysis can be repeated for the field $\Psi$. Although the field equations (1.21) have no gauge freedom, they enjoy (reducible) gauge identities, which can also be taken into account with the help of the general formula. Actually, as the system(1.21) and (1.22) are dual to each other, they are known to describe the same number of physical degrees of freedom. A rigorous proof of this fact can be found in the recent paper [2].

  1. Line after (1.42): It might be better to replace "the two-raw Young diagram of so(1,5), ...'' with "the two-raw Young subdiagram of so(1,5)''.

  2. Section 1.2 dresses the coupling to a Yang-Mills field. I think the term Yang-Mills field'' for the field $A$ is somewhat misleading in this context. Usually, Yang-Mills fields refers to both the fields and their equations of motion. The fields $A$ and $B$ define a BF-modal and carry no physical degrees of freedom, unlike YM theory. It might be better clearer to refer to $A$ as agauge vector field" or a ``connection 1-form".

  3. Line before (1.55): "this bilinear form verifies'' $\longrightarrow$ "this bilinear form satisfies''.

  4. There is no coupling constant $\mathrm{g}$ in the definition of the covariant derivative (1.56). However, this constant pops up in Eq. (1.60). Clearly, it can be absorbed by the re-definition of the field $B$.

  5. The sentence around (160) and (161) is too wordy and unclear. Please consider splitting it into two or three sentences for better clarity.

  6. The comment before Eq. (1.62) is somewhat confusing. Equations of motion know everything about their gauge symmetries, reducibility, and physical degrees of freedom. Therefore, any choice of gauge symmetry generators (1.59) -- with or without $F$-term -- would lead to reducible gauge transformations and correct number of physical degrees of freedom. Since $F\sim \delta S/\delta B$, the $F$-term vanishes on-shell, representing thus a natural arbitrariness in the choice of gauge symmetry generators.

  7. The sentence around (1.73) and (1.74) is too wordy and unclear. Please, split it into two or more sentences.

  8. Paragraph after (1.76): and that the Lie bracket is understood...'' $\longrightarrow$and the Lie bracket is understood ...''.

  9. It is worth stressing that the cubic term (1.85) is a peculiar property of chiral theory in six dimensions.

  10. In the context of higher-spin gravity, the presymplectic AKSZ action (1.95) was introduced and studied in Ref. [3]. Among other things, in that paper, the first order correction to the free presimplectic potential ($\Psi^2$-term) was explicitly constructed, see Eq. (6.43) and Footnote 23.

References:

[1]  Kaparulin, Dmitry S., Simon L. Lyakhovich, and Alexey A. Sharapov. "Consistent interactions and involution." Journal of High Energy Physics 2013.1 (2013): 1-31; arXiv preprint arXiv:1210.6821 [hep-th].

[2] Lyakhovich, Simon, and Dmitri Piontkovski. "Degree of freedom count in linear gauge invariant PDE systems." arXiv preprint arXiv:2501.16042 (2025).

[3] Sharapov, Alexey, and Evgeny Skvortsov. "Higher spin gravities and presymplectic AKSZ models." Nuclear Physics B 972 (2021): 115551; arXiv preprint arXiv:2102.02253 [hep-th].

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Recommendation

Publish (surpasses expectations and criteria for this Journal; among top 10%)

  • validity: top
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: excellent

Report #2 by Anonymous (Referee 1) on 2025-1-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2409.12800v1, delivered 2025-01-30, doi: 10.21468/SciPost.Report.10574

Strengths

1) The author proposed a new approach to massless higher-spin fields in any even dimension (in fact, to partially-massless as well)

2) As different from many other sources that consider free fields first, the paper gives examples of interactions

3) As compared to most of the papers that pursue the Noether procedure and consider mostly cubic interactions, the paper gives examples that are consistent to all orders

4) It seems to be the first example of an interacting theory with mixed-symmetry fields

Weaknesses

This is not an actual weakness, but some of the interactions are somewhat of topological nature, see also my last question below

Report

After the sudden ``death'' of Vasiliev's equations the vacuum was soon filled in by the chiral higher-spin theory. It has become more and more apparent that, in general, higher-spin symmetries lead to nonlocalities there is not much difference between the flat and anti-de Sitter spaces. Nevertheless, several exotic examples of higher-spin theories that are local in some sense do exist.

The present paper is an excellent attempt to push the idea behind chiral higher-spin theory to higher dimensions. The simplest case of $D=6$ is discussed in detail. It is also shown how to extend the results to all even dimensions.

The paper is well-written and the author clearly has put a lot of efforts into it (for instance, different colors for the diagrams of different groups).

The list of strong points and my list of confusions and questions can be found in the other windows.

Requested changes

It is not a exactly the list of requested changes (I leave to the author to decide).

1) One confusion that I have is that the orange diagrams are actually of $su^*(4)$ instead of $sl(2,H)$, right? It is difficult to imagine anti-symmetric tensors such as in (1.5) with indices taking two values only. Maybe it is an interesting exercise on its own to realized the quaternionic basis...

2) I am also slightly confused while reading the paragraph after (1.74). $F=0$ describes a topological field, but why one needs to consider $H\wedge A=0$ at all? The dynamical field is a component of $A$ modulo Stueckelberg symmetries. What does it have to do with $H\wedge A=0$?

3) Can one combine the BF-theory with the zero-form interactions?

4) What about the simplest type of current interactions ? Can they also be realized?

Recommendation

Publish (surpasses expectations and criteria for this Journal; among top 10%)

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: perfect
  • grammar: excellent

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