Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper aims for a construction of free Lagrangians for  fields, that belong to representations of a d-dimensional  Anti de Sitter group with a mixed symmetry.   The problem is very nontrivial from the technical point of view and the authors successfully solve it. They also discus a possible generalization of their results to cubic interactions. In my opinion the paper is interesting and should be published in present form.

Author Response

We are thankful for the Reviewer for the acceptance of our manuscript for publication in the Universe.

Reviewer 2 Report

Comments and Suggestions for Authors

The paper is devoted to the investigation of the higher spin theories with the help of complicated mathematical constructions. Higher spin fields naturally appear in string theories, so that it is really important to understand their features. The BRST approach used by the authors is a very powerful tool for this. It is developed during a long time by various researchers, including the authors. Its application to the higher spin fields in various representations on the AdS background in combination with the group theory methods considered in the paper seems for me rather interesting. However, I have to make some important comments. First, the paper is written too mathematically and is very difficult for understanding. In my opinion, it is necessary at least to reduce the abstract and rewrite it more clearly. In particular, the new results should be properly highlighted. Next, it is necessary to cite the classical papers devoted to the higher spin fields and the BRST symmetry, namely, C.Fronsdal, Phys.Rev. D18 (1978), 3624; J.Fang, C.Fronsdal, Phys. Rev. D18 (1978), 3630; C.Becchi, A.Rouet, R.Stora, Comm.Math.Phys. 42, (1975) 127; Ann.Phys. 98 (1976), 287; I.V.Tyutin, Lebedev Institute preprint N39 (1975). Moreover, the paper is not written accurately. There are large empty spaces (e.g., on pages 19, 20, 29, 58), which can presumably be removed by the command \allowdisplaybreaks. The figure 1 does not fit on the page. Certainly, this should be corrected. After that, I could recommend the paper for publication.

Author Response

Reply on the  Comments and Suggestions for Authors (universe-2710140).

We are very thankful for the Referee for the comments and suggestions to improve the paper.

Following these  recommendations we modify abstract to preserve all main results in it as follows:

 “We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group -- subject to a Young tableaux $Y(s_1,\ldots ,s_k)$ with $k\geq 2$ rows -- in a $d$-dimensional anti-de-Sitter space. Auxiliary representations for a  deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic $sp(2k)$ algebra used for mixed-symmetry HS fields in flat spaces [60]. Polynomial deformations of the $su(1,1)$ algebra related to the Bethe ansatz are studied as a by-product. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for $Y(s_1, s_2)$ is found, with non-vanishing terms (resolving the Jacobi identities) of third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced. Following the results of [1,2], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new  reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin $(1,1)$ is obtained.”

we added 6  references to the list of references (with new enumeration):

[32] C. Becchi, A. Rouet, R. Stora, Renormalization of the Abelian Higgs-Kibble Model, Comm. Math. Phys. 42, (1975) 127–162.

[33] C. Becchi, A. Rouet, R. Stora, Renormalization of Gauge Theories, Ann.Phys. 98 (1976) 287–321.

[34] I.V. Tyutin, Gauge Invariance in Field Theory and Statistical Physics in Oper-ator Formalism, Lebedev Institute preprint N39 (1975), [arXiv:0812.0580[hep-th]].

46] C. Fronsdal, Massless Fields with Integer Spin, Phys.Rev.D 18 (1978) 3624–3629.

[47] J. Fang, C. Fronsdal, Massless Fields with Half Integral Spin, Phys.Rev.D 18 (1978) 3630–3636.

[60] I.L. Buchbinder, A.A. Reshetnyak,  General Lagrangian Formulation for Higher Spin Fields with Arbitrary Index Symmetry. I. Bosonic fields, Nucl. Phys. B 862  (2012) 270, [arXiv:1110.5044[hep-th]].

We modify the references:

[9] A. Sharapov, E. Skvortsov, A. Sukhanov,  Minimal model of Chiral Higher Spin Gravity, JHEP 09 (2022) 134, [arXiv:2205.07794[hep-th]].

[97]  C. Burdik, A. Pashnev, M. Tsulaia, Auxiliary representations of Lie algebras and the BRST constructions Mod. Phys. Lett. A 15 (2000) 281-292, [arxiv:hep-th/0001195].

We remove the reference (in old enumeration):

[92] C. Burdik, O. Navratil, A. Pashnev, Proc. of XVI Max Born Symposium \Supersymmetries and Quantum Symmetries" (SQS01), Karpacz, Poland, September 21{25, 2001. Dubna, 2002, P. 11.

We correct the appearance of new and corrected references.

On p. 2 in the Introduction

The sentence “At the level of free fields, there exist two efficient approaches to the above objectives, known as BRST methods (initially developed to quantize constrained dynamical systems using the BFV{BRST procedure [32, 33, 34]), with respective complete (e.g., [1, 35, 36]) and incomplete [37] BRST operators (implied by String Field Theory [38, 39]), whose Lagrangian descriptions for one and the same higher-spin field in a d-dimensional Minkowski space-time are shown to be equivalent [40].”  

is changed on

“At the level of free fields, there exist two efficient approaches to the above objectives, known as BRST methods (initially developed to quantize gauge field theories [32, 33, 34], constrained dynamical systems using the BFV–BRST procedure [35, 36, 37]), with respective complete (e.g., [1, 38, 39]) and incomplete [40] BRST operators (implied by String Field Theory [41, 42]), whose Lagrangian descriptions for one and the same higher-spin field in a d-dimensional Minkowski space-time are shown to be equivalent [43].”

As to the large empty spaces “ (e.g., on pages 19, 20, 29, 58),”  then the respective source tex-file is not compiled on our personal devices. The same is valid for the figure 1. In our pdf-file all is correct. To solve the problem we may cooperate with respective assistants from the  Editorial Board to change step by step the filling of empty spaces. To fill the pages in our pdf-file we have used additional inserting of the couple of Latex-operators:

“\end{eqnarray}

\begin{eqnarray}”

to provide transition of  a part of formulae on the next or previous  pages. E.g. it was done for the formulas (23)-(25); (38)-(40); (58)-(63);

(83)-(85);  (88); (109)-(111); (125); (148)-(163);  (179)-(183); (C7)-(C8) ; (C19); (C23); (C28); (C31); (C38); (C40); (C42)-(C45); (C50); (C58).

We think that an assistant from the  Editorial Board with our cooperation  could solve  this technical problem, being related to the peculiarities of latex system of the Journal. We are grateful to the referee for comments and advises and ask him to accept the paper for publication in Universe

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper studies interactions of higher spin fields with mixed symmetry in AdS space, which is represented by a Young tableau. The string theory or higher-spin gravity is a natural generalization of general relativity. It is very important to determine Lagrangians that have gauge symmetries like BRST symmetry.  

 

The authors deduced a higher-spin symmetry algebra in AdS space for an arbitrary Young tableau. Then, they focused on a Young tableau with two rows and formulated a BRST operator and Lagrangian. They also examined general cubic interactions of totally symmetric fields.

 

I think that their results are reasonable. Although they mainly focused on a Young tableau with two rows, their methods are specific. In particular, they showed some examples in Section 6. The final Lagrangian actions in these examples may not be so novel, but the gauge transformations along the way may be useful in the future. So, I think that this paper can be published in Universe.

 

 

 

 

Author Response

We are grateful for the Referee for the acceptance of our manuscript for publication in Universe.