三维自旋投影算子与高自旋棉花张量的简洁表示

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本文主要探讨了三维空间中的自旋投影算子及其在高自旋棉花张量（higher-spin Cotton tensors）理论中的应用。自旋投影算子在物理学中扮演着关键角色，尤其是在描述高维场论中的自旋场性质时，它们简化了表达式并有助于理解复杂度较高的理论结构。在四维（4D）背景下，Weyl张量的线性化可以通过预势函数hα1αs˙α1˙αs简洁地表示，这是高自旋场的一个基本特征，参考文献[1]和[3]的章节6.9对此有所阐述。然而，当将注意力转向三维（3D）时，尽管复杂性可能增加，研究者们依然寻求类似的简单关系和表示形式。作者Evgeny I. Buchbinder、Sergei M. Kuzenko、James LaFontaine和Michael Ponds在《物理快报B》（Physics Letters B, Vol. 790, 2019年）中详细介绍了如何在三维空间中构造和利用自旋投影算子。他们首先对这些算子进行了深入的理论探讨，确保了在三维几何框架下的适用性和一致性。通过这些算子，他们能够推导出线性化高自旋棉花张量的新表示，这种表示对于理解三维空间中高自旋场的行为和相互作用至关重要。高自旋棉花张量是一种特殊的张量，它在描述引力理论中的高阶效应，如高阶引力波以及在弦理论中的作用方面具有重要意义。在三维，由于空间维度的差异，这些张量的结构可能会更为复杂，但通过巧妙地运用自旋投影算子，作者们找到了一个能够捕捉到高自旋物理本质的数学工具。该研究工作不仅拓展了对三维自旋场的理解，也为相关领域的理论计算提供了一个实用的工具。此外，作为开放获取文章，它促进了学术界的交流与合作，让更多学者有机会参与到这一前沿领域的讨论中。通过引用Creative Commons BY 4.0 许可协议，作者鼓励其他研究者基于此成果进行进一步的创新和探索。这篇论文是三维自旋场论和高自旋棉花张量研究的重要里程碑，它揭示了如何在三维空间中利用自旋投影算子来简化和深化对这些物理概念的理解。对于任何在这些领域工作的研究人员，理解和掌握这些方法都将对他们的研究工作产生深远影响。

Physics Letters B 790 (2019) 389–395

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Spin projection operators and higher-spin Cotton tensors in three

dimensions

Evgeny I. Buchbinder, Sergei M. Kuzenko

∗

, James La Fontaine, Michael Ponds

Department of Physics M013, The University of Western Australia, 35 Stirling Highway, Crawley, W.A. 6009, Australia

a r t i c l e i n f o a b s t r a c t

Article history:

Received

10 January 2019

Accepted

18 January 2019

Available

online 24 January 2019

Editor:

M. Cveti

We elaborate on the spin projection operators in three dimensions and use them to derive a new

representation for the linearised higher-spin Cotton tensors.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

In four dimensions (4D), there exists a remarkably simple ex-

pression

for the linearised higher-spin Weyl tensors in terms of

gauge prepotentials h

...α

...

= h

(α

...α

)(

...

)

, see e.g. [1,2]

and

section 6.9 of [3]. In the case of an integer spin s ≥1, it reads

...α

= ∂

(α

...∂

s+1

...α

)

...

. (1.1)

For a half-integer spin s +

≥

, with s = 1, 2, ..., we have

...α

2s+1

= ∂

(α

...∂

s+1

...α

2s+1

)

...

. (1.2)

It follows from (1.1) and (1.2) that the higher-spin Weyl tensors are

invariant under gauge transformations of the form

...α

...

= ∂

(α

(

...α

)

...

)

, (1.3)

...α

s+1

...

= ∂

(α

(

...α

s+1

)

...

)

. (1.4)

It should be remarked that there are two ways for the bosonic

gauge ﬁeld h

α(s)

= h

...α

...

to occur in higher-spin gauge

theories.

Firstly, h

α(s)

is one of the two gauge prepotentials

α(s)

, h

α(s−2)

} in the Fronsdal massless integer-spin mod-

els

[4,5](see section 6.9 of [3]for a review). Secondly, it is the

gauge ﬁeld in the Fradkin–Tseytlin conformal integer-spin theo-

ries [6].

In the former theories, C

α(2s)

and its conjugate are the

Corresponding author.

E-mail

addresses: evgeny.buchbinder@uwa.edu.au (E.I. Buchbinder),

sergei.kuzenko@uwa.edu.au (S.M. Kuzenko), 21319182@student.uwa.edu.au

(J. La Fontaine),

michael.ponds@research.uwa.edu.au (M. Ponds).

The story with the fermionic gauge ﬁeld ψ

α(s+1)

α(s)

is analogous.

The compensator h

α(s−2)

transforms under (1.3)by the rule

α(s−2)

∝ ∂

βα(s−2)

α(s−2)

only gauge invariant ﬁeld strengths which survive on the mass

shell. In the latter theories, the gauge-invariant action may be for-

mulated

in terms of C

α(2s)

and its conjugate [1,2].

three dimensions, the Weyl tensor vanishes identically and

all information about the conformal geometry of spacetime is en-

coded

in the Cotton tensor. Spacetime is conformally ﬂat if and

only if the Cotton tensor vanishes [7](see [8]for a modern

proof). Linearised higher-spin extensions of the Cotton tensor in

Minkowski space were constructed in [9] and [10]in the bosonic

and fermionic cases, respectively. In terms of a gauge prepotential

...α

(α

...α

)

, with n > 1, the linearised Cotton tensor is given

by the expression [10]

α(n)

(h) =

n−1

n/2



j=0



2 j



×2

∂

(α

...∂

n−2 j−1

n−2 j

...α

)β

...β

n−2 j−1

(1.5)

where x denotes the ﬂoor function and returns the integer part

of a real number x ≥ 0. The fundamental properties of C

α(n)

are

the following:

(i) C

α(n)

is invariant under gauge transformations of the form

α(n)

= ∂

(α

...α

)

=⇒ δ

α(n)

= 0 ; (1.6)

(ii) C

α(n)

is divergenceless

∂

βγ

βγα

...α

n−2

=0 . (1.7)

Unlike the 4D relations (1.1)and (1.2), the expression for C

α(n)

given by (1.5)is not illuminating. It is not obvious from (1.5) that

https://doi.org/10.1016/j.physletb.2019.01.040

0370-2693/

SCOAP

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