扭转矢量方程与黑洞对称性：3型场的可分性研究

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"Principal Tensor Strikes Again: Separability of Vector Equations with Torsion" 这篇研究文章发表在《Physics Letters B》795期（2019年），主要探讨了具有3型场的黑洞时空中的隐藏对称性，这种对称性与主Killing-Yano张量的扭曲推广有关。主Killing-Yano张量在决定这些黑洞的基本特性方面发挥着关键作用，并且是Hamilton-Jacobi、Klein-Gordon以及扭转向量场方程（修正的Dirac场方程）可分性的基础。文章中特别关注了一个具体的例子——D = 5的Chong-Cvetič-Lü-Pope黑洞，这是五维最小规范超重力理论中的一个解决方案。研究者们证明，在这个背景下，扭转向量场方程也是可分离的，这进一步强调了主张量在解的结构和可分性中的核心地位。同时，为了提供对比，作者还探讨了Proca场在高维Kerr-NUT-AdS时空中的可分性，包括在奇数维度中的新显式结果。扭转向量方程的可分性是理论物理学中的一个重要概念，因为它允许将复杂的问题分解成更易于管理的部分，这对于理解和求解这些高度非线性系统的动态至关重要。在黑洞物理中，这种可分性使得可能对黑洞的性质有更深入的理解，例如通过分离变量来求解运动粒子或波的轨迹和传播。 Killing-Yano张量是描述空间时间对称性的工具，当它带有扭转时，意味着存在额外的隐藏对称性，这在理解黑洞的动态和量子效应中是极其重要的。对于Chong-Cvetič-Lü-Pope黑洞的研究，不仅深化了我们对这些特殊黑洞的认识，也展示了扭曲Killing-Yano张量在多维物理系统中的应用。此外，文章还包括了Proca场的分析，这是一种具有质量的标量场，其在高维背景下的可分性研究提供了额外的洞察，特别是在AdS（Anti-de Sitter）空间中，这在AdS/CFT对应和量子引力研究中具有重要意义。这篇论文在黑洞物理和高能物理领域提供了新的理论进展，强调了隐藏对称性和可分性在理解和计算复杂物理系统中的核心地位。它不仅贡献了理论成果，也为未来研究扭曲张量和扭转向量方程在其他物理模型中的应用奠定了基础。

Physics Letters B 795 (2019) 650–656

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Principal tensor strikes again: Separability of vector equations with

torsion

Ramiro Cayuso

a,b

, Finnian Gray

a,b,∗

, David Kubiz

nák

a,b

, Aoibheann Margalit

a,b

Renato Gomes Souza

a,b,c

, Leander Thiele

a,b

Perimeter Institute, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada

Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

ICTP South American Institute for Fundamental Research, IFT-UNESP, Sao Paulo, SP, 01140-070, Brazil

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

Article history:

Received

28 June 2019

Accepted

3 July 2019

Available

online 5 July 2019

Editor:

M. Cveti

Many black hole spacetimes with a 3-form ﬁeld exhibit a hidden symmetry encoded in a torsion

generalization of the principal Killing–Yano tensor. This tensor determines basic properties of such black

holes while also underlying the separability of the Hamilton–Jacobi, Klein–Gordon, and (torsion-modiﬁed)

Dirac ﬁeld equations in their background. As a speciﬁc example, we consider the Chong–Cveti

c–Lü–

Pope

black hole of D = 5 minimal gauged supergravity and show that the torsion-modiﬁed vector

ﬁeld equations can also be separated, with the principal tensor playing a key role in the separability

ansatz. For comparison, separability of the Proca ﬁeld in higher-dimensional Kerr–NUT–AdS spacetimes

(including new explicit formulae in odd dimensions) is also presented.

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

1. Introduction

Hidden symmetries of dynamics play a crucial role for the study

of diverse physical systems that can be relativistic or not, with

or without gravity, classical or quantum [1]. Particularly useful for

black hole physics is the hidden symmetry of the principal Killing–

Yano

tensor [2]. Such a tensor not only generates other explicit and

hidden symmetries, it also determines the algebraic type of the

solution, and allows for a separation of variables when studying

physical ﬁeld equations in black hole spacetimes.

The

Golden Era of ﬁeld equation separability in Kerr geome-

try

spanned the late 60s and 70s. During this period, separable

solutions were found for particles following geodesic motion, test

scalar and spinor ﬁelds, as well as massless vector (spin 1) and

tensor (spin 2) ﬁelds [3–8]. Much later, after the generalization

of Kerr geometry to higher dimensions by Myers and Perry [9], a

ﬂurry of papers extended the results for geodesic motion [10], and

test scalar [11] and spinor [12]ﬁelds to arbitrary dimensions—all

thanks to the principal tensor. However, the appropriate separa-

Corresponding author.

E-mail

addresses: rcayuso@perimeterinstitute.ca (R. Cayuso),

fgray@perimeterinstitute.ca (F. Gray), dkubiznak@perimeterinstitute.ca

(D. Kubiz

nák),

amargalit@perimeterinstitute.ca (A. Margalit),

rsouza@perimeterinstitute.ca (R.G. Souza), lthiele@perimeterinstitute.ca (L. Thiele).

tion scheme for vector and tensor in dimensions D > 4 remained

elusive.

breakthrough on this front came in 2017 when Lunin [13]

demonstrated

the separability of Maxwell’s equations in Myers–

Perry-(A)dS

geometry [14]. Lunin’s approach was novel in that it

provided a separable ansatz for the vector potential rather than the

ﬁeld strength, a method that had previously seen success in D = 4

dimensions

[7,8]. In 2018, Frolov, Krtous, and Kubiznak showed

that Lunin’s ansatz can be written in a covariant form, in terms of

the principal tensor [15,16], allowing them to extend Lunin’s result

to general (possibly off-shell) Kerr–NUT–AdS spacetimes [17]. The

separation of massive vector (Proca) ﬁeld perturbations in these

spacetimes (an achievement previously absent even for the four-

dimensional

Kerr geometry) followed shortly after that [18], see

also [19–21].

The

separability of the vector ﬁeld hinges on the existence of

the principal tensor. Such a tensor: i) determines the canonical

(preferred) coordinates in which the separation occurs, ii) gener-

ates

the towers of explicit and hidden symmetries linked to the

‘symmetry operators’ of the separated vector equation, and iii) ex-

plicitly

enters the separation ansatz for the vector potential P .

Namely, this ansatz can be written in the following covariant form:

= B

∇

Z , (1)

https://doi.org/10.1016/j.physletb.2019.07.007

0370-2693/

SCOAP

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