近地平线几何学：共伴轨道与量子化研究

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"这篇论文详细探讨了近地平线极值几何中的辛对称性和共伴轨道的量化。NHEG代数，即近地平线极值几何代数，是一个在[arXiv:1503.07861]中提出的Virasoro代数的扩展，用于描述具有SL(2, ℝ)×U(1)^n+1同胚的（n+4）维近地平线几何的对称性质。研究者构建了NHEG群并分类了其在相空间上的共伴轨道。这些轨道可视为从n维托拉斯到Virasoro群的映射，形成了T^n上的标准Virasoro共伴轨道的束。此外，论文还阐述了这些轨道的量化应遵循的统一表示形式，以及它们的特性。作者证明了利用扭曲的Sugawara构造可以从u(1)电流构建NHEG代数。" 这篇科研论文由R. Javadinezhad、B. Oblak和M. M. Sheikh-Jabbari合作完成，分别来自纽约大学物理系、苏黎世联邦理工学院理论物理研究所和伊朗德黑兰基础科学研究所物理学院。论文于2018年1月10日提交，4月2日接受，并于4月5日发表在JHEP04(2018)025期刊上，由Springer出版，且标记为“Open Access”，意味着公众可以免费获取。文章的核心概念是NHEG代数，它扩展了Virasoro代数，这在研究黑洞物理和广义相对论的极端情况时至关重要。Virasoro代数在二维共形场论中扮演着重要角色，而NHEG代数则将其应用于更高维度的近地平线极值几何，这是一种在极限旋转或电荷情况下出现的特殊空间时间结构。通过分析NHEG群的共伴轨道，作者能够更深入地理解这种几何结构的对称性质。共伴轨道的分类揭示了它们与Virasoro群的关系，这为理解NHEG群的动态提供了新的视角。共伴表示是量子力学中对经典对称性进行量化的一种方法，因此，研究这些轨道的量子化是探索近地平线极值几何量子理论的关键步骤。扭曲的Sugawara构造则提供了一种构造NHEG代数的新途径，它涉及u(1)电流的组合，这在凝聚态物理和弦理论中也有着广泛的应用。这篇论文对理解和研究高维空间中极端物理条件下的对称性、几何结构以及相应的量子理论有着重要的贡献。它不仅深化了对NHEG代数的理解，也促进了共伴轨道和量子化的理论发展。

JHEP04(2018)025

or rings with k

= 1/k

[34, 35, 44]. Note that under SL(2, R) isometries which keep the t, r

parts of the metric intact, rdt transforms by a closed form, rdt → rdt+dξ, where ξ depends

on the details of the transformation. Hence SL(2, R) isometries also involve translations of

along k

, φ

→ φ

+ k

ξ, so the vector

k also aﬀects the generators (Killing vectors) of

the SL(2,R) isometry [36, 37].

NHEGs are not black holes (they have no event horizon), but they do have inﬁnitely

many bifurcate Killing horizons at constant t, r, all at the same Frolov-Thorne [46] tem-

perature 1/2π [36, 37]; see [26, section 2.1] for details. The metric on each bifurcation

surface is

= Γ(θ)



dθ

+ γ

(θ)dφ

dφ



. (2.2)

It is smooth for all values of θ; although

k does not appear here, it is a measurable physical

parameter as its components are related to the angular momentum of the black hole —

see [26, 36, 37]. Thus, on the horizon, and of course on the whole NHEG (2.1), one is

dealing with an anisotropic torus — a torus with a preferred direction speciﬁed by

k. In

principle, the components of

k may take any value; in practice however, the ratios of those

components are directly related to ratios of components of angular momentum. Assuming

that the latter is quantized, this implies that

k is proportional to a vector with integer

entries, i.e. an element of the dual lattice of T

n+1

. Throughout this work we will always

assume that this condition is satisﬁed, as it will be necessary to ensure smoothness of the

NHEG group.

Let us then consider an anisotropic torus whose

k belongs to the dual lattice. In that

case one can use the SL(n + 1, Z) volume-preserving symmetry of the torus to bring

possibly up to normalization, to the convenient form

k = (0, . . . , 0, 1). (2.3)

However, note that smoothness of the metric (2.2) generally prevents one from ﬁnding a

global frame where

k has this form for all values of θ. This fact will make the connection

between NHEG orbits and the corresponding space-time metrics somewhat subtle; see

section 7.

Symplectic symmetries. Consider a metric g

µν

solving Einstein’s equations and let

, χ

be vector ﬁelds; the corresponding symplectic density is a two-form in ﬁeld space,

ω(δ

µν

, δ

µν

; g

µν

), where δ

µν

= L

µν

. This density can be of the Lee-Wald [12, 13]

or Barnich-Brandt [5] type, or either of them up to a boundary term (see [26] for de-

tails). When ω vanishes on-shell for suitable vector ﬁelds, the latter generate symplectic

symmetries [26–28]. In such cases, the integrability condition on charge variations is usu-

ally satisﬁed automatically and surface charges can be deﬁned by integration over generic

compact, space-like, codimension-two surfaces. In contrast to the perhaps more familiar

asymptotic symmetries, these surfaces need not be at inﬁnity; for the NHEG (2.1) they

can be located at arbitrary (t, r). Furthermore, one can view the charges as generators of

symplectomorphisms on a phase space built by acting on a background metric with ﬁnite

diﬀeomorphisms generated by χ’s. Each such phase space is an orbit of the symplectic

symmetry group.

– 4 –

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