Local supersymmetry and the square roots of Bondi-Metzner-Sachs supertranslations
Local supersymmetry and the square roots of Bondi-Metzner-Sachs supertranslations
复制标题
局部超对称性和 Bondi-Metzner-Sachs 超平移的平方根
作者：
Oscar Fuentealba;M. Henneaux;Sucheta Majumdar;Javier Matulich;Turmoli Neogi
超BMS 4代数（英语：Super-BMS 4 algebra），也称为BMS 4超代数，是BMS 4代数的分次扩张。它们可以是两种不同的类型：它们可以包含有限个或无限个费米子生成元。我们在这封信中表明，适当的边界条件的引力子和引力子场在空间无穷远，超引力在渐近平坦的空间拥有作为超代数的渐近对称性的（非线性）超BMS 4代数包含无限数量的费米子生成元，我们表示SBMS 4。这些边界条件不仅在SBMS 4下是不变的，而且还导致了对超对称性的完全一致的正则描述，这些超对称性特别具有根据非线性SBMS 4代数闭合的定义良好的哈密顿生成元。人们特别发现，费米子发生器之间的分级括号产生了所有的BMS 4超平移，因此它们提供了“平方根”。对无穷大引力场的研究揭示了某种意想不到的无穷维渐近对称群的出现。这一现象首先在四维时空的渐近平坦背景下被展示出来，其中包含闵可夫斯基空间的庞加莱等距群作为子群的无限维BMS 4群被证明在无穷远处出现为渐近对称群[1-4]（最近的评论见[5，6]）。后来，三个时空维中的反德西特引力也被证明是反德西特群的无限维扩展[7]。虽然在AdS/CFT对应[8，9]的背景下，反德西特代数的无限维增强的意义是自然的，但无限维BMS代数的物理含义仍然是深入研究的主题（见[10]早期的工作，[11-13]的形式主义的一个有趣的扩展，包括超旋转，和[14]供回顾和参考触发当前活动的最近令人兴奋的发展）。在量子理论中，状态自然地定义在一般柯西超曲面上[15-17]。渐近对称是由作用于物理希尔伯特空间的算子产生的，并形成渐近对称代数的表示，直到代数允许的可能中心项。量子理论的一个直接途径是基于与量子结构密切相关的哈密顿形式。在标准描述中，系统的经典状态完全由柯西超曲面上的动力学变量的值指定（包括辐射，如果有的话），柯西超曲面渐近类空无穷大。一个令人满意的配方需要一个规格的相空间变量在空间无穷远的下降，这应该是这样的行动和变分原理是明确定义的。对称性和哈密顿生成元之间的联系则由经典力学的标准定理给出。人们特别发现，对称性具有辛作用，并且被矩映射（可能在中心电荷出现时定义在中心扩展代数上）捕获。这种与量子公式的密切相似是使哈密顿形式主义具有指导意义的原因之一。满足上述良定性要求的类空超曲面上的BMS 4对称性的哈密顿公式在文献[18- 20]中得到了发展。这是通过两组不同的边界条件来实现的。在[18]中，在空间无限远展开中，场的首阶的宇称条件被认为是不同的，并且与[21]的宇称条件不等价，甚至直到坐标变换。在[19，20]中，场的首阶上的奇偶性条件仅通过具有指定下降的坐标变换而与[21]中的奇偶性条件不同。在这两种情况下，BMS 4群都是理论的渐近对称群。第一组奇偶性条件（[18]）代表了[21]的奇偶性条件的更剧烈的扭曲，因为双同态不变对象，如Weyl张量，服从不等价的条件。在这封信中，我们展示了如何将引力渐近结构的哈密顿分析扩展到超引力。事实证明，[18]和[19，20]的两组宇称条件都允许具有无限个费米子生成元的BMS 4代数的超对称扩展，但是我们将在这里关注的[18]的宇称条件导致了一个具有比[19，20]更丰富结构的超代数。特别是，费米子发生器之间的分级括号产生了所有的BMS 4超平移，而不仅仅是普通的时空平移。费米子发生器可以被认为是BMS 4超平移发生器的“平方根”。早期关于BMS 4代数的超对称扩展的工作考虑了只有有限个费米子生成元的费米子扩展，即零无穷大和空间无穷大的标准超荷[22]。涉及无穷多个
Super-BMS4 algebras – also called BMS4 superalgebras – are graded extensions of the BMS4 algebra. They can be of two different types: they can contain either a finite number or an infinite number of fermionic generators. We show in this letter that, with suitable boundary conditions on the graviton and gravitino fields at spatial infinity, supergravity on asymptotically flat spaces possesses as superalgebra of asymptotic symmetries a (nonlinear) super-BMS4 algebra containing an infinite number of fermionic generators, which we denote SBMS4. These boundary conditions are not only invariant under SBMS4, but also lead to a fully consistent canonical description of the supersymmetries, which have in particular well-defined Hamiltonian generators that close according to the nonlinear SBMS4 algebra. One finds in particular that the graded brackets between the fermionic generators yield all the BMS4 supertranslations, of which they provide therefore “square roots”. The study of the gravitational field at infinity has revealed the somewhat unanticipated emergence of infinitedimensional asymptotic symmetry groups. This phenomenon was exhibited first in the asymptotically flat context in four spacetime dimensions, where the infinitedimensional BMS4 group, which contains the Poincaré group of isometries of Minkowski space as a subgroup, was shown to emerge as asymptotic symmetry group at infinity [1–4] (for recent reviews, see [5, 6]). Later and independently, anti-de Sitter gravity in three spacetime dimensions was also shown to exhibit an infinitedimensional extension of the anti-de Sitter group [7]. While the significance of the infinite-dimensional enhancement of the anti-de Sitter algebra takes a natural place in the context of the AdS/CFT correspondence [8, 9], the physical implications of the infinitedimensional BMS algebra are still a subject of intense study (see [10] for earlier work, [11–13] for an intriguing extension of the formalism to include super-rotations, and [14] for review and references to the more recent exciting developments that triggered the current activity). In the quantum theory, states are naturally defined on general Cauchy hypersurfaces [15–17]. The asymptotic symmetries are generated by operators that act on the physical Hilbert space and form a representation of the asymptotic symmetry algebra, up to possible central terms when these are algebraically permitted. One direct access to the quantum theory is based on the Hamiltonian formalism, which closely parallels the quantum structure. In the standard description, the classical state of the system is completely specified (including radiation if any) by the values of the dynamical variables on Cauchy hypersufaces, which asymptote spacelike infinity. A satisfactory formulation needs a specification of the fall-off of the phase space variables at spatial infinity, which should be such that the action and the variational principle are well-defined. The connection between symmetries and Hamiltonian generators is then given by standard theorems of classical mechanics. One finds in particular that the symmetries have a symplectic action and are captured by the moment map (possibly defined on the centrally extended algebra when central charges occur). This close parallel with the quantum formulation is one of reasons that make the Hamiltonian formalism instructive. A Hamiltonian formulation of the BMS4 symmetry on spacelike hypersurfaces fulfilling the above welldefinedness requirement was developed in the papers [18– 20]. This was achieved through two distinct sets of boundary conditions. In [18], the parity conditions on the leading orders of the fields in an expansion at spatial infinity were taken to be different and inequivalent to the parity conditions of [21], even up to a coordinate transformation. In [19, 20], the parity conditions on the leading orders of the fields were taken to merely differ from those of [21] by a coordinate transformation with specified fall-off. In both cases, the BMS4 group emerges as asymptotic symmetry group of the theory. The first set of parity conditions (of [18]) represents a more drastic twist of the parity conditions of [21] because diffeomorphism invariant objects, such as the Weyl tensor, obey inequivalent conditions. We show in this letter how to extend the Hamiltonian analysis of the asymptotic structure of gravity to cover supergravity. It turns out that both sets of parity conditions those of [18] and those of [19, 20] admit a supersymmetric extension of the BMS4 algebra with an infinite number of fermionic generators, but that those of [18], on which we shall focus here, lead to a superalgebra with a richer structure that those of [19, 20]. In particular, the graded brackets between the fermionic generators yield all BMS4 supertranslations and not just the ordinary spacetime translations. The fermionic generators may be considered for that reason as being the “square roots” of the BMS4 supertranslation generators. Earlier work on the supersymmetric extensions of the BMS4 algebra considered fermionic extensions with only a finite number of fermionic generators the standard supercharges both at null infinity [22] and spatial infinity [23]. Extensions involving an infinite number of
影响因子：
5
作者：
Gautam Satishchandran;R. Wald
通讯作者：
Gautam Satishchandran;R. Wald
影响因子：
5.4
作者：
A. Fotopoulos;S. Stieberger;T. R. Taylor;B. Zhu
通讯作者：
A. Fotopoulos;S. Stieberger;T. R. Taylor;B. Zhu