论文标题
重力边缘模式的基质量化
Matrix Quantization of Gravitational Edge Modes
论文作者
论文摘要
带有边界的引力子系统具有无限维对称代数的作用，对重力量子理论具有潜在的深远影响。我们开始研究该角对称代数的量化，以定位于由二维球体界定的区域的重力空间。首先要观察到，代数$ \ Mathfrak {sdiff}（s^2）$的区域保护差异性的差异性是对有限维代数$ \ mathfrak {susfrak {su}（n）$的变形，我们得出了两个重要的粒料，以奉献两种小圈子的圈地。具体而言，我们发现面积的流体动力学代数$ \ mathfrak {sdiff}（s^2）\ oplus _ {\ Mathcal {l}} \ Mathbb {r}^{s^{s^2} $作为大型 - $ n $ n $ n $ n $ limit of $ - $ n $ n $ n $ limit c）\ oplus \ mathbb {r} $和完整的区域呈现角落对称代数$ \ mathfrak {sdiff}（s^2）（s^2）\ oplus _ {\ mathcal {l}}} \ mathfrak {slfrak {sl}伪独立组$ \ Mathfrak {su}（n，n）$。我们找到了变形和连续代数的Casimir元素的匹配条件，并显示这些条件如何确定变形参数$ n $的值以及与量化局部重力相位空间相关的变形代数的表示。此外，我们提出了许多与出现各种代数有关的新结果，包括对$ \ Mathfrak {su}（n）$结构常数的渐近扩展以及完整的$ \ Mathfrak {diff}（diff}（s^2）$结构常数的明确计算。我们工作的结果是区域操作员的定义，该定义与有限$ n $时的区域保护角对称性的变形兼容。
Gravitational subsystems with boundaries carry the action of an infinite-dimensional symmetry algebra, with potentially profound implications for the quantum theory of gravity. We initiate an investigation into the quantization of this corner symmetry algebra for the phase space of gravity localized to a region bounded by a 2-dimensional sphere. Starting with the observation that the algebra $\mathfrak{sdiff}(S^2)$ of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra $\mathfrak{su}(N)$, we derive novel finite-$N$ deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra $\mathfrak{sdiff}(S^2)\oplus_{\mathcal{L}}\mathbb{R}^{S^2}$ arises as the large-$N$ limit of $\mathfrak{sl}(N,\mathbb C)\oplus\mathbb{R}$ and that the full area-preserving corner symmetry algebra $\mathfrak{sdiff}(S^2)\oplus_{\mathcal{L}}\mathfrak{sl}(2,\mathbb{R})^{S^2}$ is the large-$N$ limit of the pseudo-unitary group $\mathfrak{su}(N,N)$. We find matching conditions for the Casimir elements of the deformed and continuum algebras and show how these determine the value of the deformation parameter $N$ as well as the representation of the deformed algebra associated with a quantization of the local gravitational phase space. Additionally, we present a number of novel results related to the various algebras appearing, including a detailed analysis of the asymptotic expansion of the $\mathfrak{su}(N)$ structure constants, as well as an explicit computation of the full $\mathfrak{diff}(S^2)$ structure constants in the spherical harmonic basis. A consequence of our work is the definition of an area operator which is compatible with the deformation of the area-preserving corner symmetry at finite $N$.
