封闭矩阵世界的若干方面
Aspects of Closed Matricial Worlds
- King’s College London（伦敦国王学院）
- KU Leuven（荷语鲁汶大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究具有\(\Lambda>0\)的封闭空间截面简单引力理论的希尔伯特空间结构，通过重新审视矩阵模型文献结果，从惠勒 - 德维特方程和引力路径积分角度研究引力波函数，建立直至亏格六的结果，对比类时刘维尔理论。
AI中文摘要：
我们研究了在具有\(\Lambda>0\)的封闭空间截面的简单引力理论中的希尔伯特空间结构。我们的动机源于具有\(S^2\times\Sigma_h\)拓扑的四维\(\Lambda>0\)爱因斯坦 - 麦克斯韦理论中引力鞍点的存在，其中\(\Sigma_h\)是亏格为\(h\)的黎曼曲面。这里，作为具体起点，我们针对二维\(\Lambda>0\)量子引力探索该问题。我们重新审视并详细阐述矩阵模型文献中的精确结果。我们从惠勒 - 德维特方程和引力路径积分两个角度研究引力波函数。该设置虽简单，但展现出许多普遍感兴趣的特征，如扰乱微扰展开的大体积效应、违反精确惠勒 - 德维特方程的路径积分波函数的拓扑修正，以及\(\Lambda\)中具有非平凡结构的球面路径积分\({Z}^{(0)}_{\text{grav}}\)。我们讨论了Lian和Zuckerman发现的无限维正则引力希尔伯特空间的候选内积。通过建立直至亏格六的明确结果，我们认为在大空间尺度下亏格为\(h\)的引力盘路径积分的主导贡献模仿了二维拓扑引力的行为。顺便提及，我们表明为使\({Z}^{(0)}_{\text{grav}}\)对离散黎曼曲面产生正计数问题，它必须有一个负的前置因子。我们将我们的分析与更现实的类时刘维尔理论情况进行对比。
英文摘要：
We investigate the Hilbert space structure for simple theories of gravity with $Λ>0$ on closed spatial sections. Our motivation ties to the presence of gravitational saddles in four-dimensional $Λ>0$ Einstein-Maxwell theory with $S^2\times Σ_h$ topology, where $Σ_h$ is a genus-$h$ Riemann surface. Here, as a concrete starting point, the problem is explored for two-dimensional $Λ>0$ quantum gravity. We revisit and elaborate on exact results in the matrix model literature. We study gravitational wavefunctions from both the perspective of the Wheeler-DeWitt equation and the gravitational path integral. Though simple, the setting displays many features of general interest such as large volume effects that disrupt the perturbative expansion, topological corrections to the path-integral wavefunction that offend the exact Wheeler-DeWitt equation, and a sphere path integral ${Z}^{(0)}_{\text{grav}}$ with non-trivial structure in $Λ$. We discuss candidate inner products for the infinite-dimensional canonical gravitational Hilbert space uncovered by Lian and Zuckerman. By establishing explicit results up to genus-six, we argue that the dominant contribution to the genus-$h$ gravitational disk path integral at large spatial size mimics the behavior of two-dimensional topological gravity. In passing, we show that for ${Z}^{(0)}_{\text{grav}}$ to give rise to a positive counting problem for discretised Riemann surfaces, it must have a negative pre-factor. We contrast our analysis to the more realistic case of timelike Liouville theory.
