来自二维欧几里得量子引力数学的启示
Lessons from the Mathematics of Two-Dimensional Euclidean Quantum Gravity
AI总结：
本文回顾二维欧几里得量子引力相关的随机几何数学进展，从随机平面图离散化出发，介绍其计数、无限图处理及布朗球面连续极限构建，为高维研究提供指导。
摘要：
为欧几里得量子引力的路径积分寻找数学基础，需要在时空流形上构建随机几何。随着物理学中二维理论的发展，2-球面（2-sphere）上的随机几何近年来在数学文献中受到广泛关注，推动了二维欧几里得量子引力路径积分表述的完全严格实现。本章回顾了若干重要的数学进展，它们可作为研究更高维度欧几里得量子引力的指导原则。我们的出发点是由随机平面图（random planar maps）编码的离散几何，它实现了路径积分的格点离散化。我们通过生成函数重述了平面图的计数，并展示了与树的双射如何解释其中一些计数结果惊人的简洁性。随后我们说明了如何处理无限平面图，以及如何通过剥离过程（peeling process）分析其探索过程。上述的树为构建随机离散几何的普适连续极限——即布朗球面（Brownian sphere）——提供了基础，布朗球面代表了无物质情况下二维欧几里得量子引力背后的随机几何。
英文摘要：
The search for a mathematical foundation for the path integral of Euclidean quantum gravity calls for the construction of random geometry on the spacetime manifold. Following developments in physics on the two-dimensional theory, random geometry on the 2-sphere has in recent years received much attention in the mathematical literature, which has led to a fully rigorous implementation of the path integral formulation of two-dimensional Euclidean quantum gravity. In this chapter we review several important mathematical developments that may serve as guiding principles for approaching Euclidean quantum gravity in dimensions higher than two. Our starting point is the discrete geometry encoded by random planar maps, which realizes a lattice discretization of the path integral. We recap the enumeration of planar maps via their generating functions and show how bijections with trees explain the surprising simplicity of some of these. Then we explain how to handle infinite planar maps and to analyze their exploration via the peeling process. The aforementioned trees provide the basis for the construction of the universal continuum limit of the random discrete geometries, known as the Brownian sphere, which represents the random geometry underlying two-dimensional Euclidean quantum gravity in the absence of matter.
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