论文标题
布朗循环和liouville随机表面的中心电荷
Brownian loops and the central charge of a Liouville random surface
论文作者
论文摘要
我们在紧凑的表面上探索了拉普拉斯 - 贝特拉米操作员所谓的Zeta Zeta调控决定因素的几何含义，具有或没有边界。我们将Laplacian决定因素的$（-C/2）$ - 表面上强度$ C $的适当正规分区功能与适当的正规分区功能联系起来。这意味着，从某种意义上说，用强度$ c $的布朗环路汤来装饰随机表面，对应于$（-C/2）$ - Laplacian决定因素的The The The Surtive Law。 接下来，我们引入了一种定期化liouville量子重力（LQG）表面的方法（具有某些物质中心电荷参数$ \ mathbf {c} $）以产生光滑的表面。我们表明，通过$（ - \ Mathbf {C}'/ 2）加权该随机表面的定律 - laplacian降决定剂的$ TH具有将物质中心费用从$ \ mathbf {c} $更改为$ \ Mathbf {c} $ to $ \ mathbf {c} + \ \ \ \ \ \ \ \ \ \ \ \ \ mathbf {c}'$的效果。结合了较早的结果，这提供了一种解释物质表面中心电荷$ \ mathbf {c} $作为纯LQG表面的方法，该表面是由Brownian Loop of Intentive $ \ mathbf {C} $装饰的。 在这个想法的基础上，我们提出了有关随机平面图及其连续类似物的几个开放问题。尽管LQG的原始结构仅针对$ \ mathbf {c} \ leq 1 $定义，但是当$ \ mathbf {c}> 1 $时，某些构造和问题也很有意义。
We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace-Beltrami operator on a compact surface, with or without boundary. We relate the $(-c/2)$-th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity $c$ on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity $c$ corresponds to weighting the law of the surface by the $(-c/2)$-th power of the determinant of the Laplacian. Next, we introduce a method of regularizing a Liouville quantum gravity (LQG) surface (with some matter central charge parameter $\mathbf{c}$) to produce a smooth surface. And we show that weighting the law of this random surface by the $( -\mathbf{c}'/ 2)$-th power of the Laplacian determinant has precisely the effect of changing the matter central charge from $\mathbf{c}$ to $\mathbf{c} + \mathbf{c}'$. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge $\mathbf{c}$ as a pure LQG surface decorated by a Brownian loop soup of intensity $\mathbf{c}$. Building on this idea, we present several open problems about random planar maps and their continuum analogs. Although the original construction of LQG is well-defined only for $\mathbf{c}\leq 1$, some of the constructions and questions also make sense when $\mathbf{c}>1$.
