论文标题
Dirac合奏和Liouville量子重力的双缩放限制
Double scaling limits of Dirac ensembles and Liouville quantum gravity
论文作者
论文摘要
在本文中，我们研究了有限的真实光谱三元组合的合奏，该组合在可能的狄拉克操作员的整个空间上配备了一个积分的路径。在光谱三元组的非交通性几何设置中，狄拉克运算符以替代了歧管上的度量的中心阶段。因此，该路径积分是对指标的整合的非共同类似物，这是量子重力理论的关键特征。从所谓的双缩放限制中的这些积分中，我们从liouville共形场理论中得出了最小模型的关键指数，并加重了重力。此外，这些模型的分区函数的渐近学满足了诸如PainlevéI之类的微分方程，作为减少KDV层次结构的降低，该方程是由保形场理论预测的。所有这些都是使用随机矩阵理论的良好和严格的技术证明的。
In this paper we study ensembles of finite real spectral triples equipped with a path integral over the space of possible Dirac operators. In the noncommutative geometric setting of spectral triples, Dirac operators take the center stage as a replacement for a metric on a manifold. Thus, this path integral serves as a noncommutative analogue of integration over metrics, a key feature of a theory of quantum gravity. From these integrals in the so-called double scaling limit we derive critical exponents of minimal models from Liouville conformal field theory coupled with gravity. Additionally, the asymptotics of the partition function of these models satisfy differential equations such as Painlevé I, as a reduction of the KDV hierarchy, which is predicted by conformal field theory. This is all proven using well-established and rigorous techniques from random matrix theory.
