Geometry and Large N Limits in Quantum Hall States

量子霍尔效应(QHE)是一种显著的现象，在含有杂质的材料中发生霍尔电导的精确量子化。它最初是在低温和强磁场下的二维电子系统中发现的，最近在石墨烯中发现了它，据报道，它甚至在室温下也是如此。众所周知，QHE中的量子化现象本质上是几何起源的。这个项目解决了量子霍尔态几何理论中的一些问题，主要集中在数学方面，着眼于潜在的物理应用。该项目的主要目标是(I)定义和构造各种几何背景下的量子霍尔态，如Laughlin态和Pfaffian态；(Ii)对这些态在Riemann面模空间上的几何绝热输运进行完整的分类；(Iii)发展有效的新方法和工具来研究QH态的配分函数、关联函数、绝热曲率的大N渐近性；(Iv)揭示QH态与Liouville量子引力、Bergman核、Quillen理论和几何的新的令人兴奋的联系。为了达到这些目标，我们利用Kaehler几何中的最新方法、Bergman核的渐近分析和量子场论技术，提出了研究QH态的新方法。作为该项目的结果，我们希望发现QH态与现代几何和物理的各个方面的新的令人兴奋的联系。该项目的结果将对数学物理、Kaehler几何、Bergman核、随机矩阵和QHE社区产生潜在的兴趣。

Quantum Hall effect (QHE) is a remarkable phenomenon, where the precise quantization of Hall conductance occurs in materials with impurities. It was originally discovered in two-dimensional electron systems at low temperatures and in strong magnetic fields, and more recently in graphene, where it was reported even at room temperatures. It is well known that the quantization phenomena in QHE have essentially geometric origin. This project addresses a number of questions in geometric theory of the quantum Hall states, focusing mainly on mathematical aspects, with an eye for potential physics applications. The main goals of the project are (i) to define and construct the Quantum Hall states, such as Laughlin state and Pfaffian state on various geometric backgrounds (ii) to give a complete classification of the geometric adiabatic transport on the moduli spaces of Riemann surfaces for these states (iii) develop effective new methods and tools to study large N asymptotics of the partition functions, correlation functions, adiabatic curvatures for the QH states, (iv) uncovering new exciting connections of QH states to Liouville quantum gravity, Bergman kernels, Quillen theory and geometry. In order to reach these goals we propose new approaches to QH states, making use of recent methods in Kaehler geometry, asymptotic analysis of Bergman kernels, and quantum field theory techniques. As the outcome of the project we expect to uncover new exciting connections of the QH states with various aspects of modern geometry and physics. The results of the project will be of potential interest to the mathematical physics, Kaehler geometry, Bergman kernel, random matrix and QHE communities.