CAREER: Liouville Quantum Gravity, Two-Dimensional Random Geometry, and Conformal Field Theory

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Significant recent advances have been made in the probability theory of natural two-dimensional (2D) mathematical structures, including metrics, measures, functions, and curves on surfaces. A one-parameter family of random surfaces, called Liouville quantum gravity (LQG) surfaces, has emerged as a fruitful setting to study such structures. LQG originated from the study of 2D quantum gravity and string theory in theoretical physics. Subsequently, LQG has become an active and deep mathematical subject in probability. The research part of the project funded by this award aims to address outstanding challenges in the mathematical theory of LQG, thereby providing a firm foundation for several assumptions in theoretical physics. The educational part of the project aims to make high quality mathematical education and research more accessible, in particular, to students from underrepresented groups, researchers from geographically disadvantaged areas, and high school mathematics teachers.Quantum gravity is the physics counterpart of random geometry. It is believed in physics that 2D quantum gravity coupled with conformal matter is described by LQG, governed by a conformal field theory (CFT) called Liouville CFT. The most intuitive formulation of 2D quantum gravity is through its microscopic description, namely random planar maps. The primary research goal of the project is to show that certain classical random planar map models converge to LQG in the scaling limit, laying a mathematical foundation for this physical picture. The two most challenging cases are when the random geometry is non-uniform or when the underlying surface is non-simply-connected. The project aims to address open questions in both cases. Besides the geometric aspect, the correlation functions of Liouville CFT possess deep algebraic structures, which are expected to be computed by a schematic program called the conformal bootstrap. Another major goal of the research is to rigorously establish the conformal bootstrap program for Liouville CFT. To achieve these goals, the PI will rely on the rich interplay between LQG and a family of random planar curves called the Schramm-Loewner evolution.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.