Random conformal geometry and bosonic string theory

About This Grant

This project explores random curves and surfaces known as Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG), which are central objects in modern probability theory and mathematical physics. The goal is to resolve longstanding problems arising from theoretical physics — specifically, bosonic string theory and conformal field theory. To this end, the PI will further develop the rich interplay between SLE and LQG. These methodological contributions are highly likely to find application in related problems. Additionally, this project will provide mentorship opportunities for undergraduate and graduate researchers. There are three main directions of research. First, Brownian surfaces are special LQG surfaces arising as the scaling limits of uniform random planar maps with specified topologies. A Brownian surface has a random modulus describing its conformal structure. The aim is to determine the law of the random modulus for each topology, thereby verifying a conjecture from bosonic string theory. Second, the conformal loop ensemble is a collection of SLE loops which arises as the scaling limit of lattice loop models at criticality. The objective is to derive exact formulas for various three-point functions of the conformal loop ensemble, and thus relate them to conformal field theories. Third, in bosonic string theory in physics, a string is a loop in d-dimensional space which evolves in time, tracing out a random surface described by LQG. The goal is to use LQG to give a mathematical construction of the bosonic string, and prove that its evolution is Markovian. 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.