2021 Theses Doctoral
A renormalization approach to the Liouville quantum gravity metric
This thesis explores metric properties of Liouville quantum gravity (LQG), a random geometry with conformal symmetries introduced in the context of string theory by Polyakov in the 80’s. Formally, it corresponds to the Riemannian metric tensor “e^{γh}(dx² + dy²)” where h is a planar Gaussian free field and γ is a parameter in (0, 2). Since h is a random Schwartz distribution with negative regularity, the exponential e^{γh} only makes sense formally and the associated volume form and distance functions are not well-defined. The mathematical language to define the volume form was introduced by Kahane, also in the 80’s. In this thesis, we explore a renormalization approach to make sense of the distance function and we study its basic properties.
Files
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Falconet_columbia_0054D_16428.pdf application/pdf 2.69 MB Download File
More About This Work
- Academic Units
- Mathematics
- Thesis Advisors
- Dubedat, Julien
- Degree
- Ph.D., Columbia University
- Published Here
- April 21, 2021