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.
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/d8-f9v7-7j12 |
| Date | January 2021 |
| Creators | Falconet, Hugo Pierre |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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