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Segal Axioms and modular bootstrap for Liouville CFT

Banff International Research Station for Mathematical Innovation and Discovery

Liouville conformal field theory is a conformal field theory quantizing the uniformization of Riemann surfaces. In joint work with Kupiainen, Rhodes, Vargas, we show that Segal axioms are satisfied for Liouville Conformal Field theory on Riemann surfaces, i.e. that the correlation/partition functions can be expressed by cutting the surfaces into surfaces with boundary. This is reminiscent to topological quantum field theory approaches where one associates Hilbert spaces H to boundaries and trace class operators on H to manifolds with boundary, with the property that operators compose when we glue two manifold along one common boundary. Using our previous work on the conformal bootstrap for the 4-point function on the sphere, this allows to express the partition and correlation functions as explicit functions on the moduli space of Riemann surface with marked points in terms of the conformal blocks associated to the Virasoro algebra and the structure constant (called DOZZ). The proof is a combination of probability methods, scattering theory and the representation theory of Virasoro algebra.

Mathematics; Global Analysis; Analysis On Manifolds; Differential Geometry; Geometric Analysis

video/mp4

Author affiliation: Université Paris Saclay and CNRS

2023-10-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86338

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/