Probabilistic construction and conformal bootstrap of Imaginary Liouville theory
Romain Usciati
Conformal field theories describe systems that remain invariant under changes of scale. In two dimensions, they provide a common language for studying critical phenomena in statistical mechanics and quantum gravity. Among them, Liouville theory occupies a distinguished place: it is the best-understood non-rational conformal field theory and has become an important meeting point between conformal bootstrap, probability and random geometry. Its conventional formulation, however, applies only to central charges greater than or equal to 25. Extending Liouville theory to central charges below 1 is a long-standing problem, motivated in particular by its expected connection with non-local observables in critical statistical models and applications to string theory.
In this thesis, we propose a constructive path-integral definition of Liouville theory at central charge below 1. Our approach combines probabilistic tools with an original prescription to deal with the infrared divergence of the theory, inspired from methods for the analytic continuation of functions of the complex variable. We provide both analytic and numerical evidence that our construction reproduces the correlation functions expected in Imaginary Liouville theory.
A second contribution is the rigorous study of the theory’s local conformal structure. Our results demonstrate that the proposed path integral possesses essential structural properties expected of a conformal field theory.
Finally, we take initial steps toward the quantization and conformal bootstrap description of the theory. We present numerical investigations of four-point functions, together with a spectral analysis of a simplified Hamiltonian that reveals both discrete and continuous spectral components of the spectrum.
Jury : Denis Bernard, Benoît Estienne (invité), Colin Guillarmou (co-directeur de thèse), Alexey Litvinov (rapporteur), Titus Lupu (invité), Eveliina Peltola, Raoul Santachiara (directeur de thèse), Christian Webb (rapporteur).