Probabilistic Liouville field theory on the two-dimensional sphere

In this thesis we construct the probabilistic Liouville field theory on the two-dimensional sphere. We prove some of the symmetry properties of the theory and define the correlation functions of the vertex operators. Finally, we define the Liouville quantum gravity measure. The thesis also contains a discussion on how the theory is related to quantum field theory and scaling limits of random planar maps. Essential building block of the theory is the Gaussian free field, which can be thought of as a random Gaussian field with the covariance operator given by the inverse of the Laplacian. Another important aspect of the Liouville field theory is the exponential of the Gaussian free field. Defining this requires some work, since the Gaussian free field will turn out to be a random generalized function, and the exponential of such an object is not defined in general. We will define the exponential by using the theory of Gaussian multiplicative chaos. The thesis contains a self-contained exposition of the definitions and basic properties of the Gaussian free field and its exponential. Some basic background in analysis, probability and geometry is assumed