  In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact
formula for the density of the total mass of (sub-critical) Gaussian
multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the
unit circle. In this paper we will give a proof of this formula. In the
mathematical literature this is the first occurrence of an explicit probability
density for the total mass of a GMC measure. The key observation of our proof
is that the negative moments of the total mass of GMC determine its law and are
equal to one-point correlation functions of Liouville conformal field theory in
the disk defined by Huang, Rhodes and Vargas. The rest of the proof then
consists in implementing rigorously the framework of conformal field theory
(BPZ equations for degenerate field insertions) in a probabilistic setting to
compute the negative moments. Finally we will discuss applications to random
matrix theory, asymptotics of the maximum of the GFF and tail expansions of
GMC.
