Appears in collection : Jean Morlet Chair : Geometric and Probabilistic Aspects in Quantum Field Theory / Chaire Jean Morlet Conférence - Aspects géométriques et probabilistes de la théorie quantique des champs

We show that there exists a projective unitary representation of the Moore-Seiberg groupoid, and of the mapping class group as a consequence, into the (infinite dimensional) Hilbert bundle, over Teichm¨uller space, of Virasoro conformal blocks for central charge c > 25. The conformal blocks, in a fixed marked pair of pant decomposition, are sections of a holomorphic line bundle L over Teichm¨uller space, a choice of Riemannian metrics induces a Hermitian metric on the line bunble, and on the Hilbert bundle of blocks using in addition the DOZZ-constants. The Liouville correlation functions are square norms of conformal blocks. Contrary to usual methods, the relations between fusion kernels of elementary moves in the Moore-Seiberg groupoid need not be checked, but follow from the construction of conformal blocks as global holomorphic sections of L and the proof that conformal blocks are dense in the Hilbert bundle. This involves a combination of probabilistic (GMC theory), analytic methods (scattering theory) and the complex geometry of complex surfaces with parametrised boundary. This is joint work with Baverez, Kupiainen, Rhodes and Xie.