CMSA Interdisciplinary Science Seminar: Mathematical resolution of the Liouville conformal field theory

The Liouville conformal field theory is a well-known beautiful quantum field theory in physics describing random surfaces. Only recently a mathematical approach based on a well-defined path integral to this theory has been proposed using probability by David, Kupiainen, Rhodes, Vargas.

Many works since the 80’s in theoretical physics (starting with Belavin-Polyakov-Zamolodchikov) tell us that conformal field theories in dimension 2 are in general « Integrable », the correlations functions are solutions of PDEs and can in principle be computed explicitely by using algebraic tools (vertex operator algebras, representations of Virasoro algebras, the theory of conformal blocks). However, for Liouville Theory this was not done at the mathematical level by algebraic methods.

I’ll explain how to combine probabilistic, analytic and geometric tools to give explicit (although complicated) expressions for all the correlation functions on all Riemann surfaces in terms of certain holomorphic functions of the moduli parameters called conformal blocks, and of the structure constant (3-point function on the sphere). This gives a concrete mathematical proof of the so-called conformal bootstrap and of Segal’s gluing axioms for this CFT. The idea is to break the path integral on a closed surface into path integrals on pairs of pants and reduce all correlation functions to the 3-point correlation function on the Riemann sphere $S^2$. This amounts in particular to prove a spectral resolution of a certain operator acting on $L^2(H^{-s}(S^1))$ where $H^{-s}(S^1)$ is the Sobolev space of order -s<0 equipped with a Gaussian measure, which is viewed as the space of fields, and to construct a certain representation of the Virasoro algebra into unbounded operators acting on this Hilbert space.

This is joint work with A. Kupiainen, R. Rhodes and V. Vargas.