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I describe a systematic method for solving PDEs of conformal field theory, known as Belavin-Polyakov-Zamolodchikov (BPZ) equations. These PDEs also arise in connections with statistical physics, in the theory of Schramm-Loewner evolutions (SLEs). Our method is a correspondence associating vectors in a tensor product representation of a quantum group to Coulomb gas type integral functions, in which properties of the functions are encoded in natural, representation theoretical properties of the vectors. In particular, this hidden quantum group structure on the solution space of such PDEs enables explicit calculation of the asymptotics and monodromy properties of the solutions. This also leads us to a generalization of the Temperley-Lieb algebra, defined in terms of a diagrammatic representation, which is nothing but the commutant algebra of the quantum group in the setup of the quantum Schur-Weyl duality.

Joint work with Kalle KytölÃ_ (Aalto University) and Steven Flores (University of Helsinki).