noindent (In collaboration with R. Belliard, T. Kimura, S. Ribault.) par It has been realized recently that the conformal block of 4 point function in Liouville CFT is related to the Tau function of the Painlevé 6 integrable system. par Here we propose a general construction: starting from a very general integrable system (a Hitchin system: the moduli space of flat G-connections over a Riemann surface, with G an arbitrary semi-simple Lie group), we construct amplitudes, and we show that these amplitudes satisfy all the axioms of a CFT: they satisfy OPEs, Ward identities and crossing symmetry. Ward identities come from the flatness of the connection. par The construction is very geometrical, by defining a notion of “quantum spectral curve” attached to a flat connection, defining cycles and forms on it and showing that they satisfy Seiberg-Witten relations, related to crossing symmetry. par So this link between CFTs and integrable systems unearths a new and beautiful quantum geometry.

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