CFTs, and the (quantum) geometry of integrable systems
(IPhT CEA/Saclay & CRM Montréal)
Amhithéâtre Léon Motchane (IHES)
Amhithéâtre Léon Motchane
Le Bois Marie
35, route de Chartres
It has been realized recently that the c=1 conformal block of 4 point function in Liouville CFT is related to the Tau function of the Painlevé 6 integrable system. 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 define some "amplitudes", and we show that these amplitudes satisfy all the axioms of a CFT: they satisfy OPEs, Ward identities and crossing symmetry. The construction is very geometrical, by defining a notion of "quantum spectral curve" attached to a flat connection, defining homology and cohomology on it, and showing that amplitudes satisfy Seiberg-Witten like relations, and behave well under modular transformations.
So this link between CFTs and integrable systems unearths a new and beautiful quantum geometry.