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Description

This is joint work with B. Feigin and E. Frenkel. I will explain how to diagonalise the quantum Hamiltonians arising from the Casimir connection by using affine Kac-Moody algebras at critical level. This mirrors arising from the Casimir connection by using affine Kac-Moody algebras at critical level. This mirrors the construction of Feigin, Frenkel and Reshetikhin who diagonalised the Gaudin Hamiltonians arising from the Knizhnik-Zamolodchikov connection, and leads to a new class of quantum integrable systems generalizing the Gaudin model. Two interesting new features appear in the construction: the use of non-highest weight representations of affine Lie algebras and connections (more precisely opers) with irregular (as opposed to regular) singularities on the Riemann sphere which describe the spectrum of the algebras of quantum Hamiltonians.