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Single-valuedness of the eigenfunctions of the quantised Hitchin Hamiltoniansis proposed as a natural quantisation condition. Separation of Variables can beused to relate the classification of eigenstates to the classification ofprojective structures with real holonomy. Using complex Fenchel-Nielsencoordinates one may reformulate the quantisation conditions in terms of thegenerating function for the variety of opers. These results are interpreted asa variant of the geometric Langlands correspondence.

Quantisation conditions of the quantum Hitchin system and the real geometric Langlands correspondence