Abstract
Borrowing ideas from the relation between classical and quantum mechanics, we study a non-commutative elevation of the A D E geometries involved in building Calabi–Yau manifolds. We derive the corresponding geometric Hamiltonians and the holomorphic wave equations representing these non-commutative geometries. The spectrum of the holomorphic waves is interpreted as the quantum moduli space. Quantum A 1 geometry is analyzed in some details and is found to be linked to the Whittaker differential equation.
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