Hitchin’s connection in metaplectic quantization

Abstract

We give a differential geometric construction of a connection, which we call the Hitchin connection, in the bundle of quantum Hilbert spaces arising from metaplectically corrected geometric quantization of a prequantizable, symplectic manifold, endowed with a rigid family of Kähler structures, all of which give vanishing first Dolbeault cohomology groups. This generalizes work of both Hitchin, Scheinost and Schottenloher, and Andersen, since our construction does not need that the first Chern class is proportional to the class of the symplectic form, nor do we need compactness of the symplectic manifold in question. Furthermore, when we are in a setting similar to the moduli space, we give an explicit formula and show that this connection agrees with previous constructions. Mathematics Subject Classification (2010). 53D50, 32Q55.