Hitchin's Connection in Half-Form Quantization

Abstract

We give a differential geometric construction of a connection in the bundle of quantum Hilbert spaces arising from half-form corrected geometric quantization of a prequantizable, symplectic manifold, endowed with a rigid, family of K\"ahler structures, all of which give vanishing first Dolbeault cohomology groups. In [And1] Andersen gave an explicit construction of Hitchin's connection in the non-corrected case using additional assumptions. Under the same assumptions we also give an explicit solution in terms of Ricci potentials. Morover we show that if these are carefully chosen the construction coincides with the construction of Andersen in the non-corrected case.