Quantizable functions on Kähler manifolds and non-formal quantization (Q6058877)

The paper under review deals with the geometric/deformation quantisation of Kähler manifolds, building on previous work of the authors. In Section 2, a first construction of a deformation quantisation involves (flat) Fedosov connections on (formal) Weyl bundles over a Kähler manifold \(X\): a source of such connections arises from the quantisation of Kapranov's \(L_{\infty}\)-structure on \(X\). This leads to a natural notion of level-\(k\) quantisable observables, namely horizontal sections for the Fedosov connection. In turn, in Section 3 these are shown to define a sheaf \(C^{\infty}_{\alpha,k}\) of twisted differential operators (TDOs) on \(X\), whose characteristic class is also explicitly computed. (It depends upon the Kähler form \(\omega\), and the formal differential form \(\alpha\) defining the Fedosov connection.) Section 4 instead connects to geometric quantisation, in the usual way. Roughly, spaces of holomorphic sections of (prequantum) line bundles \(\mathcal L \to X\) are supposed to yield modules for the deformation quantisation ring, so in this example one expects the above TDOs to act. And indeed, in the case where the \(\ast\)-product above coincides with the Berezin-Toeplitz one (i.e., by choosing a suitable formal (1,1)-form \(\alpha\), involving the Ricci form of the Kähler metric of \(X\)), there is an isomorphism of \(C^{\infty}_{\alpha,k}\) with the sheaf of holomorphic differential operators acting on sections of \(\mathcal L^{\otimes k}\). \newline In particular, the corresponding differential operators are not necessarily first-order, so go beyond the standard (pre)quantum differential operators in geometric quantisation. Finally, Section 5 produces examples of (first-order) quantisable observables à la Fedosov via quantum (co)moment maps for Hamiltonian actions of Lie groups -- at any level. A solitary appendix contains the technical proof of a proposition in Section 4, required in the proof of the above isomorphism of sheaves.