A formal model of Berezin-Toeplitz quantization

Abstract

We give a new construction of symbols of the differential operators on the sections of a quantum line bundle $L$ over a Kaehler manifold $M$ using the natural contravariant connection on $L$. These symbols are the functions on the tangent bundle $TM$ polynomial on fibres. For high tensor powers of $L$, the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on $TM$ corresponding to some pseudo-Kaehler structure on $TM$. Surprisingly, this star product is intimately related to the formal symplectic groupoid with separation of variables over $M$. We extend the star product on $TM$ to generalized functions supported on the zero section of $TM$. The resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the Bergman projection operator. Using this idempotent, we define an algebra of Toeplitz elements and show that it is naturally isomorphic to the algebra of Berezin-Toeplitz deformation quantization on $M$.