Deformation Quantization via Toeplitz Operators on Geometric Quantization in Real Polarizations

Abstract

In this paper, we study quantization on a compact integral symplectic manifold X with transversal real polarizations. In the case of complex polarizations, namely X is Kähler equipped with transversal complex polarizations $$T^{1, 0}X, T^{0, 1}X$$ , geometric quantization gives $$H^0(X, L^{\otimes k})$$ ’s. They are acted upon by $${\mathcal {C}}^\infty (X, {\mathbb {C}})$$ via Toeplitz operators as $$\hbar = \tfrac{1}{k} \rightarrow 0^+$$ , determining a deformation quantization $$({\mathcal {C}}^\infty (X, {\mathbb {C}})[[\hbar ]], \star )$$ of X. We investigate the real analogue to these, comparing deformation quantization, geometric quantization and Berezin-Toeplitz quantization. The techniques used are different from the complex case as distributional sections supported on Bohr-Sommerfeld fibres are involved. By switching the roles of the two real polarizations, we obtain Fourier-type transforms for both deformation quantization and geometric quantization, and they are compatible asymptotically as $$\hbar \rightarrow 0^+$$ . We also show that the asymptotic expansion of traces of Toeplitz operators realizes a trace map on deformation quantization.