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 K\"ahler 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 $\hbar = \tfrac{ 1}{ k} \to 0^+$, determining a deformation quantization $(\mathcal{ C}^\infty(X, \mathbb{ C})[[\hbar]], \star)$ of $X$.\par 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.\par 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 $\hbar \to 0^+$. We also show that the asymptotic expansion of traces of Toeplitz operators realizes a trace map on deformation quantization.
Categories: math.SG math-ph math.DG math.MP