Deformation Quantization via Toeplitz Operators on Geometric Quantization in Real Polarizations

In this paper, we study quantization on a compact integral symplectic manifold X with transversal real polarizations. the case of complex polarizations, namely is Kähler equipped polarizations $$T^{1, 0}X, T^{0, 1}X$$ , geometric 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 deformation $$({\mathcal {C}})[[\hbar ]], \star )$$ X. We investigate analogue to these, comparing quantization, and Berezin-Toeplitz quantization. The techniques used different from distributional sections supported Bohr-Sommerfeld fibres involved. By switching roles two obtain Fourier-type transforms for both they compatible asymptotically . also show that asymptotic expansion traces realizes trace map