English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

In this paper, we study quantization on a compact integral symplecticmanifold $X$ with transversal real polarizations. In the case of complexpolarizations, namely $X$ is K\"ahler equipped with transversal complexpolarizations $T^{1, 0}X, T^{0, 1}X$, 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 adeformation quantization $(\mathcal{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$of $X$.\par We investigate the real analogue to these, comparing deformationquantization, geometric quantization and Berezin-Toeplitz quantization. Thetechniques used are different from the complex case as distributional sectionssupported on Bohr-Sommerfeld fibres are involved.\par By switching the roles ofthe two real polarizations, we obtain Fourier-type transforms for bothdeformation quantization and geometric quantization, and they are compatibleasymptotically as $\hbar \to 0^+$. We also show that the asymptotic expansionof traces of Toeplitz operators realizes a trace map on deformationquantization.

Deformation quantization via Toeplitz operators on geometric quantization in real polarizations