Probing Quantization Via Branes

TL;DR

This paper explores the relationship between geometric and brane quantization of symplectic manifolds, showing their equivalence under certain analytic continuation conditions and examining symmetry groups and examples.

Contribution

It establishes conditions under which geometric and brane quantizations agree, linking polarizations to complexifications and analyzing symmetry groups in this context.

Findings

Geometric and brane quantizations coincide when polarizations extend holomorphically.

Symmetry groups in brane quantization correspond to analytically continued symplectomorphisms.

Examples demonstrate equivalence of different polarizations in geometric quantization.

Abstract

We re-examine quantization via branes with the goal of understanding its relation to geometric quantization. If a symplectic manifold $M$ can be quantized in geometric quantization using a polarization ${\mathcal P}$, and in brane quantization using a complexification $Y$, then the two quantizations agree if ${\mathcal P}$ can be analytically continued to a holomorphic polarization of $Y$. We also show, roughly, that the automorphism group of $M$ that is realized as a group of symmetries in brane quantization of $M$ is the group of symplectomorphisms of $M$ that can be analytically continued to holomorphic symplectomorphisms of $Y$. We describe from the point of view of brane quantization several examples in which geometric quantization with different polarizations gives equivalent results.