Surveys in Differential Geometry

Volume 24 (2019)

Probing quantization via branes

Pages: 293 – 402

DOI:  https://dx.doi.org/10.4310/SDG.2019.v24.n1.a8

Authors

Davide Gaiotto (Perimeter Institute, Waterloo, Ontario, Canada)

Edward Witten (Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

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.

Published 29 December 2021