The quantization of the symplectic groupoid of the standard Podles sphere

TL;DR

This paper explicitly constructs the symplectic groupoid integrating the semiclassical standard Podles sphere and explores its quantization, revealing the role of polarizations and the modular class in the process.

Contribution

It provides an explicit form of the symplectic groupoid for the Podles sphere and connects Sheu's groupoid with Bohr-Sommerfeld leaves, advancing understanding of its quantization.

Findings

Sheu's groupoid arises as Bohr-Sommerfeld leaves

Complex polarization recovers convolution algebra

Modular class is crucial for scalar product

Abstract

We give an explicit form of the symplectic groupoid that integrates the semiclassical standard Podles sphere. We show that Sheu's groupoid, whose convolution C*-algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of the symplectic groupoid. By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.