Geometric objects in an approach to quantum geometry

TL;DR

This paper explores quantum geometry through deformation quantization, focusing on nonlinear flat connections and the moduli space of parallel sections, highlighting the treatment of singularities in algebraic structures.

Contribution

It introduces a novel approach to quantum geometry by applying deformation quantization to algebras with one generator and analyzing their moduli spaces.

Findings

Methods to handle nonlinear flat connections

Identification of bundle-like structures with complex transition functions

Insights into movable branching singularities

Abstract

Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is studied as an example of bundle-like objects with discordant (sogo) transition functions, which suggests us to treat movable branching singularities.