Quantum (Matrix) Geometry and Quasi-Coherent States

TL;DR

This paper introduces a general framework linking finite-dimensional hermitian matrices to geometric structures, enabling classical space extraction without large matrix limits, and develops quantum Kähler geometry concepts.

Contribution

It generalizes fuzzy space examples, systematizes their geometric interpretation, and introduces a quantization map for quantum Kähler geometries, including non-Kähler examples.

Findings

Framework associates geometrical structures to matrices

Quantum Kähler geometry naturally arises from the approach

Includes examples like fuzzy sphere and minimal fuzzy torus

Abstract

A general framework is described which associates geometrical structures to any set of $D$ finite-dimensional hermitian matrices $X^a, \ a=1,...,D$. This framework generalizes and systematizes the well-known examples of fuzzy spaces, and allows to extract the underlying classical space without requiring the limit of large matrices or representation theory. The approach is based on the previously introduced concept of quasi-coherent states. In particular, a concept of quantum K\"ahler geometry arises naturally, which includes the well-known quantized coadjoint orbits such as the fuzzy sphere $S^2_N$ and fuzzy $\mathbb{C} P^n_N$. A quantization map for quantum K\"ahler geometries is established. Some examples of quantum geometries which are not K\"ahler are identified, including the minimal fuzzy torus.