Quantum (Matrix) Geometry and Quasi-Coherent States

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 Kähler geometry arises naturally, which includes the well-known quantized coadjoint orbits such as the fuzzy sphere $S_N^2$ and fuzzy $\mathbb{C}{P}_N^n$. A quantization map for quantum Kähler geometries is established. Some examples of quantum geometries which are not Kähler are identified, including the minimal fuzzy torus.

35 pages. V2: minor corrections. V3: minor improvements

References in corpus (5)

• Emergent Gravity from Noncommutative Spacetime
• Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model
• Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking
• Self-intersecting fuzzy extra dimensions from squashed coadjoint orbits in $\mathcal{N}=4$ SYM and matrix models
• Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model

Cited by in corpus (9)

• A Twistorial Description of the IKKT-Matrix Model
• One-loop effective action and emergent gravity on quantum spaces in the IKKT matrix model
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• String modes, propagators and loops on fuzzy spaces
• Dirac operators for matrix algebras converging to coadjoint orbits
• Spatially flat FLRW spacetimes with a Big Bang from matrix geometry
• Spatially flat cosmological quantum spacetimes
• Oxidation, Reduction and Semi-Classical Limit for Quantum Matrix Geometries
• Exploring the gravity sector of emergent higher-spin gravity: effective action and a solution