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.

arXiv comment35 pages. V2: minor corrections. V3: minor improvements

References in corpus (32)

• String Theory and Noncommutative Geometry
• Noncommutative Perturbative Dynamics
• A Large-N Reduced Model as Superstring
• Classical and Quantum Mechanics of Free \k Relativistic Systems
• Noncommutative Yang-Mills in IIB Matrix Model
• Emergent Geometry and Gravity from Matrix Models: an Introduction
• Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions
• On Finite 4D Quantum Field Theory in Non-Commutative Geometry
• IIB Matrix Model
• Emergent Gravity from Noncommutative Spacetime
• Emergent Gravity and Noncommutative Branes from Yang-Mills Matrix Models
• 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
• Bi-local Fields in Noncommutative Field Theory
• Intersecting branes and a standard model realization in matrix models
• Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model
• Matrix embeddings on flat $R^3$ and the geometry of membranes
• The fuzzy 4-hyperboloid $H_n^4$ and higher-spin in Yang-Mills matrix models
• The Fuzzy Disc
• Matrix Geometry and Coherent States
• Measuring finite Quantum Geometries via Quasi-Coherent States
• Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids
• String states, loops and effective actions in noncommutative field theory and matrix models
• 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
• The world as quantized minimal surfaces
• Noncommutative spaces and matrix embeddings on flat R^{2n+1}
• Realizing three generations of the Standard Model fermions in the type IIB matrix model
• Kähler structure in the commutative limit of matrix geometry
• On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model
• Scalar and Spinor Field Actions on Fuzzy $S^4$: fuzzy $C{P}^3$ as a $S_F^2$ bundle over $S_F^4$
• Intersecting branes, Higgs sector, and chirality from $\mathcal{N}=4$ SYM with soft SUSY breaking

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• 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