Matrix algebras converge to the sphere for quantum Gromov--Hausdorff distance

Abstract

On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2-sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with ``quantum metric spaces''. We show how to make these ideas precise by means of Berezin quantization using coherent states. We work in the general setting of integral coadjoint orbits for compact Lie groups.

arXiv comment28 pages. various minor improvements.To appear Memoirs Amer. Math. Soc

References in corpus (20)

• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory
• Lattice Gauge Fields and Discrete Noncommutative Yang-Mills Theory
• Metrics on State Spaces
• Noncommutative Gauge Theory on Fuzzy Sphere from Matrix Model
• Metrics on states from actions of compact groups
• Scaling Limits of the Fuzzy Sphere at one Loop
• Fuzzy CP2
• Monge Metric on the Sphere and Geometry of Quantum States
• Fuzzy Spheres in AdS/CFT Correspondence and Holography from Noncommutativity
• Fuzzy Cosets and their Gravity Duals
• Non-Commutative Geometry on a Discrete Periodic Lattice and Gauge Theory
• Fuzzy Complex Grassmannian Spaces and their Star Products
• Fuzzy Actions and their Continuum Limits
• Fuzzy Line Bundles, the Chern Character and Topological Charges over the Fuzzy Sphere
• Strict quantization of coadjoint orbits
• Topology at the Planck Length
• Towards a Nonperturbative Covariant Regularization in 4D Quantum Field Theory
• Generalised coherent states and the diagonal representation for operators
• Almost Kaehler deformation quantization

Cited by in corpus (39)

• Higher Dimensional Geometries from Matrix Brane constructions
• The spectral distance on the Moyal plane
• Spectral geometry with a cut-off: topological and metric aspects
• On Gromov-Hausdorff convergence for operator metric spaces
• Poisson boundary of the dual of SUq(n)
• Nilpotent group C*-algebras as compact quantum metric spaces
• Matricial bridges for "Matrix algebras converge to the sphere"
• theta-deformations as compact quantum metric spaces
• Matrix Bases for Star Products: a Review
• The Podles sphere as a spectral metric space
• Symbol correspondences for spin systems
• C*-algebraic quantum Gromov-Hausdorff distance
• Gromov-Hausdorff convergence of state spaces for spectral truncations
• Group C*-algebras as compact quantum metric spaces
• Dynamics of compact quantum metric spaces
• Gromov-Hausdorff Distance for Quantum Metric Spaces
• Non-Commutative Resistance Networks
• The Podles spheres converge to the sphere
• Vector bundles for "Matrix algebras converge to the sphere"
• Harmonic analysis approach to Gromov--Hausdorff convergence for noncommutative tori
• On multimatrix models motivated by random noncommutative geometry II: A Yang-Mills-Higgs matrix model
• Dirac operators for coadjoint orbits of compact Lie groups
• Noncommutative Ricci flow in a matrix geometry
• Convergence of Fourier truncations for compact quantum groups and finitely generated groups
• Berezin quantization of noncommutative projective varieties
• The problem of completeness for Gromov-Hausdorff metrics on C*-algebras
• Twist star products and Morita equivalence
• Pythagoras Theorem in Noncommutative Geometry
• Quantum flag manifolds as quotients of degenerate quantized universal enveloping algebras
• Berezin Quantization From Ergodic Actions of Compact Quantum Groups, and Quantum Gromov-Hausdorff Distance
• Quantum metrics on noncommutative spaces
• Quantized Gromov-Hausdorff distance
• Vietoris topology on hyperspaces associated to a noncommutative compact space
• Matricial quantum Gromov-Hausdorff distance
• MSc Thesis: Presentation of Certain New Trends in Noncommutative Geometry
• Lipschitzness of *-homomorphisms between C*-metric algebras
• Convergence of Spectral Triples on Fuzzy Tori to Spectral Triples on Quantum Tori
• Order-unit quantum Gromov-Hausdorff distance
• An operator Arzelà-Ascoli theorem