Fuzzy Complex Projective Spaces and their Star-products

Abstract

We derive an explicit expression for an associative *-product on fuzzy complex projective spaces. This generalises previous results for the fuzzy 2-sphere and gives a discrete non-commutative algebra of functions on fuzzy complex projective spaces, represented by matrix multiplication. The matrices are restricted to ones whose dimension is that of the totally symmetric representations of SU(N). In the limit of infinite dimensional matrices we recover the commutative algebra of functions on ordinary projective space. Derivatives on the fuzzy projective space are also expressed as matrix commutators.

20 pages, typeset in plain TeX

References in corpus (5)

• Scaling Limits of the Fuzzy Sphere at one Loop
• Poisson sigma models and deformation quantization
• Perturbative dynamics on fuzzy S^2 and RP^2
• Fuzzy Complex Grassmannian Spaces and their Star Products
• On the Polarization of Unstable D0-Branes into Non-Commutative Odd Spheres

Cited by in corpus (80)

• Quantum Hall Effect in Higher Dimensions, Matrix Models and Fuzzy Geometry
• Finite Gauge Theory on Fuzzy CP^2
• Dimensional Reduction over Fuzzy Coset Spaces
• A Gauge-Invariant UV-IR Mixing and The Corresponding Phase Transition For U(1) Fields on the Fuzzy Sphere
• Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere
• Scalar Field Theory on Fuzzy S^4
• Fuzzy Scalar Field Theory as a Multitrace Matrix Model
• Simulation of a scalar field on a fuzzy sphere
• Monopole Bundles over Fuzzy Complex Projective Spaces
• A matrix phase for the phi^4 scalar field on the fuzzy sphere
• Fuzzy Complex Grassmannian Spaces and their Star Products
• The Fuzzy Disc
• Duality symmetry for star products
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras
• Fiber Bundles and Matrix Models
• Matrix Geometry and Coherent States
• The Star Product on the Fuzzy Supersphere
• Noncommutative gravity: fuzzy sphere and others
• On the Origin of the UV-IR Mixing in Noncommutative Matrix Geometry
• On the quantum structure of space-time, gravity, and higher spin in matrix models
• Dynamical generation of gauge groups in the massive Yang-Mills-Chern-Simons matrix model
• Fuzzy Complex Quadrics and Spheres
• A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces
• Measuring finite Quantum Geometries via Quasi-Coherent States
• Split noncommutativity and compactified brane solutions in matrix models
• A Fuzzy Three Sphere and Fuzzy Tori
• Noncommutative vector bundles over fuzzy CP^N and their covariant derivatives
• Signature change in matrix model solutions
• Drinfeld Twist and General Relativity with Fuzzy Spaces
• Fuzzy Toric Geometries
• The Chern-Simons One-form and Gravity on a Fuzzy Space
• Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
• Noncommutative Chiral Anomaly and the Dirac-Ginsparg-Wilson Operator
• Construction of fuzzy S^4
• Bosonization of the lowest Landau level in arbitrary dimensions: edge and bulk dynamics
• Magnetic operations: a little fuzzy physics?
• Information metric, Berry connection and Berezin-Toeplitz quantization for matrix geometry
• The Standard Model Fermion Spectrum From Complex Projective spaces
• The beat of a fuzzy drum: fuzzy Bessel functions for the disc
• On Lie-algebraic solutions of the type IIB matrix model
• A Twistorial Description of the IKKT-Matrix Model
• Large N reduction on coset spaces
• Higher-Dimensional Unification with continuous and fuzzy coset spaces as extra dimensions
• Understanding truncated non-commutative geometries through computer simulations
• The Matrix Chern-Simons One-form as a Universal Chern-Simons Theory
• One-loop stabilization of the fuzzy four-sphere via softly broken SUSY
• Quantum states to brane geometries via fuzzy moduli spaces of giant gravitons
• Emergent geometry from random multitrace matrix models
• Scalar and Spinor Field Actions on Fuzzy $S^4$: fuzzy $C{P}^3$ as a $S_F^2$ bundle over $S_F^4$
• Supersymmetric Monopole Quantum Mechanics on Sphere
• Fuzzy $C{P}^2$ Space-Times
• Holographic Hierarchy in the Gaussian Matrix Model via the Fuzzy Sphere
• Explicit Formulae for Noncommutative Deformations of CP^N and CH^N
• A complete solution of a Constrained System: SUSY Monopole Quantum Mechanics
• Towards Noncommutative Fuzzy QED
• Gauge Theories in Noncommutative Homogeneous Kähler Manifolds
• Quantum Gravity as a Multitrace Matrix Model
• A Perturbative Approach to Fuzzifying Field Theories
• Noncommutative Spherically Symmetric Spaces
• The Fuzzy Sphere: From The Uncertainty Relation To The Stereographic Projection
• Asymptotic commutativity of quantized spaces: the case of ${\mathbb{CP}}^{p,q}$
• Monopoles On $S_F^2$ From The Fuzzy Conifold
• Quantized Noncommutative Geometry from Multitrace Matrix Models
• A projective Dirac operator on CP^2 within fuzzy geometry
• The fuzzy BTZ
• The Fuzzy Sphere: Star Product Induced From Generalized Squeezed States
• On plane wave and vortex-like solutions of noncommutative Maxwell-Chern-Simons theory
• Remark on a Group-Theoretical Formalism for Quantum Mechanics and the Quantum-to-Classical Transition
• Contravariant geometry and emergent gravity from noncommutative gauge theories
• Dequantization via quantum channels
• Non-commutative Complex Projective Spaces and the Standard Model
• Fuzzy Torus and q-Deformed Lie Algebra
• Wilson RG of Noncommutative ${\Phi }_4^4$
• Fuzzy Torus via q-Parafermion
• Fuzzy Conifold $Y_F^6$ and Monopoles on $S_F^2\times S_F^2$
• Emergence of longitudinal 7-branes and fuzzy S^4 in compactification scenarios of M(atrix) theory
• Remarks on the eigenvalues distributions of D\leq 4 Yang-Mills matrix models
• Explicit formula of deformation quantization with separation of variables for complex two-dimensional locally symmetric Kähler manifold
• Deformation of Matrix Geometry via Landau Level Evolution