Diffeomorphisms on Fuzzy Sphere

Abstract

Diffeomorphisms can be seen as automorphisms of the algebra of functions. In the matrix regularization, functions on a smooth compact manifold are mapped to finite size matrices. We consider how diffeomorphisms act on the configuration space of the matrices through the matrix regularization. For the case of the fuzzy $S^2$, we construct the matrix regularization in terms of the Berezin-Toeplitz quantization. By using this quantization map, we define diffeomorphisms on the space of matrices. We explicitly construct the matrix version of holomorphic diffeomorphisms on $S^2$. We also propose three methods of constructing approximate invariants on the fuzzy $S^2$. These invariants are exactly invariant under area-preserving diffeomorphisms and only approximately invariant (i.e. invariant in the large-$N$ limit) under the general diffeomorphisms.

arXiv comment34 pages, 3 figures; v4: appendix F added, section 6 improved, minor modifications

References in corpus (32)

• String Theory and Noncommutative Geometry
• M Theory As A Matrix Model: A Conjecture
• A Large-N Reduced Model as Superstring
• The Spectral Action Principle
• A Four Dimensional Generalization of the Quantum Hall Effect
• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Noncommutative Geometry and Gravity
• Matrix Perturbation Theory For M-theory On a PP-Wave
• Emergent Geometry and Gravity from Matrix Models: an Introduction
• Monopoles and Solitons in Fuzzy Physics
• Topologically nontrivial field configurations in noncommutative geometry
• Longitudinal 5-branes as 4-spheres in Matrix theory
• Describing Curved Spaces by Matrices
• N=4 Super Yang-Mills from the Plane Wave Matrix Model
• D-branes, Matrix Theory and K-homology
• Toeplitz operators on symplectic manifolds
• Embedding of theories with SU(2|4) symmetry into the plane wave matrix model
• The SPIN${}^c$ Dirac operator on high tensor powers of a line bundle
• Matrix embeddings on flat $R^3$ and the geometry of membranes
• Matrix Geometry and Coherent States
• Measuring finite Quantum Geometries via Quasi-Coherent States
• Relativistic Landau Models and Generation of Fuzzy Spheres
• Relating branes and matrices
• Information metric, Berry connection and Berezin-Toeplitz quantization for matrix geometry
• Noncommutative spaces and matrix embeddings on flat R^{2n+1}
• $SO(4)$ Landau Models and Matrix Geometry
• Noncommutativity and Tachyon Condensation
• Kähler structure in the commutative limit of matrix geometry
• Heat kernel coefficients for compact fuzzy spaces
• Geometry from Matrices via D-branes
• Effective local geometric quantities in fuzzy spaces from heat kernel expansions
• Thermofield dynamics and Gravity

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