Asymptotic commutativity of quantized spaces: the case of ${\mathbb{CP}}^{p,q}$

Abstract

We present a procedure for quantizing complex projective spaces ${\mathbb{CP}}^{p,q}$, $q\ge 1$, as well as construct relevant star products on these spaces. The quantization is made unique with the demand that it preserves the full isometry algebra of the metric. Although the isometry algebra, namely $su(p+1,q)$, is preserved by the quantization, the Killing vectors generating these isometries pick up quantum corrections. The quantization procedure is an extension of one applied recently to Euclidean $Ad{S}_2$, where it was found that all quantum corrections to the Killing vectors vanish in the asymptotic limit, in addition to the result that the star product trivializes to pointwise product in the limit. In other words, the space is asymptotically anti-de Sitter making it a possible candidate for the $AdS/CFT$ correspondence principle. In this article, we find indications that the results for quantized Euclidean $Ad{S}_2$ can be extended to quantized ${\mathbb{CP}}^{p,q}$, i.e., noncommutativity is restricted to a limited neighborhood of some origin, and these quantum spaces approach ${\mathbb{CP}}^{p,q}$ in the asymptotic limit.

References in corpus (33)

• The Large N Limit of Superconformal Field Theories and Supergravity
• Large N Field Theories, String Theory and Gravity
• Comments on the Sachdev-Ye-Kitaev model
• The quantum structure of spacetime at the Planck scale and quantum fields
• AdS2 Quantum Gravity and String Theory
• Fuzzy Complex Projective Spaces and their Star-products
• Generalized Coherent State Approach to Star Products and Applications to the Fuzzy Sphere
• Fuzzy Spheres in AdS/CFT Correspondence and Holography from Noncommutativity
• Twisted Noncommutative Field Theory with the Wick-Voros and Moyal Products
• Noncommutative field theory with the Wick-Voros product
• Spectral truncations in noncommutative geometry and operator systems
• Covariant cosmological quantum space-time, higher-spin and gravity in the IKKT matrix model
• Quantized open FRW cosmology from Yang-Mills matrix models
• 2D fuzzy Anti-de Sitter space from matrix models
• The fuzzy 4-hyperboloid $H_n^4$ and higher-spin in Yang-Mills matrix models
• The Fuzzy Disc
• Matrix Model Approach to Cosmology
• On the quantum structure of space-time, gravity, and higher spin in matrix models
• Large N Expansion From Fuzzy AdS_2
• Spectral geometry with a cut-off: topological and metric aspects
• Absence of the Holographic Principle in Noncommutative Chern-Simons Theory
• Cosmological space-times with resolved Big Bang in Yang-Mills matrix models
• Non-Compact Hopf Maps and Fuzzy Ultra-Hyperboloids
• Signature change in matrix model solutions
• Matrix Model Cosmology in Two Space-time Dimensions
• Higher-spin gravity and torsion on quantized space-time in matrix models
• The beat of a fuzzy drum: fuzzy Bessel functions for the disc
• Non-commutative $Ad{S}_2/CF{T}_1$ duality: the case of massless scalar fields
• From the fuzzy disc to edge currents in Chern-Simons Theory
• A New Class of Two-Dimensional Noncommutative Spaces
• Edge States from Defects on the Noncommutative Plane
• W-Infinity Algebras from Noncommutative Chern-Simons Theory
• Non-commutative $Ad{S}_2/CF{T}_1$ duality: the case of massive and interacting scalar fields

Cited by in corpus (3)

• Exact solutions for scalars and spinors on quantized (Euclidean) $Ad{S}_2$ and the correspondence principle
• Noncommutative $Ad{S}_2$ I: Exact solutions
• Noncommutative $Ad{S}_2$ II: The Correspondence Principle