Hidden classical symmetry in quantum spaces at roots of unity : From q-sphere to fuzzy sphere

Abstract

We study relations between different kinds of non-commutative spheres which have appeared in the context of ADS/CFT correspondences recently, emphasizing the connections between spaces that have manifest quantum group symmetry and spaces that have manifest classical symmetry. In particular we consider the quotient $S{U}_q(2)/U(1)$ at roots of unity, and find its relations with the fuzzy sphere with manifest classical SU(2) symmetry. Deformation maps between classical and quantum symmetry, the $U_q(SU(2))$ module structure of quantum spheres and the structure of indecomposable representations of $U_q(SU(2))$ at roots of unity conspire in an interesting way to allow the relation between manifestly $U_q(SU(2)$ symmetric spheres and manifestly U(SU(2)) symmetric spheres. The relation suggests that a subset of field theory actions on the q-sphere are equivalent to actions on the fuzzy sphere. The results here are compatible with the proposal that quantum spheres at roots of unity appear as effective geometries which account for finite N effects in the ADS/CFT correspondence.

arXiv comment19 pages in harvmac big, 5 figures; v2: refs added ; v3: more refs added

References in corpus (16)

• Dielectric-Branes
• AdS3 Black Holes and a Stringy Exclusion Principle
• Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres
• Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
• String Theory and the Space-Time Uncertainty Principle
• Marginal and Relevant Deformations of N=4 Field Theories and Non-Commutative Moduli Spaces of Vacua
• Gravity from CFT on $S^N(X)$ CFT: Symmetries and Interactions
• AdS_3 x S^3 (Un)twisted and Squashed, and an O(2,2;Z) Multiplet of Dyonic Strings
• Giant Gravitons, BPS bounds and Noncommutativity
• Harmonic Analysis on the quantum Lorentz group
• Matrix Theory, AdS/CFT and Higgs-Coulomb Equivalence
• Large N Expansion From Fuzzy AdS_2
• Quantum Spacetimes and Finite N Effects in 4D Super Yang-Mills Theories
• Noncommutative spheres and the AdS/CFT correspondence
• Quantum Anti-de Sitter space and sphere at roots of unity
• Drinfeld twist for quantum $su(2)$ in the adjoint representation

Cited by in corpus (6)

• Nonassociative Star Product Deformations for D-brane Worldvolumes in Curved Backgrounds
• Supersymmetric states of N=4 Yang-Mills from giant gravitons
• Fuzzy Sphere from Matrix Model
• Fluctuating Fuzzy Funnels
• Statistical entropy of two-dimensional dilaton de Sitter space
• Multiparametric Quantum Algebras and the Cosmological Constant