Higher (Odd) Dimensional Quantum Hall Effect and Extended Dimensional Hierarchy

Abstract

We demonstrate dimensional ladder of higher dimensional quantum Hall effects by exploiting quantum Hall effects on arbitrary odd dimensional spheres. Non-relativistic and relativistic Landau models are analyzed on $S^{2k-1}$ in the $SO(2k-1)$ monopole background. The total sub-band degeneracy of the odd dimensional lowest Landau level is shown to be equal to the winding number from the base-manifold $S^{2k-1}$ to the one-dimension higher $SO(2k)$ gauge group. Based on the chiral Hopf maps, we clarify the underlying quantum Nambu geometry for odd dimensional quantum Hall effect and the resulting quantum geometry is naturally embedded also in one-dimension higher quantum geometry. An origin of such dimensional ladder connecting even and odd dimensional quantum Hall effects is illuminated from a viewpoint of the spectral flow of Atiyah-Patodi-Singer index theorem in differential topology. We also present a BF topological field theory as an effective field theory in which membranes with different dimensions undergo non-trivial linking in odd dimensional space. Finally, an extended version of the dimensional hierarchy for higher dimensional quantum Hall liquids is proposed, and its relationship to quantum anomaly and D-brane physics is discussed.

1+49 pages, 8 figures, 4 tables, typos corrected

References in corpus (12)

• Classification of topological insulators and superconductors in three spatial dimensions
• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Modeling Multiple M2's
• Quantum Hall Effect in Higher Dimensions, Matrix Models and Fuzzy Geometry
• Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras
• D-Algebra Structure of Topological Insulators
• Electromagnetic and thermal responses of Z topological insulators and superconductors in odd spatial dimensions
• On Time-dependent Collapsing Branes and Fuzzy Odd-dimensional Spheres
• Quantum Hall Effect on Odd Spheres

Cited by in corpus (12)

• 6D Fractional Quantum Hall Effect
• Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator
• Gauge Field Induced Chiral Zero Mode in Five-dimensional Yang Monopole Metamaterials
• $SO(4)$ Landau Models and Matrix Geometry
• $SO(5)$ Landau Models and Nested Nambu Matrix Geometry
• $SO(5)$ Landau Model and 4D Quantum Hall Effect in The $SO(4)$ Monopole Background
• Mod-two APS index and domain-wall fermion
• A Unified Construction of Skyrme-type Non-linear sigma Models via The Higher Dimensional Landau Models
• Higher bracket structure of density operators in Weyl fermion systems and topological insulators
• Understanding the index theorems with massive fermions
• Quantum matrix geometry in the lowest Landau level and higher Landau levels
• Deformation of Matrix Geometry via Landau Level Evolution