Moduli space and structure of noncommutative 3-spheres

Abstract

We analyse the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the $C^{\ast }$-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric spaceof unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with 8 points in common. We show that generically these curves are the same as the characteristic variety of the associated quadratic algebra. We then apply the general theory of central quadratic forms to show that the noncommutative 3-spheres admit a natural ramified covering $\pi$ by a noncommutative 3-dimensional nilmanifold. This yields the differential calculus. We then compute the Jacobian of the ramified covering $\pi$ by pairing the direct image of the fundamental class of the noncommutative 3--dimensional nilmanifold with the Chern character of the defining unitary and obtain the answer as the product of a period (of an elliptic integral) by a rational function...

50 pages. References added

References in corpus (2)

• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples

Cited by in corpus (24)

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• From Physics to Number Theory via Noncommutative Geometry. Part I: Quantum Statistical Mechanics of Q-lattices
• The Ricci flow on noncommutative two-tori
• Generalized Kahler and hyper-Kahler quotients
• Free actions of compact quantum group on unital C*-algebras
• Liberations and twists of real and complex spheres
• Yang-Mills and some related algebras
• A walk in the noncommutative garden
• Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces
• Unitary easy quantum groups: geometric aspects
• Non commutative finite dimensional manifolds II. Moduli space and structure of non commutative 3-spheres
• Noncommutative Euclidean spaces
• Metric aspects of noncommutative homogeneous spaces
• Hypergeometric function and modular curvature
• Multilinear forms and graded algebras
• Lectures on Noncommutative Geometry
• On the Absence of Continuous Symmetries for Noncommutative 3-Spheres
• The Noncommutative Geometry Generalization of Fundamental Group
• Examples of noncommutative manifolds: complex tori and spherical manifolds
• Lectures on the three--dimensional non--commutative spheres
• Exotic Elliptic Algebras of dimension 4 (with an Appendix by Derek Tomlin)
• Graded algebras and multilinear forms
• Arithmetic structures on noncommutative tori with real multiplication
• BV Formalism and Partition Functions