Fuzzy Complex Quadrics and Spheres

TL;DR

This paper constructs matrix algebras approximating functions on complex quadrics and spheres, enabling finite truncations of harmonic expansions and a method for defining functional integrals over fuzzy spheres of any dimension.

Contribution

It introduces a new matrix algebra framework for finite approximations of function algebras on complex quadrics and spheres, including a procedure for functional integrals over fuzzy spheres.

Findings

Matrix algebras approximate functions on complex quadrics and spheres.

Finite truncations of harmonic expansions are possible within this framework.

A method for constructing functional integrals over fuzzy spheres of any dimension is demonstrated.

Abstract

A matrix algebra is constructed which consists of the necessary degrees of freedom for a finite approximation to the algebra of functions on the family of orthogonal Grassmannians of real dimension 2N, known as complex quadrics. These matrix algebras contain the relevant degrees of freedom for describing truncations of harmonic expansions of functions on N-spheres. An Inonu-Wigner contraction of the quadric gives the co-tangent bundle to the commutative sphere in the continuum limit. It is shown how the degrees of freedom for the sphere can be projected out of a finite dimensional functional integral, using second-order Casimirs, giving a well-defined procedure for construction functional integrals over fuzzy spheres of any dimension.