Fuzzy complex quadrics and spheres

A matrix algebra is constructed which consists of the necessary degrees of freedom for a finite approximation to the algebra of func-tions 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 ex-pansions of functions on N-spheres. An Inönü-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 inte-gral, using second-order Casimirs, giving a well-defined procedure for construction functional integrals over fuzzy spheres of any dimension