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 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

Matrix ϕ4\phi^4 Models on the Fuzzy Sphere and their Continuum Limits

We demonstrate that the UV/IR mixing problems found recently for a scalar $\phi^4$ theory on the fuzzy sphere are localized to tadpole diagrams and can be overcome by a suitable modification of the action. This modification is equivalent to normal ordering the $\phi^4$ vertex. In the limit of the commutative sphere, the perturbation theory of this modified action matches that of the commutative theory.Comment: 19 pages of LaTeX, with 3 figure

Matrix Models on the Fuzzy Sphere

Field theory on a fuzzy noncommutative sphere can be considered as a particular matrix approximation of field theory on the standard commutative sphere.We investigate from this point of view the scalar Ø4 theory. We demonstrate that the UV/IR mixing problems of this theory are localized to the tadpole diagrams and can be removed by an appropiate (fuzzy) normal ordering of the Ø4 vertex. The perturbative expansion of this theory reduces in the commutative limit to that on the commutative sphere