TOPOLOGY CHANGE FOR FUZZY PHYSICS: FUZZY SPACES AS HOPF ALGEBRAS

Fuzzy spaces are obtained by quantizing adjoint orbits of compact semi-simple Lie groups. Fuzzy spheres emerge from quantizing S2and are associated with the group SU (2) in this manner. They are useful for regularizing quantum field theories and modeling space–times by noncommutative manifolds. We show that fuzzy spaces are Hopf algebras and in fact have more structure than the latter. They are thus candidates for quantum symmetries. Using their generalized Hopf algebraic structures, we can also model processes where one fuzzy space splits into several fuzzy spaces. For example we can discuss the quantum transition where the fuzzy sphere for angular momentum J splits into fuzzy spheres for angular momenta K and L.

INSTANTONS AND CHIRAL ANOMALY IN FUZZY PHYSICS

In continuum physics, there are important topological aspects like instantons, θ-terms and the axial anomaly. Conventional lattice discretizations often have difficulties in treating one or the other of these aspects. In this paper, we develop discrete quantum field theories on fuzzy manifolds using noncommutative geometry. Basing ourselves on previous treatments of instantons and chiral fermions (without fermion doubling) on fuzzy spaces and especially fuzzy spheres, we present discrete representations of θ-terms and topological susceptibility for gauge theories and derive axial anomaly on the fuzzy sphere. Our gauge field action for four dimensions is bounded by a constant times the modulus of the instanton number as in the continuum.