Quantum space-times and finite N effects in 4D super Yang-Mills theories

The truncation in the number of single-trace chiral primary operators of N = 4 SYM and its conjectured connection with gravity on quantum space-times are elaborated. The model of quantum space-time we use is AdS(q)(5) X S-q(5 ) for q a root of unity. The quantum sphere is defined as a homogeneous spa ce with manifest SUq(3) symmetry, but as anticipated from the field theory correspondence, we show that there is a hidden SOq(6) symmetry in the const ruction. We also study some properties of quantum space quotients as candid ate models for the quantum space-time relevant for some Z(n) quiver quotien ts of the N = 4 theory which break SUSY to N = 2. We find various qualitati ve agreements between the proposed models and the properties of the corresp onding finite N gauge theories. (C) 2000 Elsevier Science B.V. All rights r eserved.