Matrix compactification on orientifolds

Generalizing previous results for orbifolds, in this paper we describe the compactification of the matrix model on an orientifold which is a quotient space ${\mathbf{R}}^d/\Gamma$ as a Yang-Mills theory residing on a quantum space. The information of the compactification is encoded in the action of the discrete symmetry group $\Gamma$ on Euclidean space ${\mathbf{R}}^d$ and a projective representation U of $\Gamma .$ The choice of Hilbert space on which the algebra of U is realized as an operator algebra corresponds to the choice of a physical background for the compactification. All these data are summarized in the spectral triple of the quantum space.