Towards the supersymmetric Standard Model from intersecting D6-branes on the Z6′ orientifold

We construct N=1 supersymmetric fractional branes on the Z6′ orientifold. Intersecting stacks of such branes are needed to build a supersymmetric Standard Model. If a,b are the stacks that generate the SU(3)c and SU(2)L gauge particles, then, in order to obtain just the chiral spectrum of the (supersymmetric) Standard Model (with non-zero Yukawa couplings to the Higgs multiplets), it is necessary that the number of intersections a○b of the stacks a and b, and the number of intersections a○b′ of a with the orientifold image b′ of b satisfy (a○b,a○b′)=±(2,1) or ±(1,2). It is also necessary that there is no matter in symmetric representations of the gauge group, and not too much matter in antisymmetric representations, on either stack. We provide a number of examples having these properties. Different lattices give different solutions and different physics.