  It is shown that the quarks and leptons of the standard model, including a
right-handed neutrino, can be obtained by gauging the holonomy groups of
complex projective spaces of complex dimensions two and three. The spectrum
emerges as chiral zero modes of the Dirac operator coupled to gauge fields and
the demonstration involves an index theorem analysis on a general complex
projective space in the presence of topologically non-trivial SU(n)xU(1) gauge
fields. The construction may have applications in type IIA string theory and
non-commutative geometry.
