D-branes on orbifolds with discrete torsion and topological obstruction

We find the orbifold analog of the topological relation recently found by F reed and Witten which restricts the allowed D-brane configurations of type- II vacua with a topologically non-trivial at B-field. The result relies in Douglas proposal - which we derive from worldsheet consistency conditions - of embedding projective representations on open string Chan-Paton factors when considering orbifolds with discrete torsion. The orbifold action on op en strings gives a natural definition of the algebraic K-theory group - usi ng twisted cross products - responsible for measuring Ramond-Ramond charges in orbifolds with discrete torsion. We show that the correspondence betwee n fractional branes and Ramond-Ramond fields follows in an interesting fash ion from the way that discrete torsion is implemented on open and closed st rings.