D-branes on Orbifolds with Discrete Torsion And Topological Obstruction

We find the orbifold analog of the topological relation recently found by Freed and Witten which restricts the allowed D-brane configurations of Type II vacua with a topologically non-trivial flat $B$-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 open strings gives a natural definition of the algebraic K-theory group – using twisted cross products – responsible for measuring Ramond-Ramond charges in orbifolds with discrete torsion. We show that the correspondence between fractional branes and Ramond-Ramond fields follows in an interesting fashion from the way that discrete torsion is implemented on open and closed strings.