D-branes on Orbifolds with Discrete Torsion And Topological Obstruction

TL;DR

This paper extends topological restrictions on D-branes to orbifolds with discrete torsion, deriving from worldsheet consistency and algebraic K-theory, revealing how fractional branes relate to Ramond-Ramond charges.

Contribution

It introduces an orbifold analog of Freed and Witten's topological relation, derived from worldsheet conditions, and defines algebraic K-theory for orbifolds with discrete torsion.

Findings

Derived the orbifold topological restriction on D-branes with discrete torsion.

Defined algebraic K-theory using twisted cross products for orbifolds.

Showed the relation between fractional branes and Ramond-Ramond fields.

Abstract

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.