Discrete torsion

In this article we explain discrete torsion. Put simply, discrete torsion is the choice of orbifold group action on the B field. We derive the classification $H^2(\Gamma ,U(1\left)\mathrm{}\right),$ the twisted sector phases appearing in string loop partition functions, Douglas’s description of discrete torsion for D-branes in terms of a projective representation of the orbifold group, and outline how the results of Vafa and Witten fit into this framework. In addition, we observe that additional degrees of freedom (known as shift orbifolds) appear in describing orbifold group actions on B fields, in addition to those classified by $H^2(\Gamma ,U(1\left)\mathrm{}\right),$ and explain how these degrees of freedom appear in terms of twisted sector contributions to partition functions and in terms of orbifold actions on D-brane worldvolumes. This paper represents a technically simplified version of prior papers by the author on discrete torsion. We repeat here technically simplified versions of results from those papers, and have included some new material.