  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(G, U(1)), we derive the twisted sector phases appearing in
string loop partition functions, we derive M. Douglas's description of discrete
torsion for D-branes in terms of a projective representation of the orbifold
group, and we outline how the results of Vafa-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(G, U(1)), and explain how these new degrees of
freedom appear in terms of twisted sector contributions to partition functions
and in terms of orbifold group 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.
