One of the remarkable insights of orbifold string theory is an indication
of the existence of a new cohomology theory of orbifolds containing the
so-called twisted sectors as the contribution of singularities. Mathematically,
such an orbifold cohomology theory has been constructed by
Chen-Ruan [CR]. The author believes that there is a "stringy" geometry
and topology of orbifolds whose core is orbifold cohomology. One
aspect of this new geometry and topology is the twisted orbifold cohomology
and its relation to discrete torsion. Again, the twisting process
has its roots in physics. Physicists usually work over a global quotient
X = Y/G only, where G is a finite group acting smoothly on Y. A
discrete torsion is a cohomology class α ∈ H2(G, U(1)). Physically, a
discrete torsion counts the freedom to choose a phase factor to weight
the path integral over each twisted sector without destroying the consistency
of string theory. For each α, Vafa-Witten [VW] constructed
the twisted orbifold cohomology group H*orb,α(X/G,ℂ).