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).
"Discrete Torsion and Twisted Orbifold Cohomology." J. Symplectic Geom. 2 (1) 001 - 024, October, 2003.