Discrete torsion and twisted orbifold cohomology (Q1883973)

A problem suggested by \textit{C. Vafa} and \textit{E. Witten} [J. Geom. Phys. 15, 189--214 (1995; Zbl 0816.53053)] reads as follows. Discrete torsion and twisted orbifold cohomology relate to desingularization: a sequence of resolutions and deformations of an orbifold leading to a smooth manifold. Such a desingularization may not exist in dimensions higher than three, and in this case, the desingularization is allowed to be an orbifold. Vafa-Witten proposed that discrete torsion is a parameter for deformation, and that the cohomology of the desingularization is the twisted orbifold cohomology of discrete torsion plus the possible contributions of exceptional loci of the small resolution. This proposal ran into trouble because the number of desingularizations is much larger than the number of discrete torsions. In the paper under review, the author addresses the problem of counting the missing desingularizations. The author addresses also the theory over general orbifolds because it is well known that most orbifolds are not global quotients. In particular, the author determines the appropriate notion of discrete torsion for a general orbifold. The paper is a sequel to the article by \textit{W. Chen} and \textit{Y. Ruan} [Commun. Math. Phys. 248, No. 1, 1--31 (2004; Zbl 1063.53091)].