group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
What is known in the literature as discrete torsion are phenomena of equivariant ordinary differential cohomology, specifically of circle 2-bundles with connection modelling the B-field in string theory over orbifold base spaces.
Discussion of classification includes
The identification of “discrete torsion” in type II string theory as a choice of orbifold equivariance on a principal 2-bundle/bundle gerbe is due to
Discrete Torsion and Gerbes I (arXiv:hep-th/9909108)
Discrete Torsion and Gerbes II (arXiv:hep-th/9909120)
Discrete Torsion, Quotient Stacks, and String Orbifolds (arXiv:math/0110156)
Generalization to heterotic string theory is discussed in
and the analog for the C-field 2-gerbe in M-theory in
See also