group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
What is known in the literature as discrete torsion (Vafa 86) are phenomena of equivariant ordinary differential cohomology, specifically of circle 2-bundles with connection (e.g. represented as bundle gerbes) modelling the B-field in string theory over orbifold spacetimes (Sharpe 99) and of circle 3-bundles with connection (e.g. represented as bundle 2-gerbes) modelling the supergravity C-field on orbifolds (Sharpe 00), as in M-theory on G2-manifolds with ADE-singularities.
Early discussion of classification in 2d CFTs includes
and more specifically for orbifolds in string theory in
Cumrun Vafa, Modular Invariance and Discrete Torsion on Orbifolds, Nucl. Phys. B273 (1986) 592-606 (spire:227126)
Cumrun Vafa, Edward Witten, On Orbifolds with Discrete Torsion, J. Geom. Phys. 15:189-214, 1995 (arXiv:hep-th/9409188)
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
based on
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, in Orbifolds in Mathematics and Physics (arXiv:math/0110156)
In relation to twisted Chen-ruan orbifold cohomology:
See also
Bo Feng, Amihay Hanany, Yang-Hui He, Nikolaos Prezas, Discrete Torsion, Non-Abelian Orbifolds and the Schur Multiplier, JHEP 0101:033, 2001 (arXiv:hep-th/0010023)
Kiyonori Gomi, Yuji Terashima, Discrete Torsion Phases as Topological Actions, Commun. Math. Phys. (2009) 287: 889 (doi:10.1007/s00220-009-0736-1)
The case of heterotic string theory is discussed in
The higher version of discrete torsion for circle 3-bundles describing the supergravity C-field is discussed in
Eric Sharpe, Analogues of Discrete Torsion for the M-Theory Three-Form, Phys. Rev. D68 (2003) 126004 (arXiv:hep-th/0008170)
Shigenori Seki, Discrete Torsion and Branes in M-theory from Mathematical Viewpoint, Nucl. Phys. B606 (2001) 689-698 (arXiv:hep-th/0103117)
Jan de Boer, Robbert Dijkgraaf, Kentaro Hori, Arjan Keurentjes, John Morgan, David Morrison, Savdeep Sethi, section 4.6.2 of Triples, Fluxes, and Strings, Adv. Theor. Math. Phys. 4 (2002) 995-1186 (arXiv:hep-th/0103170)
and applied to discussion of black M2-brane worldvolume field theory (BLG model/ABJM model, see at fractional M2-brane) in
Savdeep Sethi, A Relation Between $N=8$ Gauge Theories in Three Dimensions, JHEP 9811:003,1998 (arXiv:hep-th/9809162)
Neil Lambert, David Tong, Membranes on an Orbifold, Phys. Rev. Lett. 101:041602, 2008 (arXiv:0804.1114)
Ofer Aharony, Oren Bergman, Daniel Louis Jafferis, Fractional M2-branes, JHEP 0811:043,2008 (arXiv:0807.4924)
Mauricio Romo, Aspects of ABJM orbifolds with discrete torsion, J. High Energ. Phys. (2011) 2011 (arXiv:1011.4733)
See also at finite subgroup of SU(2) the section on group cohomology.