nLab
discrete torsion

Idea

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.

References

For the B-field

Early discussion of classification in 2d CFTs includes

• Maximilian Kreuzer, A. N. Schellekens, Simple currents versus orbifolds with discrete torsion – a complete classification, Nucl.Phys. B411 (1994) 97-121 (arXiv:hep-th/9306145)

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:http://inspirehep.net/record/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

• Eric Sharpe, Discrete torsion, Phys.Rev. D68 (2003) 126003 (arXiv:hep-th/0008154)

based on

• Eric Sharpe,

  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)

See also

• 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

• Eric Sharpe, Discrete Torsion in Perturbative Heterotic String Theory, Phys.Rev. D68 (2003) 126005 (arXiv:hep-th/0008184)

For the C-field

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) 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.