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.

Classification

Discrete torsion first arose in the context of 2d orbifold partition functions in Vafa 86, where it was noted that the twisted sectors could be multiplied by nontrivial $U(1)$ phases. By imposing requirements on these phases, such as respecting modular invariance, and multi-loop factorization (that the genus $2$ phase factor is equal to the product of two genus $1$ phase factors whenever the generators $a_{1(2)}$ of the fundamental group commute with the generators $b_{1(2)}$, respectively), it was observed that consistent phases are described by group 2-cocycles

$$\omega\in Z^2(G,U(1))$$

for $G$ the orbifold group with trivial action on $U(1)$, and that 2-coboundaries yield trivial phases. This gives a classification of discrete torsion in terms of the second group cohomology group $H^2(G,U(1))$. Concretely, the orbifold partition function of a genus $1$ surface with discrete torsion $\omega\in Z^2(G,U(1))$ is

$$Z = \sum_{g,h\in G} \frac{\omega(g,h)}{\omega(h,g)} Z_{g,h}$$

for $g,h$ commuting pairs. The phase factor, even though it is formulated in terms of cocycles, only depends on the cohomology class of discrete torsion, since it is clearly invariant under shifts by coboundaries.

This classification was observed again in Sharpe 99, where such 2-cocycles were interpreted as actions of the orbifold group $G$ on a U(1) 2-bundle, and extended to the three-dimensional case in Sharpe 00, with the actions now being classified by the third cohomology group $H^3(G,U(1))$ (under simplifying assumptions).

Related concepts

• equivariant ordinary differential cohomology

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: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, doi:10.1103/PhysRevD.68.126003)

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, in Orbifolds in Mathematics and Physics (arXiv:math/0110156)

In relation to twisted Chen-ruan orbifold cohomology:

• Yongbin Ruan, Discrete torsion and twisted orbifold cohomology, J. Sympl. Geom. 2 (2003), 1–24 (arXiv:math/0005299)

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

• 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, 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.