nLab discrete torsion

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Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Differential cohomology

Contents

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)U(1) phases. By imposing requirements on these phases, such as respecting modular invariance, and multi-loop factorization (that the genus 22 phase factor is equal to the product of two genus 11 phase factors whenever the generators a 1(2)a_{1(2)} of the fundamental group commute with the generators b 1(2)b_{1(2)}, respectively), it was observed that consistent phases are described by group 2-cocycles

ωZ 2(G,U(1)) \omega\in Z^2(G,U(1))

for GG the orbifold group with trivial action on U(1)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))H^2(G,U(1)). Concretely, the orbifold partition function of a genus 11 surface with discrete torsion ωZ 2(G,U(1))\omega\in Z^2(G,U(1)) is

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

for g,hg,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 GG 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))H^3(G,U(1)) (under simplifying assumptions).

References

For the B-field

Early discussion of classification in 2d CFTs includes

and more specifically for orbifolds in string theory in

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

In relation to twisted Chen-ruan orbifold cohomology:

See also

The case of heterotic string theory is discussed in

For the C-field

The higher version of discrete torsion for circle 3-bundles describing the supergravity C-field is discussed in

and applied to discussion of black M2-brane worldvolume field theory (BLG model/ABJM model, see at fractional M2-brane) in

See also at finite subgroup of SU(2) the section on group cohomology.

Last revised on February 14, 2024 at 22:21:09. See the history of this page for a list of all contributions to it.