Discrete Torsion and Shift Orbifolds
E. Sharpe
TL;DR
The paper shows that discrete torsion in string orbifolds is complemented by additional, non-discrete-torsion degrees of freedom arising from momentum/translation shifts in $B$-field orbifolds, which induce characteristic twisted-sector phases of the form $ ext{exp}( p_L a_R - p_R a_L )$ on toroidal sectors. It provides a first-principles derivation of these phases, demonstrates their manifest $SL(2,b Z)$-invariance, and analyzes their realization on D-branes as tensoring with line bundles, yielding a consistent open-closed string picture via Douglas and Gomis-type checks. The work then extends these ideas to the M-theory regime, showing that IIA discrete torsion has a natural M-theory dual encoded in the C-field, with the relevant data arising from flat 1-gerbes and their holonomies; wrapped membranes reproduce the IIA phase factors, confirming the uplift. Together, these results illuminate a richer structure of orbifold group actions beyond $H^2(G,U(1))$, clarifying the M-theory embedding and offering concrete consistency checks in multiple settings.
Abstract
In this paper we make two observations related to discrete torsion. First, we observe that an old obscure degree of freedom (momentum/translation shifts) in (symmetric) string orbifolds is related to discrete torsion. We point out how our previous derivation of discrete torsion from orbifold group actions on B fields includes these momentum lattice shift phases, and discuss how they are realized in terms of orbifold group actions on D-branes. Second, we describe the M theory dual of IIA discrete torsion, a duality relation to our knowledge not previously understood. We show that IIA discrete torsion is encoded in analogues of the shift orbifolds above for the M theory C field.